auto-oscillateurs thermoacoustiques : effets non linéaires et
TRANSCRIPT
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Auto-oscillateurs thermoacoustiques effets
non lineacuteaires et comportement dynamique
Guillaume Penelet
Laboratoire drsquoAcoustique de lrsquoUniversiteacute du Maine UMR CNRS 6613avenue Olivier Messiaen 72085 Le Mans cedex 9 France
guillaumepeneletuniv-lemansfr
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 2 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 3 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Quelle thermoacoustique
Thermoacoustique = interaction thermiqueacoustique
rArr de quelle thermoacoustique parle-t-on ici
Tomographie thermoacoustique
Haut-parleurs thermoacoustiques
Effet piston
Instabiliteacutes de combustion
Machines thermoacoustiques (pompes agrave chaleur et moteurs thermoacoustiques)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 4 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Tomographie Thermoacoustique
Tomographie thermoacoustiqueMeacutethode drsquoimagerie meacutedicale baseacutee sur la reacuteponse de certainstissus agrave une impulsion EM
chauffage =gt deacutetente thermoeacutelastique =gt onde acoustique
Scheacutema de principe de la tomographie TA
Haut-parleurs thermoacoustiques
Effet piston
Instabiliteacutes de combustion
Machines thermoacoustique
Image 3D drsquoun rein drsquoagneauobtenu par tomographie TA
(CD) compareacutee agrave celleobtenue par IRM Drsquoapregraves [1]
[1] Kruger RA et al Radiology 216 279-283 2000
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 5 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Haut-parleur TA
Tomographie thermoacoustique
Haut-parleurs thermoacoustiqueschauffage fluctuant drsquoune surface solide
=gt rayonnement acoustique
Haut-parleur thermoacoustique Drsquoapregraves [1]
Effet piston Instabiliteacutes de combustion Machines thermoacoustiques
Reacuteponse en freacutequence drsquoun HP TADrsquoapregraves [1]
Ecouteurs thermoacoustiques Drsquoapregraves [2]
[1] Nyskacen et al Appl Phys Let 95 163102 2009 [2] US Patent n 8625822 Tsinghua University (Beijing)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 6 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Effet piston
Tomographie thermoacoustique Haut-parleurs thermoacoustiques
Effet pistonUn 4e mode de transport de la chaleur
Autour du point critique χ rarr infin et κ rarr 0Increacutement de chauffage localrArr expansion abrupte de la CL (effet piston)rArr compression adiabatique de la cellule amp ondes TArArr T uarr avec un temps caracteacuteristique τPE ≪ τκ
Instabiliteacutes de combustion Machines thermoacoustiques
Mesure des ondes TA dans du CO2 aupoint critique1
Reacuteponse ∆ρρ agrave un pulse de chauffage1
∆ρ(x t) au cours du chauffage1
[1] Y Miura et al Phys Rev E 74 010101 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 7 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Instabiliteacutes de combustion
Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston
Instabiliteacutes de combustionGeacuteneacuteration spontaneacutee drsquooscillations acoustiques reacutesul-tant drsquoune interaction complexe drsquoune flamme et drsquouneacutecoulement dans une chambre de combustion
Machines thermoacoustiques
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 8 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 2 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 3 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Quelle thermoacoustique
Thermoacoustique = interaction thermiqueacoustique
rArr de quelle thermoacoustique parle-t-on ici
Tomographie thermoacoustique
Haut-parleurs thermoacoustiques
Effet piston
Instabiliteacutes de combustion
Machines thermoacoustiques (pompes agrave chaleur et moteurs thermoacoustiques)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 4 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Tomographie Thermoacoustique
Tomographie thermoacoustiqueMeacutethode drsquoimagerie meacutedicale baseacutee sur la reacuteponse de certainstissus agrave une impulsion EM
chauffage =gt deacutetente thermoeacutelastique =gt onde acoustique
Scheacutema de principe de la tomographie TA
Haut-parleurs thermoacoustiques
Effet piston
Instabiliteacutes de combustion
Machines thermoacoustique
Image 3D drsquoun rein drsquoagneauobtenu par tomographie TA
(CD) compareacutee agrave celleobtenue par IRM Drsquoapregraves [1]
[1] Kruger RA et al Radiology 216 279-283 2000
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 5 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Haut-parleur TA
Tomographie thermoacoustique
Haut-parleurs thermoacoustiqueschauffage fluctuant drsquoune surface solide
=gt rayonnement acoustique
Haut-parleur thermoacoustique Drsquoapregraves [1]
Effet piston Instabiliteacutes de combustion Machines thermoacoustiques
Reacuteponse en freacutequence drsquoun HP TADrsquoapregraves [1]
Ecouteurs thermoacoustiques Drsquoapregraves [2]
[1] Nyskacen et al Appl Phys Let 95 163102 2009 [2] US Patent n 8625822 Tsinghua University (Beijing)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 6 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Effet piston
Tomographie thermoacoustique Haut-parleurs thermoacoustiques
Effet pistonUn 4e mode de transport de la chaleur
Autour du point critique χ rarr infin et κ rarr 0Increacutement de chauffage localrArr expansion abrupte de la CL (effet piston)rArr compression adiabatique de la cellule amp ondes TArArr T uarr avec un temps caracteacuteristique τPE ≪ τκ
Instabiliteacutes de combustion Machines thermoacoustiques
Mesure des ondes TA dans du CO2 aupoint critique1
Reacuteponse ∆ρρ agrave un pulse de chauffage1
∆ρ(x t) au cours du chauffage1
[1] Y Miura et al Phys Rev E 74 010101 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 7 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Instabiliteacutes de combustion
Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston
Instabiliteacutes de combustionGeacuteneacuteration spontaneacutee drsquooscillations acoustiques reacutesul-tant drsquoune interaction complexe drsquoune flamme et drsquouneacutecoulement dans une chambre de combustion
Machines thermoacoustiques
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 8 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 3 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Quelle thermoacoustique
Thermoacoustique = interaction thermiqueacoustique
rArr de quelle thermoacoustique parle-t-on ici
Tomographie thermoacoustique
Haut-parleurs thermoacoustiques
Effet piston
Instabiliteacutes de combustion
Machines thermoacoustiques (pompes agrave chaleur et moteurs thermoacoustiques)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 4 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Tomographie Thermoacoustique
Tomographie thermoacoustiqueMeacutethode drsquoimagerie meacutedicale baseacutee sur la reacuteponse de certainstissus agrave une impulsion EM
chauffage =gt deacutetente thermoeacutelastique =gt onde acoustique
Scheacutema de principe de la tomographie TA
Haut-parleurs thermoacoustiques
Effet piston
Instabiliteacutes de combustion
Machines thermoacoustique
Image 3D drsquoun rein drsquoagneauobtenu par tomographie TA
(CD) compareacutee agrave celleobtenue par IRM Drsquoapregraves [1]
[1] Kruger RA et al Radiology 216 279-283 2000
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 5 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Haut-parleur TA
Tomographie thermoacoustique
Haut-parleurs thermoacoustiqueschauffage fluctuant drsquoune surface solide
=gt rayonnement acoustique
Haut-parleur thermoacoustique Drsquoapregraves [1]
Effet piston Instabiliteacutes de combustion Machines thermoacoustiques
Reacuteponse en freacutequence drsquoun HP TADrsquoapregraves [1]
Ecouteurs thermoacoustiques Drsquoapregraves [2]
[1] Nyskacen et al Appl Phys Let 95 163102 2009 [2] US Patent n 8625822 Tsinghua University (Beijing)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 6 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Effet piston
Tomographie thermoacoustique Haut-parleurs thermoacoustiques
Effet pistonUn 4e mode de transport de la chaleur
Autour du point critique χ rarr infin et κ rarr 0Increacutement de chauffage localrArr expansion abrupte de la CL (effet piston)rArr compression adiabatique de la cellule amp ondes TArArr T uarr avec un temps caracteacuteristique τPE ≪ τκ
Instabiliteacutes de combustion Machines thermoacoustiques
Mesure des ondes TA dans du CO2 aupoint critique1
Reacuteponse ∆ρρ agrave un pulse de chauffage1
∆ρ(x t) au cours du chauffage1
[1] Y Miura et al Phys Rev E 74 010101 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 7 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Instabiliteacutes de combustion
Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston
Instabiliteacutes de combustionGeacuteneacuteration spontaneacutee drsquooscillations acoustiques reacutesul-tant drsquoune interaction complexe drsquoune flamme et drsquouneacutecoulement dans une chambre de combustion
Machines thermoacoustiques
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 8 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Quelle thermoacoustique
Thermoacoustique = interaction thermiqueacoustique
rArr de quelle thermoacoustique parle-t-on ici
Tomographie thermoacoustique
Haut-parleurs thermoacoustiques
Effet piston
Instabiliteacutes de combustion
Machines thermoacoustiques (pompes agrave chaleur et moteurs thermoacoustiques)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 4 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Tomographie Thermoacoustique
Tomographie thermoacoustiqueMeacutethode drsquoimagerie meacutedicale baseacutee sur la reacuteponse de certainstissus agrave une impulsion EM
chauffage =gt deacutetente thermoeacutelastique =gt onde acoustique
Scheacutema de principe de la tomographie TA
Haut-parleurs thermoacoustiques
Effet piston
Instabiliteacutes de combustion
Machines thermoacoustique
Image 3D drsquoun rein drsquoagneauobtenu par tomographie TA
(CD) compareacutee agrave celleobtenue par IRM Drsquoapregraves [1]
[1] Kruger RA et al Radiology 216 279-283 2000
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 5 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Haut-parleur TA
Tomographie thermoacoustique
Haut-parleurs thermoacoustiqueschauffage fluctuant drsquoune surface solide
=gt rayonnement acoustique
Haut-parleur thermoacoustique Drsquoapregraves [1]
Effet piston Instabiliteacutes de combustion Machines thermoacoustiques
Reacuteponse en freacutequence drsquoun HP TADrsquoapregraves [1]
Ecouteurs thermoacoustiques Drsquoapregraves [2]
[1] Nyskacen et al Appl Phys Let 95 163102 2009 [2] US Patent n 8625822 Tsinghua University (Beijing)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 6 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Effet piston
Tomographie thermoacoustique Haut-parleurs thermoacoustiques
Effet pistonUn 4e mode de transport de la chaleur
Autour du point critique χ rarr infin et κ rarr 0Increacutement de chauffage localrArr expansion abrupte de la CL (effet piston)rArr compression adiabatique de la cellule amp ondes TArArr T uarr avec un temps caracteacuteristique τPE ≪ τκ
Instabiliteacutes de combustion Machines thermoacoustiques
Mesure des ondes TA dans du CO2 aupoint critique1
Reacuteponse ∆ρρ agrave un pulse de chauffage1
∆ρ(x t) au cours du chauffage1
[1] Y Miura et al Phys Rev E 74 010101 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 7 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Instabiliteacutes de combustion
Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston
Instabiliteacutes de combustionGeacuteneacuteration spontaneacutee drsquooscillations acoustiques reacutesul-tant drsquoune interaction complexe drsquoune flamme et drsquouneacutecoulement dans une chambre de combustion
Machines thermoacoustiques
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 8 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Tomographie Thermoacoustique
Tomographie thermoacoustiqueMeacutethode drsquoimagerie meacutedicale baseacutee sur la reacuteponse de certainstissus agrave une impulsion EM
chauffage =gt deacutetente thermoeacutelastique =gt onde acoustique
Scheacutema de principe de la tomographie TA
Haut-parleurs thermoacoustiques
Effet piston
Instabiliteacutes de combustion
Machines thermoacoustique
Image 3D drsquoun rein drsquoagneauobtenu par tomographie TA
(CD) compareacutee agrave celleobtenue par IRM Drsquoapregraves [1]
[1] Kruger RA et al Radiology 216 279-283 2000
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 5 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Haut-parleur TA
Tomographie thermoacoustique
Haut-parleurs thermoacoustiqueschauffage fluctuant drsquoune surface solide
=gt rayonnement acoustique
Haut-parleur thermoacoustique Drsquoapregraves [1]
Effet piston Instabiliteacutes de combustion Machines thermoacoustiques
Reacuteponse en freacutequence drsquoun HP TADrsquoapregraves [1]
Ecouteurs thermoacoustiques Drsquoapregraves [2]
[1] Nyskacen et al Appl Phys Let 95 163102 2009 [2] US Patent n 8625822 Tsinghua University (Beijing)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 6 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Effet piston
Tomographie thermoacoustique Haut-parleurs thermoacoustiques
Effet pistonUn 4e mode de transport de la chaleur
Autour du point critique χ rarr infin et κ rarr 0Increacutement de chauffage localrArr expansion abrupte de la CL (effet piston)rArr compression adiabatique de la cellule amp ondes TArArr T uarr avec un temps caracteacuteristique τPE ≪ τκ
Instabiliteacutes de combustion Machines thermoacoustiques
Mesure des ondes TA dans du CO2 aupoint critique1
Reacuteponse ∆ρρ agrave un pulse de chauffage1
∆ρ(x t) au cours du chauffage1
[1] Y Miura et al Phys Rev E 74 010101 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 7 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Instabiliteacutes de combustion
Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston
Instabiliteacutes de combustionGeacuteneacuteration spontaneacutee drsquooscillations acoustiques reacutesul-tant drsquoune interaction complexe drsquoune flamme et drsquouneacutecoulement dans une chambre de combustion
Machines thermoacoustiques
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 8 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Haut-parleur TA
Tomographie thermoacoustique
Haut-parleurs thermoacoustiqueschauffage fluctuant drsquoune surface solide
=gt rayonnement acoustique
Haut-parleur thermoacoustique Drsquoapregraves [1]
Effet piston Instabiliteacutes de combustion Machines thermoacoustiques
Reacuteponse en freacutequence drsquoun HP TADrsquoapregraves [1]
Ecouteurs thermoacoustiques Drsquoapregraves [2]
[1] Nyskacen et al Appl Phys Let 95 163102 2009 [2] US Patent n 8625822 Tsinghua University (Beijing)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 6 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Effet piston
Tomographie thermoacoustique Haut-parleurs thermoacoustiques
Effet pistonUn 4e mode de transport de la chaleur
Autour du point critique χ rarr infin et κ rarr 0Increacutement de chauffage localrArr expansion abrupte de la CL (effet piston)rArr compression adiabatique de la cellule amp ondes TArArr T uarr avec un temps caracteacuteristique τPE ≪ τκ
Instabiliteacutes de combustion Machines thermoacoustiques
Mesure des ondes TA dans du CO2 aupoint critique1
Reacuteponse ∆ρρ agrave un pulse de chauffage1
∆ρ(x t) au cours du chauffage1
[1] Y Miura et al Phys Rev E 74 010101 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 7 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Instabiliteacutes de combustion
Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston
Instabiliteacutes de combustionGeacuteneacuteration spontaneacutee drsquooscillations acoustiques reacutesul-tant drsquoune interaction complexe drsquoune flamme et drsquouneacutecoulement dans une chambre de combustion
Machines thermoacoustiques
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 8 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Effet piston
Tomographie thermoacoustique Haut-parleurs thermoacoustiques
Effet pistonUn 4e mode de transport de la chaleur
Autour du point critique χ rarr infin et κ rarr 0Increacutement de chauffage localrArr expansion abrupte de la CL (effet piston)rArr compression adiabatique de la cellule amp ondes TArArr T uarr avec un temps caracteacuteristique τPE ≪ τκ
Instabiliteacutes de combustion Machines thermoacoustiques
Mesure des ondes TA dans du CO2 aupoint critique1
Reacuteponse ∆ρρ agrave un pulse de chauffage1
∆ρ(x t) au cours du chauffage1
[1] Y Miura et al Phys Rev E 74 010101 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 7 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Instabiliteacutes de combustion
Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston
Instabiliteacutes de combustionGeacuteneacuteration spontaneacutee drsquooscillations acoustiques reacutesul-tant drsquoune interaction complexe drsquoune flamme et drsquouneacutecoulement dans une chambre de combustion
Machines thermoacoustiques
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 8 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Instabiliteacutes de combustion
Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston
Instabiliteacutes de combustionGeacuteneacuteration spontaneacutee drsquooscillations acoustiques reacutesul-tant drsquoune interaction complexe drsquoune flamme et drsquouneacutecoulement dans une chambre de combustion
Machines thermoacoustiques
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 8 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Machines thermoacoustiques
Tomographie thermoacoustique Haut-parleurs thermoacoustiques Effet piston Instabiliteacutes de combustion
Machines thermoacoustiqueMachine thermique cyclique faisant usage drsquoun travailmeacutecanique de nature acoustique
Congeacutelateur thermoacoustique reacutealiseacute agrave lrsquouniversiteacute de Penn
State pour le glacier ben amp Jerryrsquos1
Machine TA tritherme (moteur+frigo) reacutea-
liseacute au Los Alamos National Laboratory2
[1] httpwwwacspsueduthermoacousticsrefrigerationbenandjerryshtm [2] G W Swift J J WollanGasTIPS Volume 8 Number 4 pages 21-26 (Fall 2002)
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Machines thermoacoustiques
Elementary sources of sound emission unsteady heat release
The linearized equations of motion for an inviscid and non-heat-conducting idealgas
parttρprime + ρ0divv = Q(r t) Q mass addition
time
pulsating sphere
ρ0parttv + gradp = f(r t) f force
time
oscillating sphere
ρ0Cpparttτ minus parttp = q(r t) q Heat release
time
sphere with oscillating temperature
The resulting inhomogeneous wave equation (using p = RT0ρprime + Rρ0τ) is
∆p minus1
c2
0
part2
ttp = minuspartt q
T0Cp
minus ρ0parttQ + divf
Contrarily to Q and f q is difficult to derive in realistic cases because The heat conductivity of the gas should be considered Generally q = q (p v) which may lead to self-sustained oscillations(eg Rijke tube etc )
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The Rayleigh criterium
The Rayleigh criterium (int
pqdt gt 0)
ldquo at the phase of greatest condensation heat is receivedby the air and at the phase of greatest rarefaction heat isgiven up from it and thus there is a tendency to maintainthe vibrations rdquo JW Strutt (Lord Rayleigh) laquo The theory ofsound raquo Dover New York 2nd edition sect 322i p 231 1945)
1ρ
pq
TA Laser Rijke tube
Q in
Wout
T(x)
Q in
WoutWout
V0
wire
q = f (dxT p vx stack ) q = f (Twire vx V0 )
⋆ not acoustically compact ⋆ acoustically compact(q asymp q(t)δ(x minus xwire ))
⋆ amplification of the first mode ⋆ amplification of the first mode R
pqdt gt 0 =gt dxT lt 0R
pqdt gt 0 =gt xw lt L2
Common features⋆ Autonomous oscillators driven by heat (which both fill the ldquoRayleigh criteriumrdquo)⋆ Simple in terms of geometry but complicated operation above threshold
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TA Laser a prototypical example
Heat input =gt sound output (self-sustained acoustic oscillations at the frequency of themost unstable mode(s))
Keywords heat engine (prime-movers and heat pumps) autonomous oscillator Paradigmatic of the challenges that need to be taken up to understand more deeply the
processes controlling the operation of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 12 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Auto-oscillateur thermo-acoustique contexte et probleacutematique
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
⋆ Application drsquoun nablaT le long drsquoun mateacuteriau poreux placeacute dans un reacuteseau de guidedrsquoonde rArr auto-oscillations agrave la freacutequence du mode le plus instable
⋆ Geacuteomeacutetrie plusmn simple mais fonctionnement complexe au delagrave du seuil (pompage θac ldquovent acoustiquerdquo perte de charge singuliegravere propa NL )
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 13 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 14 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic principles
The TA Laser a conceptual very simplified approach
⋆ Simplistic description =gt Lagrangian approach follow a gas parcel submitted to a standingwave through a channel with an axial temperature gradient dxT0
δκ
ρ1
x
Hot Heat eXchanger (HHX) Cold Heat eXchanger (CHX)
Q Qrsquo
2 411
33
v
21 3 4 1τp
p 2
13
4
ξ max2
1 ~ adiabatic compression2 ~ isobaric expansion3 ~ adiabatic relaxation4 ~ isobaric contraction
=gt Work production if |dxT |ξmax gt τmax and dxT lt 0
⋆ But actual processes are actually more complex (heat exchange occurs continuously)⋆ A key point need of an imperfect thermal contact
=gtldquoefficientrdquo gas parcels should be at about δκ from the wall where δκ =p
2κωis the (frequency-dependent) thermal boundary layer thickness
⋆ Threshold of thermoacoustic instability (for a given mode)
=gt Sound amplification must be larger than losses
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 15 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Plane wave propagation through a viscous and heat-conducting gas along a ductsubmitted to a temperature gradient
⋆ Both the production of acoustic work and the ldquobucket briga-derdquo heat transport by sound along the duct are then described bysecond-order quantities
the ldquobucket brigaderdquo
xR
y
z
r
C T (x)s ss sρ λ pC T (x)ρ λ (x) (x)
θ
Main assumptions⋆ Low amplitudes ⋆ Typical wavelength raquo R ⋆ No mean flow
P(x r t) = P0 + p(x r t) =gt planes waves ⋆ ideal gasρ(x r t) = ρ0(x) + ρprime(x r t) p(x r t) = p(x t) ⋆ ρsCs ≫ ρ0Cp
T (x r t) = T0(x) + τ (x r t) =gtldquoboundary layer approximationrdquo ⋆ λs ≫ λv(x r t) ≪ c0 |partrζ| ≫ |partxζ|
S(x r t) = S0(x) + s(x r t) (ζ = p ρprime τ v s)[1] N Rott Adv Appl Mech 1980 [2] GW Swift J Acoust Soc Am 1988
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 16 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Equations fondamentales lineacuteariseacutees
ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )
parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0
ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )
dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0
dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p
⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0
vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]
τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]
ρprime = 1ω2
[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p
s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]
Fνκ(r) =J0(kνκr)J0(kνκR)
kνκ = 1minusiδνκ
δν =q
2νω δκ =
q2κω
fνκ = 〈Fνκ〉 = 2kνκR
J1(kνκR)J0(kνκR)
0 05 1 15 20
02
04
06
08
1
(1minusFν)(1minusfν)
rR
R=100δν
R=10δν
R=δν
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 18 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 20 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Machines thermoacoustiques
Elementary sources of sound emission unsteady heat release
The linearized equations of motion for an inviscid and non-heat-conducting idealgas
parttρprime + ρ0divv = Q(r t) Q mass addition
time
pulsating sphere
ρ0parttv + gradp = f(r t) f force
time
oscillating sphere
ρ0Cpparttτ minus parttp = q(r t) q Heat release
time
sphere with oscillating temperature
The resulting inhomogeneous wave equation (using p = RT0ρprime + Rρ0τ) is
∆p minus1
c2
0
part2
ttp = minuspartt q
T0Cp
minus ρ0parttQ + divf
Contrarily to Q and f q is difficult to derive in realistic cases because The heat conductivity of the gas should be considered Generally q = q (p v) which may lead to self-sustained oscillations(eg Rijke tube etc )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 10 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The Rayleigh criterium
The Rayleigh criterium (int
pqdt gt 0)
ldquo at the phase of greatest condensation heat is receivedby the air and at the phase of greatest rarefaction heat isgiven up from it and thus there is a tendency to maintainthe vibrations rdquo JW Strutt (Lord Rayleigh) laquo The theory ofsound raquo Dover New York 2nd edition sect 322i p 231 1945)
1ρ
pq
TA Laser Rijke tube
Q in
Wout
T(x)
Q in
WoutWout
V0
wire
q = f (dxT p vx stack ) q = f (Twire vx V0 )
⋆ not acoustically compact ⋆ acoustically compact(q asymp q(t)δ(x minus xwire ))
⋆ amplification of the first mode ⋆ amplification of the first mode R
pqdt gt 0 =gt dxT lt 0R
pqdt gt 0 =gt xw lt L2
Common features⋆ Autonomous oscillators driven by heat (which both fill the ldquoRayleigh criteriumrdquo)⋆ Simple in terms of geometry but complicated operation above threshold
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 11 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TA Laser a prototypical example
Heat input =gt sound output (self-sustained acoustic oscillations at the frequency of themost unstable mode(s))
Keywords heat engine (prime-movers and heat pumps) autonomous oscillator Paradigmatic of the challenges that need to be taken up to understand more deeply the
processes controlling the operation of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 12 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Auto-oscillateur thermo-acoustique contexte et probleacutematique
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
⋆ Application drsquoun nablaT le long drsquoun mateacuteriau poreux placeacute dans un reacuteseau de guidedrsquoonde rArr auto-oscillations agrave la freacutequence du mode le plus instable
⋆ Geacuteomeacutetrie plusmn simple mais fonctionnement complexe au delagrave du seuil (pompage θac ldquovent acoustiquerdquo perte de charge singuliegravere propa NL )
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 13 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 14 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic principles
The TA Laser a conceptual very simplified approach
⋆ Simplistic description =gt Lagrangian approach follow a gas parcel submitted to a standingwave through a channel with an axial temperature gradient dxT0
δκ
ρ1
x
Hot Heat eXchanger (HHX) Cold Heat eXchanger (CHX)
Q Qrsquo
2 411
33
v
21 3 4 1τp
p 2
13
4
ξ max2
1 ~ adiabatic compression2 ~ isobaric expansion3 ~ adiabatic relaxation4 ~ isobaric contraction
=gt Work production if |dxT |ξmax gt τmax and dxT lt 0
⋆ But actual processes are actually more complex (heat exchange occurs continuously)⋆ A key point need of an imperfect thermal contact
=gtldquoefficientrdquo gas parcels should be at about δκ from the wall where δκ =p
2κωis the (frequency-dependent) thermal boundary layer thickness
⋆ Threshold of thermoacoustic instability (for a given mode)
=gt Sound amplification must be larger than losses
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 15 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Plane wave propagation through a viscous and heat-conducting gas along a ductsubmitted to a temperature gradient
⋆ Both the production of acoustic work and the ldquobucket briga-derdquo heat transport by sound along the duct are then described bysecond-order quantities
the ldquobucket brigaderdquo
xR
y
z
r
C T (x)s ss sρ λ pC T (x)ρ λ (x) (x)
θ
Main assumptions⋆ Low amplitudes ⋆ Typical wavelength raquo R ⋆ No mean flow
P(x r t) = P0 + p(x r t) =gt planes waves ⋆ ideal gasρ(x r t) = ρ0(x) + ρprime(x r t) p(x r t) = p(x t) ⋆ ρsCs ≫ ρ0Cp
T (x r t) = T0(x) + τ (x r t) =gtldquoboundary layer approximationrdquo ⋆ λs ≫ λv(x r t) ≪ c0 |partrζ| ≫ |partxζ|
S(x r t) = S0(x) + s(x r t) (ζ = p ρprime τ v s)[1] N Rott Adv Appl Mech 1980 [2] GW Swift J Acoust Soc Am 1988
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Equations fondamentales lineacuteariseacutees
ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )
parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0
ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )
dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0
dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p
⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0
vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]
τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]
ρprime = 1ω2
[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p
s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]
Fνκ(r) =J0(kνκr)J0(kνκR)
kνκ = 1minusiδνκ
δν =q
2νω δκ =
q2κω
fνκ = 〈Fνκ〉 = 2kνκR
J1(kνκR)J0(kνκR)
0 05 1 15 20
02
04
06
08
1
(1minusFν)(1minusfν)
rR
R=100δν
R=10δν
R=δν
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The Rayleigh criterium
The Rayleigh criterium (int
pqdt gt 0)
ldquo at the phase of greatest condensation heat is receivedby the air and at the phase of greatest rarefaction heat isgiven up from it and thus there is a tendency to maintainthe vibrations rdquo JW Strutt (Lord Rayleigh) laquo The theory ofsound raquo Dover New York 2nd edition sect 322i p 231 1945)
1ρ
pq
TA Laser Rijke tube
Q in
Wout
T(x)
Q in
WoutWout
V0
wire
q = f (dxT p vx stack ) q = f (Twire vx V0 )
⋆ not acoustically compact ⋆ acoustically compact(q asymp q(t)δ(x minus xwire ))
⋆ amplification of the first mode ⋆ amplification of the first mode R
pqdt gt 0 =gt dxT lt 0R
pqdt gt 0 =gt xw lt L2
Common features⋆ Autonomous oscillators driven by heat (which both fill the ldquoRayleigh criteriumrdquo)⋆ Simple in terms of geometry but complicated operation above threshold
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 11 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TA Laser a prototypical example
Heat input =gt sound output (self-sustained acoustic oscillations at the frequency of themost unstable mode(s))
Keywords heat engine (prime-movers and heat pumps) autonomous oscillator Paradigmatic of the challenges that need to be taken up to understand more deeply the
processes controlling the operation of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 12 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Auto-oscillateur thermo-acoustique contexte et probleacutematique
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
⋆ Application drsquoun nablaT le long drsquoun mateacuteriau poreux placeacute dans un reacuteseau de guidedrsquoonde rArr auto-oscillations agrave la freacutequence du mode le plus instable
⋆ Geacuteomeacutetrie plusmn simple mais fonctionnement complexe au delagrave du seuil (pompage θac ldquovent acoustiquerdquo perte de charge singuliegravere propa NL )
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 13 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 14 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic principles
The TA Laser a conceptual very simplified approach
⋆ Simplistic description =gt Lagrangian approach follow a gas parcel submitted to a standingwave through a channel with an axial temperature gradient dxT0
δκ
ρ1
x
Hot Heat eXchanger (HHX) Cold Heat eXchanger (CHX)
Q Qrsquo
2 411
33
v
21 3 4 1τp
p 2
13
4
ξ max2
1 ~ adiabatic compression2 ~ isobaric expansion3 ~ adiabatic relaxation4 ~ isobaric contraction
=gt Work production if |dxT |ξmax gt τmax and dxT lt 0
⋆ But actual processes are actually more complex (heat exchange occurs continuously)⋆ A key point need of an imperfect thermal contact
=gtldquoefficientrdquo gas parcels should be at about δκ from the wall where δκ =p
2κωis the (frequency-dependent) thermal boundary layer thickness
⋆ Threshold of thermoacoustic instability (for a given mode)
=gt Sound amplification must be larger than losses
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 15 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Plane wave propagation through a viscous and heat-conducting gas along a ductsubmitted to a temperature gradient
⋆ Both the production of acoustic work and the ldquobucket briga-derdquo heat transport by sound along the duct are then described bysecond-order quantities
the ldquobucket brigaderdquo
xR
y
z
r
C T (x)s ss sρ λ pC T (x)ρ λ (x) (x)
θ
Main assumptions⋆ Low amplitudes ⋆ Typical wavelength raquo R ⋆ No mean flow
P(x r t) = P0 + p(x r t) =gt planes waves ⋆ ideal gasρ(x r t) = ρ0(x) + ρprime(x r t) p(x r t) = p(x t) ⋆ ρsCs ≫ ρ0Cp
T (x r t) = T0(x) + τ (x r t) =gtldquoboundary layer approximationrdquo ⋆ λs ≫ λv(x r t) ≪ c0 |partrζ| ≫ |partxζ|
S(x r t) = S0(x) + s(x r t) (ζ = p ρprime τ v s)[1] N Rott Adv Appl Mech 1980 [2] GW Swift J Acoust Soc Am 1988
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 16 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Equations fondamentales lineacuteariseacutees
ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )
parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0
ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )
dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0
dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p
⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0
vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]
τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]
ρprime = 1ω2
[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p
s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]
Fνκ(r) =J0(kνκr)J0(kνκR)
kνκ = 1minusiδνκ
δν =q
2νω δκ =
q2κω
fνκ = 〈Fνκ〉 = 2kνκR
J1(kνκR)J0(kνκR)
0 05 1 15 20
02
04
06
08
1
(1minusFν)(1minusfν)
rR
R=100δν
R=10δν
R=δν
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 17 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 18 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 20 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 21 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TA Laser a prototypical example
Heat input =gt sound output (self-sustained acoustic oscillations at the frequency of themost unstable mode(s))
Keywords heat engine (prime-movers and heat pumps) autonomous oscillator Paradigmatic of the challenges that need to be taken up to understand more deeply the
processes controlling the operation of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 12 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Auto-oscillateur thermo-acoustique contexte et probleacutematique
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
⋆ Application drsquoun nablaT le long drsquoun mateacuteriau poreux placeacute dans un reacuteseau de guidedrsquoonde rArr auto-oscillations agrave la freacutequence du mode le plus instable
⋆ Geacuteomeacutetrie plusmn simple mais fonctionnement complexe au delagrave du seuil (pompage θac ldquovent acoustiquerdquo perte de charge singuliegravere propa NL )
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 13 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 14 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic principles
The TA Laser a conceptual very simplified approach
⋆ Simplistic description =gt Lagrangian approach follow a gas parcel submitted to a standingwave through a channel with an axial temperature gradient dxT0
δκ
ρ1
x
Hot Heat eXchanger (HHX) Cold Heat eXchanger (CHX)
Q Qrsquo
2 411
33
v
21 3 4 1τp
p 2
13
4
ξ max2
1 ~ adiabatic compression2 ~ isobaric expansion3 ~ adiabatic relaxation4 ~ isobaric contraction
=gt Work production if |dxT |ξmax gt τmax and dxT lt 0
⋆ But actual processes are actually more complex (heat exchange occurs continuously)⋆ A key point need of an imperfect thermal contact
=gtldquoefficientrdquo gas parcels should be at about δκ from the wall where δκ =p
2κωis the (frequency-dependent) thermal boundary layer thickness
⋆ Threshold of thermoacoustic instability (for a given mode)
=gt Sound amplification must be larger than losses
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 15 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Plane wave propagation through a viscous and heat-conducting gas along a ductsubmitted to a temperature gradient
⋆ Both the production of acoustic work and the ldquobucket briga-derdquo heat transport by sound along the duct are then described bysecond-order quantities
the ldquobucket brigaderdquo
xR
y
z
r
C T (x)s ss sρ λ pC T (x)ρ λ (x) (x)
θ
Main assumptions⋆ Low amplitudes ⋆ Typical wavelength raquo R ⋆ No mean flow
P(x r t) = P0 + p(x r t) =gt planes waves ⋆ ideal gasρ(x r t) = ρ0(x) + ρprime(x r t) p(x r t) = p(x t) ⋆ ρsCs ≫ ρ0Cp
T (x r t) = T0(x) + τ (x r t) =gtldquoboundary layer approximationrdquo ⋆ λs ≫ λv(x r t) ≪ c0 |partrζ| ≫ |partxζ|
S(x r t) = S0(x) + s(x r t) (ζ = p ρprime τ v s)[1] N Rott Adv Appl Mech 1980 [2] GW Swift J Acoust Soc Am 1988
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Equations fondamentales lineacuteariseacutees
ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )
parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0
ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )
dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0
dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p
⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0
vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]
τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]
ρprime = 1ω2
[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p
s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]
Fνκ(r) =J0(kνκr)J0(kνκR)
kνκ = 1minusiδνκ
δν =q
2νω δκ =
q2κω
fνκ = 〈Fνκ〉 = 2kνκR
J1(kνκR)J0(kνκR)
0 05 1 15 20
02
04
06
08
1
(1minusFν)(1minusfν)
rR
R=100δν
R=10δν
R=δν
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 17 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 18 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 20 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 21 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Auto-oscillateur thermo-acoustique contexte et probleacutematique
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
⋆ Application drsquoun nablaT le long drsquoun mateacuteriau poreux placeacute dans un reacuteseau de guidedrsquoonde rArr auto-oscillations agrave la freacutequence du mode le plus instable
⋆ Geacuteomeacutetrie plusmn simple mais fonctionnement complexe au delagrave du seuil (pompage θac ldquovent acoustiquerdquo perte de charge singuliegravere propa NL )
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 13 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 14 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic principles
The TA Laser a conceptual very simplified approach
⋆ Simplistic description =gt Lagrangian approach follow a gas parcel submitted to a standingwave through a channel with an axial temperature gradient dxT0
δκ
ρ1
x
Hot Heat eXchanger (HHX) Cold Heat eXchanger (CHX)
Q Qrsquo
2 411
33
v
21 3 4 1τp
p 2
13
4
ξ max2
1 ~ adiabatic compression2 ~ isobaric expansion3 ~ adiabatic relaxation4 ~ isobaric contraction
=gt Work production if |dxT |ξmax gt τmax and dxT lt 0
⋆ But actual processes are actually more complex (heat exchange occurs continuously)⋆ A key point need of an imperfect thermal contact
=gtldquoefficientrdquo gas parcels should be at about δκ from the wall where δκ =p
2κωis the (frequency-dependent) thermal boundary layer thickness
⋆ Threshold of thermoacoustic instability (for a given mode)
=gt Sound amplification must be larger than losses
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 15 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Plane wave propagation through a viscous and heat-conducting gas along a ductsubmitted to a temperature gradient
⋆ Both the production of acoustic work and the ldquobucket briga-derdquo heat transport by sound along the duct are then described bysecond-order quantities
the ldquobucket brigaderdquo
xR
y
z
r
C T (x)s ss sρ λ pC T (x)ρ λ (x) (x)
θ
Main assumptions⋆ Low amplitudes ⋆ Typical wavelength raquo R ⋆ No mean flow
P(x r t) = P0 + p(x r t) =gt planes waves ⋆ ideal gasρ(x r t) = ρ0(x) + ρprime(x r t) p(x r t) = p(x t) ⋆ ρsCs ≫ ρ0Cp
T (x r t) = T0(x) + τ (x r t) =gtldquoboundary layer approximationrdquo ⋆ λs ≫ λv(x r t) ≪ c0 |partrζ| ≫ |partxζ|
S(x r t) = S0(x) + s(x r t) (ζ = p ρprime τ v s)[1] N Rott Adv Appl Mech 1980 [2] GW Swift J Acoust Soc Am 1988
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 16 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Equations fondamentales lineacuteariseacutees
ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )
parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0
ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )
dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0
dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p
⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0
vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]
τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]
ρprime = 1ω2
[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p
s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]
Fνκ(r) =J0(kνκr)J0(kνκR)
kνκ = 1minusiδνκ
δν =q
2νω δκ =
q2κω
fνκ = 〈Fνκ〉 = 2kνκR
J1(kνκR)J0(kνκR)
0 05 1 15 20
02
04
06
08
1
(1minusFν)(1minusfν)
rR
R=100δν
R=10δν
R=δν
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 18 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 20 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 14 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic principles
The TA Laser a conceptual very simplified approach
⋆ Simplistic description =gt Lagrangian approach follow a gas parcel submitted to a standingwave through a channel with an axial temperature gradient dxT0
δκ
ρ1
x
Hot Heat eXchanger (HHX) Cold Heat eXchanger (CHX)
Q Qrsquo
2 411
33
v
21 3 4 1τp
p 2
13
4
ξ max2
1 ~ adiabatic compression2 ~ isobaric expansion3 ~ adiabatic relaxation4 ~ isobaric contraction
=gt Work production if |dxT |ξmax gt τmax and dxT lt 0
⋆ But actual processes are actually more complex (heat exchange occurs continuously)⋆ A key point need of an imperfect thermal contact
=gtldquoefficientrdquo gas parcels should be at about δκ from the wall where δκ =p
2κωis the (frequency-dependent) thermal boundary layer thickness
⋆ Threshold of thermoacoustic instability (for a given mode)
=gt Sound amplification must be larger than losses
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 15 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Plane wave propagation through a viscous and heat-conducting gas along a ductsubmitted to a temperature gradient
⋆ Both the production of acoustic work and the ldquobucket briga-derdquo heat transport by sound along the duct are then described bysecond-order quantities
the ldquobucket brigaderdquo
xR
y
z
r
C T (x)s ss sρ λ pC T (x)ρ λ (x) (x)
θ
Main assumptions⋆ Low amplitudes ⋆ Typical wavelength raquo R ⋆ No mean flow
P(x r t) = P0 + p(x r t) =gt planes waves ⋆ ideal gasρ(x r t) = ρ0(x) + ρprime(x r t) p(x r t) = p(x t) ⋆ ρsCs ≫ ρ0Cp
T (x r t) = T0(x) + τ (x r t) =gtldquoboundary layer approximationrdquo ⋆ λs ≫ λv(x r t) ≪ c0 |partrζ| ≫ |partxζ|
S(x r t) = S0(x) + s(x r t) (ζ = p ρprime τ v s)[1] N Rott Adv Appl Mech 1980 [2] GW Swift J Acoust Soc Am 1988
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 16 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Equations fondamentales lineacuteariseacutees
ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )
parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0
ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )
dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0
dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p
⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0
vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]
τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]
ρprime = 1ω2
[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p
s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]
Fνκ(r) =J0(kνκr)J0(kνκR)
kνκ = 1minusiδνκ
δν =q
2νω δκ =
q2κω
fνκ = 〈Fνκ〉 = 2kνκR
J1(kνκR)J0(kνκR)
0 05 1 15 20
02
04
06
08
1
(1minusFν)(1minusfν)
rR
R=100δν
R=10δν
R=δν
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 17 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 18 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 20 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 21 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic principles
The TA Laser a conceptual very simplified approach
⋆ Simplistic description =gt Lagrangian approach follow a gas parcel submitted to a standingwave through a channel with an axial temperature gradient dxT0
δκ
ρ1
x
Hot Heat eXchanger (HHX) Cold Heat eXchanger (CHX)
Q Qrsquo
2 411
33
v
21 3 4 1τp
p 2
13
4
ξ max2
1 ~ adiabatic compression2 ~ isobaric expansion3 ~ adiabatic relaxation4 ~ isobaric contraction
=gt Work production if |dxT |ξmax gt τmax and dxT lt 0
⋆ But actual processes are actually more complex (heat exchange occurs continuously)⋆ A key point need of an imperfect thermal contact
=gtldquoefficientrdquo gas parcels should be at about δκ from the wall where δκ =p
2κωis the (frequency-dependent) thermal boundary layer thickness
⋆ Threshold of thermoacoustic instability (for a given mode)
=gt Sound amplification must be larger than losses
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 15 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Plane wave propagation through a viscous and heat-conducting gas along a ductsubmitted to a temperature gradient
⋆ Both the production of acoustic work and the ldquobucket briga-derdquo heat transport by sound along the duct are then described bysecond-order quantities
the ldquobucket brigaderdquo
xR
y
z
r
C T (x)s ss sρ λ pC T (x)ρ λ (x) (x)
θ
Main assumptions⋆ Low amplitudes ⋆ Typical wavelength raquo R ⋆ No mean flow
P(x r t) = P0 + p(x r t) =gt planes waves ⋆ ideal gasρ(x r t) = ρ0(x) + ρprime(x r t) p(x r t) = p(x t) ⋆ ρsCs ≫ ρ0Cp
T (x r t) = T0(x) + τ (x r t) =gtldquoboundary layer approximationrdquo ⋆ λs ≫ λv(x r t) ≪ c0 |partrζ| ≫ |partxζ|
S(x r t) = S0(x) + s(x r t) (ζ = p ρprime τ v s)[1] N Rott Adv Appl Mech 1980 [2] GW Swift J Acoust Soc Am 1988
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Equations fondamentales lineacuteariseacutees
ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )
parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0
ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )
dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0
dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p
⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0
vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]
τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]
ρprime = 1ω2
[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p
s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]
Fνκ(r) =J0(kνκr)J0(kνκR)
kνκ = 1minusiδνκ
δν =q
2νω δκ =
q2κω
fνκ = 〈Fνκ〉 = 2kνκR
J1(kνκR)J0(kνκR)
0 05 1 15 20
02
04
06
08
1
(1minusFν)(1minusfν)
rR
R=100δν
R=10δν
R=δν
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 18 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 21 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Plane wave propagation through a viscous and heat-conducting gas along a ductsubmitted to a temperature gradient
⋆ Both the production of acoustic work and the ldquobucket briga-derdquo heat transport by sound along the duct are then described bysecond-order quantities
the ldquobucket brigaderdquo
xR
y
z
r
C T (x)s ss sρ λ pC T (x)ρ λ (x) (x)
θ
Main assumptions⋆ Low amplitudes ⋆ Typical wavelength raquo R ⋆ No mean flow
P(x r t) = P0 + p(x r t) =gt planes waves ⋆ ideal gasρ(x r t) = ρ0(x) + ρprime(x r t) p(x r t) = p(x t) ⋆ ρsCs ≫ ρ0Cp
T (x r t) = T0(x) + τ (x r t) =gtldquoboundary layer approximationrdquo ⋆ λs ≫ λv(x r t) ≪ c0 |partrζ| ≫ |partxζ|
S(x r t) = S0(x) + s(x r t) (ζ = p ρprime τ v s)[1] N Rott Adv Appl Mech 1980 [2] GW Swift J Acoust Soc Am 1988
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Equations fondamentales lineacuteariseacutees
ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )
parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0
ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )
dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0
dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p
⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0
vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]
τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]
ρprime = 1ω2
[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p
s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]
Fνκ(r) =J0(kνκr)J0(kνκR)
kνκ = 1minusiδνκ
δν =q
2νω δκ =
q2κω
fνκ = 〈Fνκ〉 = 2kνκR
J1(kνκR)J0(kνκR)
0 05 1 15 20
02
04
06
08
1
(1minusFν)(1minusfν)
rR
R=100δν
R=10δν
R=δν
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 17 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 18 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 20 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 21 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Basic equations
⋆ Equations fondamentales lineacuteariseacutees
ρdtV = minusnablaP + micronabla2V + (η + micro3)nabla (nablaV) rArr ρ0iωvx = minusdx p + micro(1r)partr (rpartr vx )
parttρ + nabla (ρV) = 0 rArr iωρprime + ρ0partx vx + vxdxρ0 = 0
ρTdtS = λ∆T + O2(v) rArr ρ0T0 (iωs + vxpartxS0) = λ(1r)partr (rpartr τ )
dS = (CpT )dT minus (αρ)dP rArr T0 s = Cp τ minus pρ0
dρ = minusραdT + ρχtdP rArr ρprime = minus(ρ0T0)τ + (γc20 )p
⋆ Conditions aux limites (tuyau drsquoaxe x et de rayon R) rArr ~v(x r = R) = 0 τ(x r = R) = 0
vx (x r) = i(ωρ0)dx p [1 minus Fν (r)]
τ(x r) = p(ρ0Cp) [1 minus Fκ(r)] minus 1(ρ0ω2)dx pdxT0 [1 minus (PrFν (r) minus Fκ(r))(Pr minus 1)]
ρprime = 1ω2
[1 minus (PrFν minus Fκ)(Pr minus 1)+] dxT0T0dx p + 1c20 [1 + (γ minus 1)Fκ] p
s = minusp(ρ0T0)Fκ minus Cp(ρ0ω2)dx pdxT0T0 [1 minus (PrFν minus Fκ)(Pr minus 1)]
Fνκ(r) =J0(kνκr)J0(kνκR)
kνκ = 1minusiδνκ
δν =q
2νω δκ =
q2κω
fνκ = 〈Fνκ〉 = 2kνκR
J1(kνκR)J0(kνκR)
0 05 1 15 20
02
04
06
08
1
(1minusFν)(1minusfν)
rR
R=100δν
R=10δν
R=δν
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 17 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 18 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 21 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le point de vue thermodynamique
⋆ Onde plane stationnaire dans un tuyau ldquolargerdquo
p(x) = Pmax cos (k0x)
ρprime = 1ω2
h
1 minus PrFνminusFκPrminus1 +
idxT0T0
dx p + 1c20
[1 + (γ minus 1)Fκ] p
δκ
d
R=10
d = δκ3 d = δκ d = 3δκ
1ρ
P
1ρ
P
1ρ
P
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
minus1
minus05
0
05
1
pPmax
ρρmax
0 02 04 06 08 1minus1
minus05
0
05
1
tT
pPmax
ρρmax
dxT0 = 0 dxT0 = 0 dxT0 = 0dxT0 = minus220Km dxT0 = minus220Km dxT0 = minus220Km
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 18 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 20 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 21 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Sound amplification
⋆ From the knowledge of the acoustic variables it is possible to calculate thetime-average second order acoustic power produced (or absorbed) per unit volume w2
([w2 ] = Wmminus3)
w2 = partx(p lt vx gt)
⋆ After some calculations3
w2 = wκ + wν + wsw + wtw
wκ =1
2
(γ minus 1)ω
ρ0c2
0
image (fκ) |p|2
wν =1
2ωρ0
image(fν)
|1 minus fν |2|〈vx〉|
2
wsw = minusimage(h)dxT0
T0
1
2image (p〈vlowast
x 〉)
wtw = real(h)dx T0
T0
1
2real (p〈vlowast
x 〉)
h =fκ minus fν
(1 minus Pr)(1 minus fν)0 02 04 06 08 1 12 14 16 18 2
minus14
minus12
minus10
minus8
minus6
minus4
minus2
δκRh
minus04
minus02
0
02
04
06
08
1
real (h)
image (fν)
|1minusfν|2
image (h)image (fκ)
[3] A Tominaga Cryogenics 1995
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 19 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 20 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 21 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Heat transport by sound
⋆ From the knowledge of the acoustic variables it is also possible to calculate thethermoacoustic heat flux q2 ([q2 ] = Wmminus2)
q2 = ρ0T0(〈svx 〉) s =CpT0
τ minus 1
ρ0T0p
⋆ After some calculations
q2 = qsw + qtw + λacdxT0
qsw = minusimage(g)1
2image`p〈vlowast
x 〉acute
qtw = real(g)1
2real`p〈vlowast
x 〉acute
λac =ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2
g =f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 20 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 21 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Amplification of a standing wave
w2 = wκ + wν| z
losses
+
propminusimage(h)z|wsw +
propreal(h)z|wtw
| z production
h = (fκ minus fν )[(1 minus Pr)(1 minus fν)]δκ
Q Qrsquo
411
33
2
x
x
yz
CYLINDER
2Rh
x
yz
2Rh
2Rh
SQUARE
2ri
hR
R = 3 rh i
PINminusARRAY
⋆ In case of a standing wave one need δκ le Rh (stack) and always have |image(h)| lt 1=gt intrinsic irreversibility4due to the need of an imperfect thermal contact⋆ But in case of a travelling wave phasing we have |image(h)| asymp 1 if δκ ge Rh=gt a travelling wave phasing combined with a regenerator (δκ ge Rh) should be better
[4] J C Wheatley et al J Acoust Soc Am 1983
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
How to build an intrinsically reversible Thermoacoustic engine
w2 = wκ + wν| z
losses
+
propminusimage(h)z|
wsw +
propreal(h)z|
wtw| z
production
δκ
Qrsquo Q
1
213
3
1
2 34
4
Stirling Cycle
ρ1
p
=gt use a regenerator(δκ ge Rh) and a travelling wave phasing between p and vx
⋆ [CeperleyJASA1979]
regenerator
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does not singbecause wtw lt wν
⋆ [Yazaki et al PRL 1998]
stack
=gt closed-loop resonator
bull p vx prop eminusi(kxminusωt)
bull Z = pvx
= ρ0c0
bull f asymp c0L=gt does sing but employs a stack
⋆ [Backhaus et al Nature1999]
regenerator
=gt acoustic feedback loopbull local TW phasingbull Z gtgt ρ0c0 locallybull f ltlt c0L=gt does sing quite well
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 22 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
A Standing wave engine [Swift JASA 1992]
bull fluiddagger 138 bar Heliumbull frequency 120 Hzbull drive ratio pP0 6bull TH 1000 Kbull Wload 630 Wbull QH 7 kWbull η = 009 (013ηc )
A thermoacoustic-Stirling engine [Backhaus amp Swift Nature 1999]
bull fluiddagger 30 bar Heliumbull frequency 80 Hzbull drive ratio pP0 6-10bull TH 1000 Kbull Wload 710 Wbull QH 24 kWbull η = 03 (042ηc )
dagger Need a large γ low Pr and high P0 (in 1-bar air at 120 dBSPL Wac asymp 10minus2Wmminus2 )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 23 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Examples
Advantages of θac engines (and heat pumps)
⋆ Simple easy to build =gt potential reliability low cost⋆ Working gas = pressurized inert gas =gt harmless environmental friendly⋆ No or not much moving parts =gt resists wear⋆ Decent efficiencies(currently up to 32 in thermo-acoustic conversion for engines 5 )with potential improvements
Limitations of θac engines (and heat pumps)
⋆ Moderate output powers =gt up to a few kilowatts⋆ decent efficiencies =gt to be improved⋆ A development which remains limited to a few researchlaboratories and start-up companies
Potential applications of θac engines (and heat pumps)
⋆ Waste heat recovery micro-cogeneration⋆ Thermoacoustically driven pulse-tube refrigeration(cryogeny liquefaction of natural gas in methane tankers )⋆ Domestic refrigeration electronics cooling
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
Sketch of a thermo-acousto-electric
generator Such kind of engine has
been demonstrated to achieve a glo-
bal efficiency of 18 6
[5] Tijani et al J Appl Phys 2011 [6] Backhaus et al Appl Phys Lett 2004
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 24 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
⋆ The basic principles of design tools may be summarized as
A two-port modeling of acoustic wave propagation under an assigned T0(x)through the device leading to a characteristic equation
An energy balance under an assigned heat input QH which accounts for thethermoacoustic heat transport by sound Q2
⋆ A simplistic illustration would be for instance as follows
T (x)0
TH
TC
QH
QC
W2
PA
|p(x)|
1 Assign QH and set PA = 0 (and thusQ2 = 0)
2 Solve heat transfer =gt get T0(x)
3 Solve acoustics =gt get spatial distributionp(x) and working frequency f
4 Fix arbitrary amplitude PA
5 Calculate thermoacoustic heat transport Q2
6 Solve heat transfer =gt get T0(x)(accounting for Q2)
7 Repeat steps 3-6 until equilibrium ieQH = QC + W2 with QC = QC0 + Q2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 25 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Design of thermoacoustic engines
Limitations of the existing design tools
⋆ Up to now the design of TA engines is always realized from the linear TA theory
⋆ A famous free-downloadable software developed at Los Alamos DELTA-EC9
⋆ Efficient tool for design purposes which works quite-well for the prediction ofsteady-state operation of Low-Amplitude TA engines
But
⋆ These tools are restricted to the linear (or weakly nonlinear) regime
⋆ These tools are restricted to a 1-D description of the phenomena
⋆ They predict performance in steady-state
⋆ The user need to be experienced in the fieldlowast
[9] WC Ward et al J Acoust Soc Am 1994
lowastlaquo Intuition is required from the start and successful solutions yield intuition raquo WC Ward Acousticsrsquo08Paris 2008
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 26 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 27 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Le systegraveme consideacutereacute
⋆ Systegraveme consideacutereacute sim tube de Sondhauss1
TH
TC
Tm
Ls L
volume V
⋆ Objectif = le deacutecrire comme 1 oscillateur 1 ddl (oscillateur agrave amortissement lt 0 )
hypothegravese BF + analogies eacutelectriques
Approximation (fausse) de son fonctionnement par une DDE
Reacutesolution analytique par la meacutethode des eacutechelle multiples
Modeacutelisation de la saturation NL par pompage thermoacoustique
C Sondhauss Ann Phys(Leipzig) 79 11850
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 28 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Analogie eacutelectroacoustique
⋆ portion de stack de longueur dx ≪ λ
dp = minus iωρ0dxφS
11minusfν
u
du = minus iωφSdxγP0
[1 + (γ minus 1)fκ] p +(fκminusfν )
(1minusfν )(1minusσ)
dT0T0
u
p~
T0 T +dT0 0
p+dp~ ~
u+du~ ~u~
dx
⋆ Analogie eacutelectro-acoustique
dp = minus(iωM + Rν)u
du = minus (iωC + 1Rκ) p + Gu
M =ρ0dxφS
1minusreal(fν )
|1minusfν |2
C = φSdxγP0
(1 + (γ minus 1)real(fκ))
Rν =ωρ0dx
φSimage(minusfν )
|1minusfν |2
Rν~u(x)
p(x)~ ~p(x+dx)
~u(x+dx)
C Rκ ~ u(x)
G
M
1Rκ
= γminus1γ
ωφSdximage(minusfκ)P0
G = fκminusfν(1minusfν )(1minusσ)
dT0T0
⋆ Approximation ldquoquasi adiabatiquerdquo
fνκ sim (1 minus i)ǫνκ avec ǫνκ = δνκRs ≪ 1 (et ǫνκ prop ωminus12)
M prop (1 minus ǫν) C prop 1 + (γ minus 1)ǫκ Rν prop ωǫν 1Rκ
prop ωǫκ G prop (1 minus i)ǫκdT0T0
isin C
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 29 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Scheacutema eacutelectrique eacutequivalent
⋆ Scheacutema eacutelectrique eacutequivalent
TH
TC
Tm
Ls L
volume V
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
⋆ Equation caracteacuteristique
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
1
ω20(TH)
=
bdquo
M +ρ0Ls
ΦS
TC
Tm
laquo
C
|f | =2radic
κ
Rs
bdquoTm
Tc
laquoβ2radic
Pr
1 +radic
Pr
∆T
Tm
1
1 + LsφL
TCTm
s
1 +
bdquoradicPr
ldquo
1 +radic
Pr
rdquo Ls
φL
TC
Tm
laquo2
Arg(f ) = arctan
bdquoradicPr
ldquo
1 +radic
Pr
rdquo TC
∆T
Ls
φL
laquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 30 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Approximation par une DDE
(iω)2 u minus (iω)32 f (TH)u + ω20 u = 0
⋆ On preacutesume que la solution est un signal harmonique de pulsation ω asymp ω0 et drsquoam-plitude lentement variable En posant ω = ω0 + ∆ω avec ∆ω = (ω minus ω0) ≪ ω0 ilvient
minus (iω)32 f (TH) asymp 12 ω0
radicω0|f |eminusiω0t1 minus iω times 3
2radic
ω0|f |eminusiω0t2
avec ω0t1 = 5π4 minus Arg(f ) et ω0t2 = 7π
4 minus Arg(f )
⋆ On prend ensuite la transformeacutee de Fourier inverse en remarquantque
ueminusiω0t12 asymp ueminusiωt12 + i ∆ωω ueminusiωt12
de sorte que (sous reacuteserve drsquoune grossedagger approximation)
Fminus1h
minus (iω)32 f u(ω)i
asymp ω320 |f |
2 u(t minus t1) minus 3ω120 |f |2 dtu(t minus t2)
rArr obtention drsquoune eacutequation diffeacuterentielle agrave retard
d2ttu(t) minus 3
2radic
ω0|f |dtu(t minus t2) + 12 ω0
radicω0|f |u(t minus t1) + ω2
0u(t) asymp 0
dont les retards t12 traduisent un effet meacutemoire qui est lrsquoessence des pheacutenomegravenes misen jeu dans les couches limites viscothermiques
daggerTransformation drsquoune deacuteriveacutee fractionnaire en une fonction agrave retard
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 31 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution numeacuterique
p minus 32radic
ω0|f |p(t minus t2) + 12 ω0
radicω0|f |p(t minus t1) + ω2
0p asymp 0
rArr Utilisation drsquoun solveur de DDE et calcul de p(t ge 0) pourdiffeacuterentes valeurs de TH et pour p(t le 0) = 1 Pa
TH
TC
Tm
Ls L
volume V
Valeurs des paramegravetres de calculfluide (agrave TC = 300K) tuyau
viscositeacute micro = 184 10minus5Pas longueur L = 12cm
conductiviteacute λ = 226 10minus2Wmminus1Kminus1 section S = π times 32cm2
masse volumique ρ0 = 12kgmminus3 stackcapaciteacute calorifique Cp = 1003Jkgminus1Kminus1 longueur Ls = 4cm
coef polytropique γ = 14 rayon pore Rs = 3mm
caviteacute porositeacute φ = 095
volume V = 1l
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
0 005 01 015 02 025
minus1
minus05
0
05
1
temps s
p (P
a)
TH = 300K TH = 460K TH = 470K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 32 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples
⋆ Changement de variable θ equiv ω0t rArr
d2θθp minus 3ζdθp(θ minus θ2) + ζp(θ minus θ1) + p = 0
avec θ12 = ω0θ12 et ζ = 12
|f |radicω0
≪ 1 (ζ prop (∆TTm)(q
2νmω0
Rs) ≪ 1)
⋆ On cherche une solution sous la forme
p(θ) = p(τ s) = p0(τ s) + ζp1(τ s) + avec s equiv ζθ et τ equiv θ
dθp asymp partτp + ζpartsp
d2θθp asymp part
2ττ p + 2ζpart
2τsp
p(θ minus θ1) asymp p(τ minus θ1 s) minus ζθ1partsp(τ minus θ1 s) +
dθp(θ minus θ2) asymp partτp(τ minus θ2 s) minus ζθ2part2τsp(τ minus θ2 s) + ζpartsp(τ minus θ2 s)
⋆ Ordre ζ0
part2ττ p0 + p0 = 0 rArr p0(τ s) = A(s)e it + cc avec A(s) = R(s)e iΦ(s) isin C
⋆ Ordre ζ1
part2ττ p1 + p1 =
ldquo
minus2idsA + 3iAeminusiθ2 minus Aeminusiθ1rdquo
| z =0 condition de solvabiliteacute
e iτ + cc
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 33 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Reacutesolution analytique meacutethode des temps multiples (2)
(
dsA = 12Aldquo
3eminusiθ2 + ieminusiθ1rdquo
A(s) = R(s)e iΦ(s)rArr
dsR =`
32 cos θ2 + 1
2 sin θ1acute
RdsΦ = 1
2 [cos θ1 minus 3 sin θ2]
p(t) asymp 2R(0)eminusΩprimeprimet cos`Ωprimet + Φ(0)
acute
avec
Ωprimeprime = minus 12 (3 cos θ2 + sin θ1) ζω0
Ωprime = ω0ˆ1 +
`12 cos θ1 minus 3
2 sin θ2acute
ζ˜ et
(
ζ(TH ) = 12
|f |radicω0
prop (∆TTm)(δκRs) ≪ 1
θ1(TH ) = 5π4 minus Arg(f ) = θ2(TH ) minus π
2
300 350 400 450 500 550 6001005
101
1015
Ωrsquoω
0
TH
(K)300 350 400 450 500 550 600
minus001
minus0005
0
0005
001
Ωrsquorsquo
ω0
minus1
minus05
0
05
1
p (P
a)
DDE23
analyt
0 005 01 015 02 025 03 035 04 045 05
minus1
minus05
0
05
1
temps (s)
p (P
a)
DDE23
analyt
TH
=470K
TH
=460K
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 34 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Saturation par pompage thermoacoustique
T =cteC
HQ
T (t)H
Rν
~ uC p~
M~u
G
~(1+G)uMstackcavite tuyau
S
QH rArr ∆T րrArr auto-oscillations ր rArr Q2 prop p2 rArr ∆T ց
rArr un effet de saturation NL deacutecrit par la theacuteorie lineacuteaire
⋆ Amplification TAp(t ζt) asymp P(ζt) cos
`Ωprime(ζt)t
acute
⋆ Equation de la thermocineacutetique
ρsCsVsdtTH = QH + Qc + Q2
Qc = minusλsS∆TLs
Q2 = ρ0TcS〈svx〉 asymp minusλacS∆TLs
avec λac (ζt) prop P2(ζt)
⋆ BilanDagger
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
0
2
4
6
8
p (k
Pa)
0 15 30 450
50
100
150
temps (min)∆
T (
K)
Transitoire de deacuteclenchement et de saturation pourun chauffage QH de 15 W en prenant ρsCs =
18 105Jmminus3Kminus1 et λs = 017W mminus1Kminus1
Daggersous reacuteserve que dtTH sim ζTH ou ≪ ζTH
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 35 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autres meacutecanismes de saturation
dtP = σ(TH )P
ρsCsdtTH = QHVs minus (λs + ΓλP2)∆TL2s
rArr
8gtlt
gt
∆T |infin = ∆T |seuil
Pinfin =
s
1Γλ
bdquoQHL2
sVs∆T|infin minus λs
laquo
Ajoutons arbitrairement un terme de saturation quadratique
dtP
prime = σ(T primeH )Pprime minus αPprime2
ρsCsdtTprimeH = QHVs minus (λs + ΓλPprime2)∆T primeL2
s
et choisissons α = 10minus4 de sorte que Pprimeinfin asymp 099Pinfin
0
2
4
6
8
p (k
Pa)
5 100
2
4
p (k
Pa)
0 15 30 450
50
100
150
temps (min)
∆ T
(K
)
Transitoire de deacuteclenchement et saturation En noir seul le pompage thermoacoustique est pris en compte En
rouge ajout arbitraire drsquoune saturation quadratique avec α = 10minus4
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 36 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 37 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Autonomous oscillators =gt nonlinear saturation processes
⋆ Thermoacoustic engines = self-sustained oscillators⋆ Most of the nonlinear processes controlling wave saturation are well identified
⋆ Some of them are well-predicted by theory (at least for moderate pressure levels) ⋆ but most of them are poorly described
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 38 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Thermoacoustic heat transport by sound
A second-order effect which is described by the linear thermoacoustic theory
q2 =1
2realraquo
f lowastν minus fκ
(1 + Pr)(1 minus f lowastν )
p〈vlowastx 〉ndash
+ρ0Cp
2ω(1 minus Pr2)
image (Prf lowastν minus fκ)
|1 minus fν |2|〈vx〉|2dxT0
but actual stackregenerators employ materials of complicated geometry
mesh grids NiCr foam RVC foam which are also anisotropicrArr How to know T0(x) from Qin Is T0 uniform through a section
and finally the problem is the same for the evaluation of w2
rArr Measure the sound scattered by the TAC
to evaluate both f 1minus4νκ (with QH = 0)
to predict the onset threshold of a TAengine56 (as a function of QH) andor to evaluate the thermophysical proper-ties of the stack7
QH
QC
QC
PZE buzzer
Ac Imp Sensor
Mic3
Ref planeMic2Mic1
ThermoAcoustic Core
Sketch of the experimental set-up used by Bannwart
et al6 to measure the T-matrix of the TAC as afunction of QH
[1] Hayden et al JASA 1997 [2] Wilen JASA 2001 [3] Petculescu et al JASA 2001 [4] Y Ueda etal JASA 2009 [5] M Guedra et al JASA 2011 [6] FC Bannwart et al JASA 2013 [7] M Guedraet al submitted to Appl Therm Eng 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 39 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation
⋆ High amplitude oscillations rArr wave distorsion eg a simple wave traveling along x uarr in an inviscidnon heat conducting gas1
parttp + (c + vx ) partxp = 0 An extensively studied topic23
t + t0 ∆t0
(c+v ) tx ∆
x
acou
stic
pre
ssur
e
⋆ Impact on the operation of TA engines Outside the stack
Higher harmonics generation and even shock waves4 Increase of losses since 〈wνκ〉 prop radic
ωWithin the stack
Decrease of TA amplification (since δκ depends on ω)⋆ Some works tried to account for it
Outside the stack Solve NL propagation5
Determine optimal shape of the resonator5
Within the stack Numerical computation78(no linearization) but a lack of quantitative comparison Most calculations made with an assigned ∆T
rArr Do high amplitude effects play a key role Most TA resonatorrsquos are inharmonic p2 vx2 lead to w4 q4 rArr Other saturating processes are probably worth considering
Transient regime in a SW TA engineunder an assigned ∆T from [7]
An optimum resonatorrsquos shape thatmaximize the Q of a resonator from[6]
[1] AD Pierce Acoustics Acoustical Society of America NY 1991 [2] Rudenko amp Soluyan consultantbureau NY 1977 [3] Hamilton amp Blackstock Ac Soc Am 1998 [4] Biwa et al JASA 2011 [5]Gusev et al Acust Acta Acust 2000 [6] Ilinskii et al JASA 2001 [7] Karpov et al JASA 2002 [8]hamilton et al JASA 2002
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 40 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Higher harmonics generation shock waves
⋆ Still the study of NL propagation in TA engines is an interesting topic ⋆ Recently shock waves reported in a closed-loop stack based TA engine [Biwa et al JASA2011] and acoustic intensity measurements were performed
TW engine=gt shock wave along x uarr
SW engine=gt no shock waves
damped SWvs
amplified TW
within the TACComplex processes
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 41 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Entrance effects
⋆ Geometrical singularities =gt Vorticity
ldquoMinorrdquo losses1minus4
Basic modelingωt le π ∆pout(t) = minus 1
2 Koutρ0u2(t)
ωt ge π ∆pin(t) = 12 Kinρ0u
2(t)
Wminor propR π
ω0 ∆pudt +
R 2πω
πω
∆pudt prop U3
but thermal effects as well Abrupt transition from polytropic to adiabatic
rArr higher harmonics in τ5minus6
Heat transport accompanying vortex shedding
polytropic process adiabatic process
Vorticity field at the edge of a stack obtainedwith PIV (from [4])
Amplitudes of τ1 and τ2 as a function of the distance tothe stack obtained with CWA (from [6])
rArr Are established results for steady flows applicable to oscillating flows
[1] GW Acoustical Society of America NY 2001 [2] Blanc-Benon et al CR Meca 2003 [3] Marx etal JASA 2003 [4] Berson et alJASA 2008 [5] Gusev et al JASA 2001 [6] Berson et al Int JournHeat Mass Transf 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 42 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Start with the Navier-Stokes equation with party gtgt partx then make successive approximations(ζ = ζ0 + ζ1 + ζ2 ) up to second order and after time averaging one gets
ν0part2yyvx2 = 1ρ0partxp2 + partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acuteminus party
ldquo
ν1partyvx1
rdquo
⋆ A steady velocity vx2 generated by acoustic oscillations
- because of the convective derivative
partx
ldquo
v2x1
rdquo
+ party`vx1vy1
acute
- because ν depends on temperature (ν = ν0(T0) + ν1(τ ))
party
ldquo
ν1partyvx1
rdquo
- and actually many other sources of streaming
ν prop T1+βR 21 νdt gt
R 32 νdt
pτ
δν
2
21 3
31
net drift
inner streaming
oute
r st
ream
ing
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 43 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Since the pioneering theoretical works of Rayleigh (1883) and Schlichting (1932) many analyticalnumericalexperimental studies including
⋆ thermal effects1minus4 (dxT0 micro = micro(T ) λ = λ(T ))
⋆ acoustic streaming in closed-loop devices5minus7 (ldquoGedeon streamingrdquo)
⋆ high amplitude effects8minus11 (eg inertia effects as Renl =p
MSh asymp 1)
⋆ the coupling between u2 and dxT0 (reciprocal impact10minus11)
u2 ρ1u1ρ + 00
u2
u2
u20 ρ1u1ρ + 0
closed loop traveling wave engine standing wave engine
but still many open question ⋆ Acoustic streaming in ldquonon-emptyrdquo resonators (additional cells )
⋆ Amount of heat-transport by 〈ρ0U2 + ρ1u1〉 ⋆ What about the dynamics of streaming establishment
[1] N Rott ZAMP 1974 [2] Bailliet et al JASA 2001 [3] Hamilton et al JASA 2003 [4] Thompsonet al JASA 2005 [5] DC Gedeon Cryocoolers 1997 [6] V Gusev et al JASA 2000 [7] desjouy etal JASA 2009 [8] Menguy et al JASA2000 [9] Moreau et al JASA 2008 [10] Daru et al WaveMotion 2013 [11] Reyt et al JASA 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 44 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
⋆ Thompson et al (JASA 2004) outer streaming at high Renl (Renl =p
MSh impact oftemperature distribution and fluid inertia⋆ Moreau et al (JASA 2007) inner and outer streaming at high Renl
insulatingfoam
water
Experimental setminusup of Thompson et al
LDV measurement window
laser beam
laser beam
laser beam
uncontrolled
insulated
isothermal
u (xr=0) Re =102 nl
u (xr=0) Re =202 nl
u (xr=0) Re =402 nl
2 nlu (L4r) Re =40
2 nlu (L2r) Re =40
2u (3L4r) Re =40nl
30 δ
ν
Re =3nl
Re =13nl
Re =13nl
Re =59nl
Re =78nl
Re =132nl
measurement window
Rott
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 45 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming in a closed-loop resonator (Desjouy et alJASA2009)
u (
ms
)2
|u |
(ms
)1
driven in phase
axial position
u (
ms
)2
|u |
(ms
)1
axial position
out of phaseπ2
Acoustic streaming in SW resonators with a stack (Moreau et al JASA 2009)
nllow Re andor stack close to a velocity antiminusnode
nlRe =4
high Re andor stack close to a velocity nodenl
velocity antinode
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 46 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Acoustic streaming
Acoustic streaming from the engineering standpoint
⋆ Acoustic streaming well-described only in simple devices at low amplitudes
=gt Empirical solutions to remove acoustic streaming
=gtldquojet pumpsrdquo1(makes ρ0u2 + ρ1u1 = 0) or membranes2(makes u2 = 0) to suppress Gedeonstreaming=gtldquotaperedrdquo tube13to diminish Rayleigh streaming in the TBT
taperedtube
∆p2 right∆p2 left
∆p2 left∆p2 right
steady flow
Jet Pump elastic membraneor
135
⋆ Open questions Heat transport by ρ0v2 + ρ1v1 Is acoustic streaming always undesirablefor TA engines
[1] Backhaus et al JASA 99 [2] Tijani et al JASA 2011 [3] Olson et al Cryogenics 1997
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 47 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Transition to turbulence extensively studied but in steady flows
rArr Only a few works in oscillating flows1minus8
Hot Wire and LDV measurements only in straight ducts without considering wallrsquos roughness should depend on 2 dimensionless numbers
UD
St = ωDU amp Re = UD
ν
vxvxvx
weakly turbulentlaminar conditionaly turbulentcenter of the duct
close to the wall
tω tωtω
HW measurements of vx as a function of the oscilla-ting amplitude and distance to the wall (from [3])
103
104
105
106
10minus2
10minus1
100
101
Re
St
transition determined by Sergeevtransition determined by Merkli et Thomannlaminar for Hino et alweakly turbulent for Hino et alconditionally turbulent for Hino et altransition determined by Hino et alweakly turbulent for Clamen and Mintonlaminar for Ohmi et alweakly turbulent for Ohmi et alturbulent for Ohmi et altransition determined by Kurzweg et allaminar for Eckmann et Grotbergconditionally turbulent for Eckmann et Grotbergtransition determined by Eckmann et Grotbergtransition determined by Zhao et Chengturbulent for Akhavan et allaminar for present workturbulent for present work
10minus2
10minus1
100
101
103 104 105 106
~minusReδ 280
Reδ minus~500
St
Re
Merkli et al Moreau et al
others
quas
iminusst
atio
narit
y
quasistationarity
loss of
Compressibility
And still many open questions Evaluation of subsequent losses Turbulence near T-junctions in coiled ducts etc
[1] Merkli amp Thomann JFM 1975 [2] Sergeev Fluid Dyn 1966 [3] Hino et al JFM 1976 [4] Ohmiet alJSME 1982 [5] Kurzweg et al Phys Fluid A 1989 [6] Eckman et alJFM 1991 [7] Zhao etal Int J Heat Fluid Flow 1996 [8] S Moreau Phd Poitiers 2006
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 48 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Turbulence
Evaluation of losses due to turbulence (DELTA-EC)
=gt modification of 〈wν〉 from the time-averaging of well-known results of fully developpedsteady flows
steady flows
∆p = fMLD
12 ρ0〈vx〉2
〈wνturb〉 = ∆p〈vx〉L =
ρ0fM〈vx 〉32D
oscillating flows (Re(t) = 〈vx (t)〉Dν )
〈wνturb〉 =ρ0fM |〈vx 〉|3
2D
fM(Re(t)) asymp fM +dfMdRe
(|Re(t)| minus Remax)
fric
tion
fact
or f
M
εro
ughn
ess
64Re
Re
laminar turbulent
where both fM and dRefM are evaluated at Remax from Moody chart knowing ǫ andRe (approach considered unsatisfactory by the authors themselves )
Avoiding turbulence from the engineering standpoint
polish internal surfaces of ducts
use of ldquoflow straightenersrdquo
flow straightener(mesh grids)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 49 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Preliminary conclusion
⋆ Linear theoryDesign tools are available but
- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport- they predict steady-state operation
⋆ Nonlinear processesNL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
Q in
Wout
Wout
Q in
experience
model
⋆ How to fit experiments and theory =gt adjust any poorly known parameter
⋆ How to dissociate the role of each NL process =gt study the transient regime
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 50 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 51 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Hysteresis6
compeacutetition de modes reacutegimes QP8
Switch onoff24
ldquoFishbone-like instabilityrdquo7
[1] Penelet et al Cryogenics 2002 [2] Swift JASA 1992 [3] Zhou et al Cryogenics 1998 [4] Peneletet al Phys Let A 2006 [5] Penelet et al Int J Heat Mass Trans 2012 [6] Chen et al Cryogenics1999 [7] Yu et al JASA 2010 [8] Unni et al N3L Workshop Munich 2013
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 52 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Pourquoi eacutetudier le transitoire
Moteur thermoacoustique ldquoquart drsquoonderdquo Geacuteneacuterateur thermo-acousto-eacutelectrique
Q in
Wout
T(x)
HHX
AHX
regenerator
TB
T
flow straighteners
mov
ing
mas
s
flexu
re b
earin
gs
mag
netic
mot
or
TA
C
rArr comprendre le plus simple pour mieux appreacutehender le ldquocompliqueacuterdquo
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 53 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
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Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Description of the TAO in the time domain
⋆ Find of a Green function and make use of the kirchhoff-Helmholtz integral theorem
Rx
part2ppartx2 minus 1
c20
part2ppartt2
= q(x t)
δκ
T0(x)
⋆ But even in the limit of a quasi-adiabatic interaction the inhomogeneous wave equation inthe time domain is not simple1
part2p
partt2minus part
partx
bdquo
c20
partp
partx
laquo
asymp minus 2c20radic
ν
R
2
6664
Cpartminus12
parttminus12
part2p
partx2
| z thermoviscous effects
+ (C + CT )1
T0
dT0
dx
partminus12
parttminus12
bdquopartp
partx
laquo
| z thermoacoustic amplification
3
7775
where the fractional derivatives
partminus12f (x t)
parttminus12=
1radic
π
Z t
minusinfin
f (x τ )radic
t minus τdτ
have the meaning of memory effects which are the essence of the thermoacoustic phenomena1 ⋆ Moreover T0(x) is controlled by p (and vice versa )
rArr letrsquos try a simpler approach based on the quasi-steady state assumption(seek for a slowly varying harmonic solution )
[1] N Sugimoto J Fluid Mech 658 89-116 2010
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 54 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
TA wave amplification
⋆ Description of wave amplitude growthattenuation For a given temperature distribution T0(x) through the device each element is described byits T-matrix12
bdquop(L)u(L)
laquo
= M3 times M2 times M1 timesbdquo
p(0)u(0)
laquo
=
raquoMpp Mpu
Mup Muu
ndash
timesbdquo
p(0)u(0)
laquo
T 8T 8
1 2 3
T (x)0
TH
xC xH L0
which combined with p(0) = 0 and u(L) = 0 leads to a characteristic equation2
=gt Muu = 0
Allow the angular frequency ω to be complex2 and find ω = Ω + iǫ so that
Muu (ω T0(x) ) = 0
Quasi-steady state assumption (ie ǫ ≪ Ω)
dP
dtasymp ǫ [T0(x)] P
where P = |p(L)| and ǫ = image (ω) is the amplification rate (ǫ gt 0 amplification ǫ lt 0 attenuation)
[1] Penelet et al Acust Acta Acust 2005 [2] Guedra et al Acust Acta Acust 2012
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 55 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Unsteady Heat Transfer
⋆ Unsteady heat transfer1(simplified and summarized)
xC
xH
8T =Twall
Heating Q(t)
thermoacoustic heat transport x
8
Heat exchange with walls
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉)
Nomenclatureρf Cf (resp ρsCs ) volumetric heat capacity of fluid (resp of stack)λf (resp λs ) thermal conductivity of fluid (resp of stack)
[1] M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 56 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Summary
⋆ Modeling of the transient regime summaryEquations describing unsteady heat transfer (including thermoacoustic heat pumping)
x isin [0 xc ] cup [xh L]partT0partt = 1
ρf Cf
partpartx
ldquo
λfpartT0partx
rdquo
minus T0minusTinfinτf
x isin [xc xh]partT0partt = 1
ρsCspart
partx
ldquo
λspartT0partx
rdquo
minus T0minusTinfinτs
minus 1ρsCs
partϕacpartx
x = xc λfpartT0partx
˛˛˛xminusc
= λspartT0partx
˛˛˛x+c
minus ϕac (x+c )
x = xh λspartT0partx
˛˛˛xminush
= λfpartT0partx
˛˛˛x+h
+ ϕac (xminush
) + Q(t)πR2
x isin [xc xh] ϕac = 12
ρf T0πR2 real (〈s vlowast
x 〉) = f (P2 x)
coupled to that describing wave amplification
dP
dt= ǫ [T0(x)] P
=gt Nonlinear set of differential and partial differential equations solved using a finitedifference scheme
⋆ Main assumptions The variations of T0(x) are negligible at the time-scale of a few acoustic periods ǫ ≪ Ω T0 is uniform through a cross-sectional area
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 57 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (1)
The device
L = 05m length of the ductLs = 005m length of the stack
rs = 09mm the diameter of a pored the distance stack-rigid plug (variable)
Qin heating powert = 0 time at which heating is switched on
Comparison theoryexperiments
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
0 50 100 150 200 250 300minus15
minus1
minus05
0
05
1
15
time (s)
P (
kPa)
(a)
(b)
(a) (d=20cmQin=233 W)(d=20cmQin=186 W)
(b) (d=14cmQin=233 W)(d=20cmQin=233W)
M Guedra et al J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 58 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Comparison with experiments (2)
100 600 700 800 900 1000minus1
minus05
0
05
1
time (s)
P (
kPa)
200 210 220 230 240 250 260 270 280 290 300minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 100 200 300 400 500 600 700 800 900 100020
30
40
50
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 233 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
200 400 600 1000 1200 1400minus1
minus05
0
05
1
time (s)
P (
kPa)
400 420 440 460 480 500 520 540 560 580 600minus1
minus05
0
05
1
time (s)
P (
kPa)
(a)
50
100
150
200
T1 (
deg C
)
100
200
300
T2 (
degC)
0 500 1000 150020
30
40
50
60
time (s)
T3 (
deg C
)
(b)
(c)
(d)
d=26 cm Qin = 262 WSolid line=model dashed line=experimentsT1 temperature in the middle of the stackT2 temperature at hot side of the stack
T3 temperature 25 cm far from the stack
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 59 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Interfeacuteromeacutetrie
⋆ Pheacutenomegravene responsable du deacuteclenchement-arrecirct
rArr transport de chaleur acoustiquement induit avec temps caracteacuteristique τ 6= τampl
rArr vent acoustique pompage thermoacoustique effets de bords
⋆ Moyen expeacuterimentaux envisageables PIV LDV fil chaudfroid interfeacuteromeacutetrieoptique
CameraSpeed High
Tri CCCCamera
laser 660nm
microphone
laser 532nmM0
M2
dichroic plate
5050 cube
M1
L1 L2
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 60 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Principe de la mesure
milieu reference n8
verre nv
∆ϕAcces a y
L(y)
faisceau objet
faisceau reference
milieu drsquoetude n(xyzt)
camera CCD
z
λ∆ϕref (x y)
2π=R
L
h
nref (x y z) minus ninfini
dz + 2R
Lv[nv (x y z) minus ninfin] dz
λ∆ϕ(x y t)
2π=R
L [n(x y z t) minus ninfin] dz + 2R
Lv[nv (x y z) minus ninfin] dz
=rArr λ[∆ϕ(xyt)minus∆ϕref (xy)]2π =
R
L
h
n(x y z t) minus nref (x y z)i
dz
rArr accegraves agrave 〈n〉 = (1L)R
ndz puis 〈ρ〉 = (1L)R
ρdz via la relation de Gladstone-Dalesect
r = 23
1ρ(xyt) (n(x y t) minus 1) r = 01516m3kgminus1mminus1 pour de lrsquoairpara agrave 288 K et λ = 607 nm
sectNB si P = P0 + pac connue alors accegraves agrave T = T0 + τac (P0 = ρ0RT0 amp propa lineacuteaire)
paraNB2 r = r(λ T) rArr utilisation de deux longueurs drsquoonde
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 61 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Monteacutee en teacutempeacuterature
Avant les auto-oscillations P0 = Patm =gt T0(t) directement deacuteduit de ρ0(t)
Exemple drsquointerfeacuterogramme avec deacutefi-nition de zones drsquointeacuterecirct
Tempeacuterature en zone 1 lors du chauffage (initieacuteagrave t = 0) pour xs = 24cm et Qin = 251W
pas drsquoeffet de chromatisme
mesure de T = T0(t) + τac assureacutement fausse apregraves le deacuteclenchement
mais peut-on supposer que P = P0 + pmic cos(k0xs ) cos(ωt) (ie pas drsquoeffets de bordssur P) pour calculer T0 et τac au delagrave du seuil Dans le doute eacutevitons de le faire
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 62 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 77 s (fe = 1kHz)dureacutee=4 s
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacutee lors du deacuteclen-chement
masse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12
d(2dac )
()8 28 48 68 88 108 128 148 168 188 208 228
ρmaxρ0 () 75 96 121 129 108 103 80 55 31 17 12 09
alors mecircme que Pmax cos(k0xs)P0 asymp 07
dac asymp Pmax
ωc0sin(k0xs ) asymp 14mm
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 63 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple analyse spectrale
⋆ Analyse spectrale des signaux de masse volumique instantaneacutee
Spectres de ρ dans les zones 3 6 9 et 12 (de hauten bas)
Amplitude des harmoniques en fonction de la zonedrsquoobservation
rArr Effets de bords ldquothermiquesrdquolowastlowast
polytropic process adiabatic process
Passage abrupte drsquoune zone polytropiquevers une zone adiabatique
Mesures par fil froid des fluctuation de tempeacuteratures aux extreacute-miteacutes drsquoun stack drsquoapregraves [3]
lowastlowast[1] Gusev et al JASA1092001 [2]Marx et al JASA 118 2005 [3] Berson et al IJHMT 54 2011
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 64 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Deacuteclenchement simple moyenne glissante
⋆ moyenne glissante ρ = (1T )R
ρdt (T asymp 20 peacuteriode acoustiques)
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
ρ (k
gm
minus3 )
0 05 1 15 2 25 3 35 4070809
11112
temps (s)
ρ (k
gm
minus3 )
ρ dans les zones 3 6 9 et 12 ρ dans les zones 3 6 9 et 12
rArr variations significatives de ρ au cours du transitoire (jusqursquoagrave asymp 007ρ0 en zone 4)
⋆ Explication possibles
Reacutechauffement localiseacute (dissipation visqueusevent acoustique )
Pertes de charges singuliegraveres ρ = f (T P)
Images PIV (vorticiteacute) en bout de stackdrsquoapregraves [berson et al JASA EL122 2007]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 65 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
xs = 24 cm Qin = 194W deacutebut acqui cameacutera rapide agrave t asymp 2183 s (fe = 1kHz)dureacutee=16 s
200 210 220 230 240 250 260 270 280minus800
minus600
minus400
minus200
0
200
400
600
800
Pre
ssio
n ac
oust
ique
(P
a)
Temps (s)
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
ρ (k
gm
minus3 )
0 2 4 6 8 10 12 14 1608
1
12
temps (s)
ρ (k
gm
minus3 )
pression acoustique instantaneacuteemasse volumique instantaneacutee (〈ρ〉 = 〈ρ0〉 + 〈ρac〉) dansles zones 3 (xs minusx asymp 13mm) 6 (asymp 3mm) 9 (asymp 47mm)et 12 (asymp 64mm) de haut en bas
Quelques estimations (au plus fort des auto-oscillations )
zone 1 2 3 4 5 6 7 8 9 10 11 12d(2dac ) () 8 28 48 68 88 108 128 148 168 188 208 228ρmaxρ0 () 45 66 79 96 91 87 7 46 29 16 08 06
alors mecircme que Pmax cos(k0xs)P0 asymp 07
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 66 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
ρ m (
kgm
minus3 )
0 2 4 6 8 10 12 14 16
1
11
12
temps (s)
ρ m (
kgm
minus3 )
moyenne glissante ρm dans les zones 369et 12
Quelques estimations grossiegraveres (∆ρmρm = 2 lowast (ρmmax minus ρmmin)(ρmmax + ρmmin))
zone 1 2 3 4 5 6 7 8 9 10 11 12∆ρmρm () 22 54 72 78 92 87 68 41 27 17 18 16
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 67 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Bilan
The transient regime of TA Oscillators the big deal
What is the mechanism responsible for the switch onoff process
1D assumption for both T and p
Entrance effects impacting T
Acoustic streaming Everything together
and how can we claim understanding TA engines if we cannot reproduce such an effect
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 68 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 69 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
The TAO as an interesting dynamical system
The ThermoAcoustic Oscillator as an interesting dynamical system
⋆ An autonomous oscillator driven by heat
which is almost out of control above threshold
and which exhibits complicated dynamical behaviors (overshoot ldquointegrate andfire regimerdquo )
⋆ Investigate the nonlinear coupling between a TAO and an external sound sourcewith two objectives
revisit universal aspects of synchronization1 phenomena in the frame ofthermoacoustics (notably for teaching purpose)
investigate the active control of TA engine to ր their efficiency
[1] A Pikovsky M Rosenblum J Kurths laquo Synchronization A Universal Concept in Nonlinear Science raquo
Cambridge University Press NY 2001
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 70 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
A movie made at Lancaster University (UK) which can be easily found on youtube
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 71 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchonization Phenomena
Synchronization phenomena
⋆ First observations Huygens (1665 laquo sympathy of two pendulum clocks raquo)
Mg
coupling beam
coupling beam
f 1 f 2 1f f 2
(a) (b)
Mg2tα ( )
1tα ( )
OR
⋆ Use or observation of synchronization phenomena are nowadays abundant biology medicine(singing crickets circadian rythm cardiac pacemaker ) electronics engineering(triode generators for radio communications ) mechanics(clocks organ pipes ) physicschemistry(lasers BndashZ reaction ) social life(clapping audience)
⋆ An example in the field of acoustics synchronization of organpipes
Former study by Lord Rayleigh mutual synchronizationof two organ pipes and the quenching effect (oscillationdeath)
but even recent studies12
Sketch of the exp by Abel et al2
[1] Abel et al J Acoust Soc Am 119 2467 2006 [2] Abel et al Phys Rev Let 103 114301 2009
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 72 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization of a TAO by a loudspeaker experiment
Thermoacoustic oscillator
stack
loudspeaker enclosure
(a)
BampK Nexus 2692C
conditioning amplifier
to data acquisition cardNI PCI 6143
amplifier
power Frequency generator
Agilent 33120 A
Agilent 54624 A
Oscilloscope Audio spectrum analyzer
Stanford Research SR785
TT
L S
igna
l
NI BNC 2110
DC Power SupplyMCP Lab electronic
M10minusTPminus305minusC
connector block
Nichrome wire
Loudspeaker enclosure
ds d
(c)
(a) Photograph of the complete experimental set-up (c) Sketch of the experimental set-up
[ Penelet amp Biwa Am Journ Phys 2013]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 73 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization phase locking phase trapping beating death
Synchronization by a sound source experimental protocol
⋆ Experimental protocol Fix heat power Q=226 W (Q gt Qonset ) Fix both d and ds
Proceed to measurementsdaggerdagger by varying the driving frequencyf and the loudspeaker voltage Urms
Q=226 W
sd d
⋆ Signal processing Make both FFT and Hilbert transforms of p(t) and U(t) Quantities of interest for data analysis - Frequency spectra p(f ) and U(f )- Amplitude modulation Ap(t)- Instantaneous phase difference Ψ(t) = Φp(t) minus ΦU(t)
⋆ Different possible states(PS) (IPL) (QP) (BD)
Perfect Synchronization Imperfect Phase Locking QuasiPeriodicity Beating Death
fTAO = f (or fTAO = fn ) fTAO 6= f (or fTAO 6= f
n ) fTAO 6= f (or fTAO 6= fn )
Ap (t) = cte Ap (t) 6= cte Ap (t) 6= cte Ap (t) = cte
Ψ(t) = cte Ψ(t) 6= cte but bounded Ψ(t) 6= cte (unbounded)
daggerdaggerEach set of measurements sim 12-24 hours of total duration
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 74 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization the obtained Arnolrsquod Tongues
⋆ Stability diagram as a function of Uloudspeaker and f for d=1 mm ds=8 cm and Q=226 W
50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
1
2
3
4
5
6
7
8
9
Urm
s (
V)
frequency (Hz)
IPL
PS
QP
BD
time (s) time (s) time (s)
Stability diagram
Transition QP=gtBD
Transition PS=gtIPL=gtQP
synchronization 31
Quenching
synchronization 21
synchronisation 11
Some universal aspects of synchronization are retrieved by this experimentDaggerDagger
Some aspects are intrinsic to the TAO itself =gt why are the Arnold tongues asymetric =gt why does quenching occurs around f asymp fTAO2
DaggerDaggerOnly the 2nd experimental evidence of the phase trapping phenomenon
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 75 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
External forcing of relaxational oscillations
Other phenomena related to the external forcing of a TAO
⋆ What if the TAO is operated as a relaxa-tional oscillator
s
Q=245 W
d=4 cmd =25 cmLoudpseaker off =gt ldquointegrate and fire regi-merdquo with fTAO asymp 1771 Hz
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
ON OFF
Urms
=0V
Urms
=4mV
Urms
=10mV
⋆ Experimental procedure1- Settle f to 1769 Hz (6= fTAO)2- Switch the loudspeaker on at t=100 s3- Switch the loudspeaker off at t=150 s4- Repeat steps 1-3 with another driving vol-tage
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
0 25 50 75 100 125 150 175 200 225 250 275minus1000
minus500
0
500
1000p(
t) (
Pa)
time (s)
Urms
=105mV
Urms
=175mV
Urms
=380mV
interpretation =gt Simply quenching or more complicated processes =gt does external forcing lead to the transformation of a bistable regime (with periodic switch
between two stable states) into a stable one
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 76 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization mutuelle de 2 TAO
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 77 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the idea
And what about forcing a TAO to increase its efficiency
Basic ideas- Above threshold (and at fixed heat input Q0) TA enginesare ldquoout of controlrdquo- Is there a possibility to ր the efficiency of a TA engine withauxiliary acoustic sources A proof of concept study performed succesfully1
- Closed-loop stack based TA engine- Active control by 2 sources controlled in both amplitude andphase
thermoacoustic core
Loudspeakers
A new proof of concept study on a larger TA engine2
- ThermoAcoustic Stirling electricity generator- Active control by 1 source and a feedback loop
[1] C Desjouy et al J Appl Phys 2010 [2] C Olivier et al accepted for publication in J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 78 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the device
⋆ The device (a ldquoStirling typerdquo Thermo-acousto-electric engine working with 5 bars air)
CHX1
CHX2
Regenerator
HHX (Qh)
TBT
Coupling Duct
Alternator (Wel)
0 20 cm
Auxiliary source S (WLS)
Microphone micro
φ
G
TAC
TAC
Auxiliary source Acoustominuselectric conversion(electrodynamic alternator)
⋆ The control parameters
The heat input Qh The phase shift φ
The power supplied to the auxiliary source WLS (controlled via the voltage gain G)⋆ The output
Feedback loop switched off
The output power Wel0
The efficiency η0 = Wel0Qh
The temperature difference ∆T0
as a function of Qh
Feedback loop switched on
Wel and ∆Wel = Wel minus Wel0
η = Wel ldquo
Qh + WLS
rdquo
and ηη0
∆T and ∆T minus ∆T0
as a function of Qh φ and G
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 79 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(1)
Feedback loop switched off (G=0)
η0(Qh) (bull) ∆T0(Qh) ()
60 80 100 120 140 1600
02
04
06
Qh [W]
η0[
]
60 80 100 120 140 160400
425
450
475
∆T0[K
]
Feedback loop switched on (G 6= 0)
Qh=75 W ηη0(G φ)
0 45 90 135 180 225 270 315 3600
05
1
φ [deg]
ηη0
G = 22
G = 46
G = 10
rArr The electroacoustic feedback loop impacts the operation of the TA engine
Possibility to ր η (ie to have η gt η0)
Here φopt asymp 270
Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 80 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization of a TAO by a loudspeaker experiment
Thermoacoustic oscillator
stack
loudspeaker enclosure
(a)
BampK Nexus 2692C
conditioning amplifier
to data acquisition cardNI PCI 6143
amplifier
power Frequency generator
Agilent 33120 A
Agilent 54624 A
Oscilloscope Audio spectrum analyzer
Stanford Research SR785
TT
L S
igna
l
NI BNC 2110
DC Power SupplyMCP Lab electronic
M10minusTPminus305minusC
connector block
Nichrome wire
Loudspeaker enclosure
ds d
(c)
(a) Photograph of the complete experimental set-up (c) Sketch of the experimental set-up
[ Penelet amp Biwa Am Journ Phys 2013]
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 73 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization phase locking phase trapping beating death
Synchronization by a sound source experimental protocol
⋆ Experimental protocol Fix heat power Q=226 W (Q gt Qonset ) Fix both d and ds
Proceed to measurementsdaggerdagger by varying the driving frequencyf and the loudspeaker voltage Urms
Q=226 W
sd d
⋆ Signal processing Make both FFT and Hilbert transforms of p(t) and U(t) Quantities of interest for data analysis - Frequency spectra p(f ) and U(f )- Amplitude modulation Ap(t)- Instantaneous phase difference Ψ(t) = Φp(t) minus ΦU(t)
⋆ Different possible states(PS) (IPL) (QP) (BD)
Perfect Synchronization Imperfect Phase Locking QuasiPeriodicity Beating Death
fTAO = f (or fTAO = fn ) fTAO 6= f (or fTAO 6= f
n ) fTAO 6= f (or fTAO 6= fn )
Ap (t) = cte Ap (t) 6= cte Ap (t) 6= cte Ap (t) = cte
Ψ(t) = cte Ψ(t) 6= cte but bounded Ψ(t) 6= cte (unbounded)
daggerdaggerEach set of measurements sim 12-24 hours of total duration
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 74 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization the obtained Arnolrsquod Tongues
⋆ Stability diagram as a function of Uloudspeaker and f for d=1 mm ds=8 cm and Q=226 W
50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
1
2
3
4
5
6
7
8
9
Urm
s (
V)
frequency (Hz)
IPL
PS
QP
BD
time (s) time (s) time (s)
Stability diagram
Transition QP=gtBD
Transition PS=gtIPL=gtQP
synchronization 31
Quenching
synchronization 21
synchronisation 11
Some universal aspects of synchronization are retrieved by this experimentDaggerDagger
Some aspects are intrinsic to the TAO itself =gt why are the Arnold tongues asymetric =gt why does quenching occurs around f asymp fTAO2
DaggerDaggerOnly the 2nd experimental evidence of the phase trapping phenomenon
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 75 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
External forcing of relaxational oscillations
Other phenomena related to the external forcing of a TAO
⋆ What if the TAO is operated as a relaxa-tional oscillator
s
Q=245 W
d=4 cmd =25 cmLoudpseaker off =gt ldquointegrate and fire regi-merdquo with fTAO asymp 1771 Hz
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
ON OFF
Urms
=0V
Urms
=4mV
Urms
=10mV
⋆ Experimental procedure1- Settle f to 1769 Hz (6= fTAO)2- Switch the loudspeaker on at t=100 s3- Switch the loudspeaker off at t=150 s4- Repeat steps 1-3 with another driving vol-tage
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
0 25 50 75 100 125 150 175 200 225 250 275minus1000
minus500
0
500
1000p(
t) (
Pa)
time (s)
Urms
=105mV
Urms
=175mV
Urms
=380mV
interpretation =gt Simply quenching or more complicated processes =gt does external forcing lead to the transformation of a bistable regime (with periodic switch
between two stable states) into a stable one
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 76 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization mutuelle de 2 TAO
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 77 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the idea
And what about forcing a TAO to increase its efficiency
Basic ideas- Above threshold (and at fixed heat input Q0) TA enginesare ldquoout of controlrdquo- Is there a possibility to ր the efficiency of a TA engine withauxiliary acoustic sources A proof of concept study performed succesfully1
- Closed-loop stack based TA engine- Active control by 2 sources controlled in both amplitude andphase
thermoacoustic core
Loudspeakers
A new proof of concept study on a larger TA engine2
- ThermoAcoustic Stirling electricity generator- Active control by 1 source and a feedback loop
[1] C Desjouy et al J Appl Phys 2010 [2] C Olivier et al accepted for publication in J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 78 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the device
⋆ The device (a ldquoStirling typerdquo Thermo-acousto-electric engine working with 5 bars air)
CHX1
CHX2
Regenerator
HHX (Qh)
TBT
Coupling Duct
Alternator (Wel)
0 20 cm
Auxiliary source S (WLS)
Microphone micro
φ
G
TAC
TAC
Auxiliary source Acoustominuselectric conversion(electrodynamic alternator)
⋆ The control parameters
The heat input Qh The phase shift φ
The power supplied to the auxiliary source WLS (controlled via the voltage gain G)⋆ The output
Feedback loop switched off
The output power Wel0
The efficiency η0 = Wel0Qh
The temperature difference ∆T0
as a function of Qh
Feedback loop switched on
Wel and ∆Wel = Wel minus Wel0
η = Wel ldquo
Qh + WLS
rdquo
and ηη0
∆T and ∆T minus ∆T0
as a function of Qh φ and G
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 79 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(1)
Feedback loop switched off (G=0)
η0(Qh) (bull) ∆T0(Qh) ()
60 80 100 120 140 1600
02
04
06
Qh [W]
η0[
]
60 80 100 120 140 160400
425
450
475
∆T0[K
]
Feedback loop switched on (G 6= 0)
Qh=75 W ηη0(G φ)
0 45 90 135 180 225 270 315 3600
05
1
φ [deg]
ηη0
G = 22
G = 46
G = 10
rArr The electroacoustic feedback loop impacts the operation of the TA engine
Possibility to ր η (ie to have η gt η0)
Here φopt asymp 270
Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 80 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization phase locking phase trapping beating death
Synchronization by a sound source experimental protocol
⋆ Experimental protocol Fix heat power Q=226 W (Q gt Qonset ) Fix both d and ds
Proceed to measurementsdaggerdagger by varying the driving frequencyf and the loudspeaker voltage Urms
Q=226 W
sd d
⋆ Signal processing Make both FFT and Hilbert transforms of p(t) and U(t) Quantities of interest for data analysis - Frequency spectra p(f ) and U(f )- Amplitude modulation Ap(t)- Instantaneous phase difference Ψ(t) = Φp(t) minus ΦU(t)
⋆ Different possible states(PS) (IPL) (QP) (BD)
Perfect Synchronization Imperfect Phase Locking QuasiPeriodicity Beating Death
fTAO = f (or fTAO = fn ) fTAO 6= f (or fTAO 6= f
n ) fTAO 6= f (or fTAO 6= fn )
Ap (t) = cte Ap (t) 6= cte Ap (t) 6= cte Ap (t) = cte
Ψ(t) = cte Ψ(t) 6= cte but bounded Ψ(t) 6= cte (unbounded)
daggerdaggerEach set of measurements sim 12-24 hours of total duration
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 74 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization the obtained Arnolrsquod Tongues
⋆ Stability diagram as a function of Uloudspeaker and f for d=1 mm ds=8 cm and Q=226 W
50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
1
2
3
4
5
6
7
8
9
Urm
s (
V)
frequency (Hz)
IPL
PS
QP
BD
time (s) time (s) time (s)
Stability diagram
Transition QP=gtBD
Transition PS=gtIPL=gtQP
synchronization 31
Quenching
synchronization 21
synchronisation 11
Some universal aspects of synchronization are retrieved by this experimentDaggerDagger
Some aspects are intrinsic to the TAO itself =gt why are the Arnold tongues asymetric =gt why does quenching occurs around f asymp fTAO2
DaggerDaggerOnly the 2nd experimental evidence of the phase trapping phenomenon
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 75 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
External forcing of relaxational oscillations
Other phenomena related to the external forcing of a TAO
⋆ What if the TAO is operated as a relaxa-tional oscillator
s
Q=245 W
d=4 cmd =25 cmLoudpseaker off =gt ldquointegrate and fire regi-merdquo with fTAO asymp 1771 Hz
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
ON OFF
Urms
=0V
Urms
=4mV
Urms
=10mV
⋆ Experimental procedure1- Settle f to 1769 Hz (6= fTAO)2- Switch the loudspeaker on at t=100 s3- Switch the loudspeaker off at t=150 s4- Repeat steps 1-3 with another driving vol-tage
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
0 25 50 75 100 125 150 175 200 225 250 275minus1000
minus500
0
500
1000p(
t) (
Pa)
time (s)
Urms
=105mV
Urms
=175mV
Urms
=380mV
interpretation =gt Simply quenching or more complicated processes =gt does external forcing lead to the transformation of a bistable regime (with periodic switch
between two stable states) into a stable one
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 76 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization mutuelle de 2 TAO
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 77 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the idea
And what about forcing a TAO to increase its efficiency
Basic ideas- Above threshold (and at fixed heat input Q0) TA enginesare ldquoout of controlrdquo- Is there a possibility to ր the efficiency of a TA engine withauxiliary acoustic sources A proof of concept study performed succesfully1
- Closed-loop stack based TA engine- Active control by 2 sources controlled in both amplitude andphase
thermoacoustic core
Loudspeakers
A new proof of concept study on a larger TA engine2
- ThermoAcoustic Stirling electricity generator- Active control by 1 source and a feedback loop
[1] C Desjouy et al J Appl Phys 2010 [2] C Olivier et al accepted for publication in J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 78 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the device
⋆ The device (a ldquoStirling typerdquo Thermo-acousto-electric engine working with 5 bars air)
CHX1
CHX2
Regenerator
HHX (Qh)
TBT
Coupling Duct
Alternator (Wel)
0 20 cm
Auxiliary source S (WLS)
Microphone micro
φ
G
TAC
TAC
Auxiliary source Acoustominuselectric conversion(electrodynamic alternator)
⋆ The control parameters
The heat input Qh The phase shift φ
The power supplied to the auxiliary source WLS (controlled via the voltage gain G)⋆ The output
Feedback loop switched off
The output power Wel0
The efficiency η0 = Wel0Qh
The temperature difference ∆T0
as a function of Qh
Feedback loop switched on
Wel and ∆Wel = Wel minus Wel0
η = Wel ldquo
Qh + WLS
rdquo
and ηη0
∆T and ∆T minus ∆T0
as a function of Qh φ and G
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 79 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(1)
Feedback loop switched off (G=0)
η0(Qh) (bull) ∆T0(Qh) ()
60 80 100 120 140 1600
02
04
06
Qh [W]
η0[
]
60 80 100 120 140 160400
425
450
475
∆T0[K
]
Feedback loop switched on (G 6= 0)
Qh=75 W ηη0(G φ)
0 45 90 135 180 225 270 315 3600
05
1
φ [deg]
ηη0
G = 22
G = 46
G = 10
rArr The electroacoustic feedback loop impacts the operation of the TA engine
Possibility to ր η (ie to have η gt η0)
Here φopt asymp 270
Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 80 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization the obtained Arnolrsquod Tongues
⋆ Stability diagram as a function of Uloudspeaker and f for d=1 mm ds=8 cm and Q=226 W
50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
1
2
3
4
5
6
7
8
9
Urm
s (
V)
frequency (Hz)
IPL
PS
QP
BD
time (s) time (s) time (s)
Stability diagram
Transition QP=gtBD
Transition PS=gtIPL=gtQP
synchronization 31
Quenching
synchronization 21
synchronisation 11
Some universal aspects of synchronization are retrieved by this experimentDaggerDagger
Some aspects are intrinsic to the TAO itself =gt why are the Arnold tongues asymetric =gt why does quenching occurs around f asymp fTAO2
DaggerDaggerOnly the 2nd experimental evidence of the phase trapping phenomenon
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 75 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
External forcing of relaxational oscillations
Other phenomena related to the external forcing of a TAO
⋆ What if the TAO is operated as a relaxa-tional oscillator
s
Q=245 W
d=4 cmd =25 cmLoudpseaker off =gt ldquointegrate and fire regi-merdquo with fTAO asymp 1771 Hz
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
ON OFF
Urms
=0V
Urms
=4mV
Urms
=10mV
⋆ Experimental procedure1- Settle f to 1769 Hz (6= fTAO)2- Switch the loudspeaker on at t=100 s3- Switch the loudspeaker off at t=150 s4- Repeat steps 1-3 with another driving vol-tage
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
0 25 50 75 100 125 150 175 200 225 250 275minus1000
minus500
0
500
1000p(
t) (
Pa)
time (s)
Urms
=105mV
Urms
=175mV
Urms
=380mV
interpretation =gt Simply quenching or more complicated processes =gt does external forcing lead to the transformation of a bistable regime (with periodic switch
between two stable states) into a stable one
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 76 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization mutuelle de 2 TAO
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 77 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the idea
And what about forcing a TAO to increase its efficiency
Basic ideas- Above threshold (and at fixed heat input Q0) TA enginesare ldquoout of controlrdquo- Is there a possibility to ր the efficiency of a TA engine withauxiliary acoustic sources A proof of concept study performed succesfully1
- Closed-loop stack based TA engine- Active control by 2 sources controlled in both amplitude andphase
thermoacoustic core
Loudspeakers
A new proof of concept study on a larger TA engine2
- ThermoAcoustic Stirling electricity generator- Active control by 1 source and a feedback loop
[1] C Desjouy et al J Appl Phys 2010 [2] C Olivier et al accepted for publication in J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 78 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the device
⋆ The device (a ldquoStirling typerdquo Thermo-acousto-electric engine working with 5 bars air)
CHX1
CHX2
Regenerator
HHX (Qh)
TBT
Coupling Duct
Alternator (Wel)
0 20 cm
Auxiliary source S (WLS)
Microphone micro
φ
G
TAC
TAC
Auxiliary source Acoustominuselectric conversion(electrodynamic alternator)
⋆ The control parameters
The heat input Qh The phase shift φ
The power supplied to the auxiliary source WLS (controlled via the voltage gain G)⋆ The output
Feedback loop switched off
The output power Wel0
The efficiency η0 = Wel0Qh
The temperature difference ∆T0
as a function of Qh
Feedback loop switched on
Wel and ∆Wel = Wel minus Wel0
η = Wel ldquo
Qh + WLS
rdquo
and ηη0
∆T and ∆T minus ∆T0
as a function of Qh φ and G
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 79 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(1)
Feedback loop switched off (G=0)
η0(Qh) (bull) ∆T0(Qh) ()
60 80 100 120 140 1600
02
04
06
Qh [W]
η0[
]
60 80 100 120 140 160400
425
450
475
∆T0[K
]
Feedback loop switched on (G 6= 0)
Qh=75 W ηη0(G φ)
0 45 90 135 180 225 270 315 3600
05
1
φ [deg]
ηη0
G = 22
G = 46
G = 10
rArr The electroacoustic feedback loop impacts the operation of the TA engine
Possibility to ր η (ie to have η gt η0)
Here φopt asymp 270
Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 80 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
External forcing of relaxational oscillations
Other phenomena related to the external forcing of a TAO
⋆ What if the TAO is operated as a relaxa-tional oscillator
s
Q=245 W
d=4 cmd =25 cmLoudpseaker off =gt ldquointegrate and fire regi-merdquo with fTAO asymp 1771 Hz
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
ON OFF
Urms
=0V
Urms
=4mV
Urms
=10mV
⋆ Experimental procedure1- Settle f to 1769 Hz (6= fTAO)2- Switch the loudspeaker on at t=100 s3- Switch the loudspeaker off at t=150 s4- Repeat steps 1-3 with another driving vol-tage
minus1000
minus500
0
500
1000
p(t)
(P
a)
minus1000
minus500
0
500
1000
p(t)
(P
a)
0 25 50 75 100 125 150 175 200 225 250 275minus1000
minus500
0
500
1000p(
t) (
Pa)
time (s)
Urms
=105mV
Urms
=175mV
Urms
=380mV
interpretation =gt Simply quenching or more complicated processes =gt does external forcing lead to the transformation of a bistable regime (with periodic switch
between two stable states) into a stable one
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 76 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization mutuelle de 2 TAO
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 77 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the idea
And what about forcing a TAO to increase its efficiency
Basic ideas- Above threshold (and at fixed heat input Q0) TA enginesare ldquoout of controlrdquo- Is there a possibility to ր the efficiency of a TA engine withauxiliary acoustic sources A proof of concept study performed succesfully1
- Closed-loop stack based TA engine- Active control by 2 sources controlled in both amplitude andphase
thermoacoustic core
Loudspeakers
A new proof of concept study on a larger TA engine2
- ThermoAcoustic Stirling electricity generator- Active control by 1 source and a feedback loop
[1] C Desjouy et al J Appl Phys 2010 [2] C Olivier et al accepted for publication in J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 78 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the device
⋆ The device (a ldquoStirling typerdquo Thermo-acousto-electric engine working with 5 bars air)
CHX1
CHX2
Regenerator
HHX (Qh)
TBT
Coupling Duct
Alternator (Wel)
0 20 cm
Auxiliary source S (WLS)
Microphone micro
φ
G
TAC
TAC
Auxiliary source Acoustominuselectric conversion(electrodynamic alternator)
⋆ The control parameters
The heat input Qh The phase shift φ
The power supplied to the auxiliary source WLS (controlled via the voltage gain G)⋆ The output
Feedback loop switched off
The output power Wel0
The efficiency η0 = Wel0Qh
The temperature difference ∆T0
as a function of Qh
Feedback loop switched on
Wel and ∆Wel = Wel minus Wel0
η = Wel ldquo
Qh + WLS
rdquo
and ηη0
∆T and ∆T minus ∆T0
as a function of Qh φ and G
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 79 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(1)
Feedback loop switched off (G=0)
η0(Qh) (bull) ∆T0(Qh) ()
60 80 100 120 140 1600
02
04
06
Qh [W]
η0[
]
60 80 100 120 140 160400
425
450
475
∆T0[K
]
Feedback loop switched on (G 6= 0)
Qh=75 W ηη0(G φ)
0 45 90 135 180 225 270 315 3600
05
1
φ [deg]
ηη0
G = 22
G = 46
G = 10
rArr The electroacoustic feedback loop impacts the operation of the TA engine
Possibility to ր η (ie to have η gt η0)
Here φopt asymp 270
Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 80 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Synchronization mutuelle de 2 TAO
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 77 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the idea
And what about forcing a TAO to increase its efficiency
Basic ideas- Above threshold (and at fixed heat input Q0) TA enginesare ldquoout of controlrdquo- Is there a possibility to ր the efficiency of a TA engine withauxiliary acoustic sources A proof of concept study performed succesfully1
- Closed-loop stack based TA engine- Active control by 2 sources controlled in both amplitude andphase
thermoacoustic core
Loudspeakers
A new proof of concept study on a larger TA engine2
- ThermoAcoustic Stirling electricity generator- Active control by 1 source and a feedback loop
[1] C Desjouy et al J Appl Phys 2010 [2] C Olivier et al accepted for publication in J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 78 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the device
⋆ The device (a ldquoStirling typerdquo Thermo-acousto-electric engine working with 5 bars air)
CHX1
CHX2
Regenerator
HHX (Qh)
TBT
Coupling Duct
Alternator (Wel)
0 20 cm
Auxiliary source S (WLS)
Microphone micro
φ
G
TAC
TAC
Auxiliary source Acoustominuselectric conversion(electrodynamic alternator)
⋆ The control parameters
The heat input Qh The phase shift φ
The power supplied to the auxiliary source WLS (controlled via the voltage gain G)⋆ The output
Feedback loop switched off
The output power Wel0
The efficiency η0 = Wel0Qh
The temperature difference ∆T0
as a function of Qh
Feedback loop switched on
Wel and ∆Wel = Wel minus Wel0
η = Wel ldquo
Qh + WLS
rdquo
and ηη0
∆T and ∆T minus ∆T0
as a function of Qh φ and G
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 79 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(1)
Feedback loop switched off (G=0)
η0(Qh) (bull) ∆T0(Qh) ()
60 80 100 120 140 1600
02
04
06
Qh [W]
η0[
]
60 80 100 120 140 160400
425
450
475
∆T0[K
]
Feedback loop switched on (G 6= 0)
Qh=75 W ηη0(G φ)
0 45 90 135 180 225 270 315 3600
05
1
φ [deg]
ηη0
G = 22
G = 46
G = 10
rArr The electroacoustic feedback loop impacts the operation of the TA engine
Possibility to ր η (ie to have η gt η0)
Here φopt asymp 270
Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 80 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the idea
And what about forcing a TAO to increase its efficiency
Basic ideas- Above threshold (and at fixed heat input Q0) TA enginesare ldquoout of controlrdquo- Is there a possibility to ր the efficiency of a TA engine withauxiliary acoustic sources A proof of concept study performed succesfully1
- Closed-loop stack based TA engine- Active control by 2 sources controlled in both amplitude andphase
thermoacoustic core
Loudspeakers
A new proof of concept study on a larger TA engine2
- ThermoAcoustic Stirling electricity generator- Active control by 1 source and a feedback loop
[1] C Desjouy et al J Appl Phys 2010 [2] C Olivier et al accepted for publication in J Appl Phys 2014
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 78 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the device
⋆ The device (a ldquoStirling typerdquo Thermo-acousto-electric engine working with 5 bars air)
CHX1
CHX2
Regenerator
HHX (Qh)
TBT
Coupling Duct
Alternator (Wel)
0 20 cm
Auxiliary source S (WLS)
Microphone micro
φ
G
TAC
TAC
Auxiliary source Acoustominuselectric conversion(electrodynamic alternator)
⋆ The control parameters
The heat input Qh The phase shift φ
The power supplied to the auxiliary source WLS (controlled via the voltage gain G)⋆ The output
Feedback loop switched off
The output power Wel0
The efficiency η0 = Wel0Qh
The temperature difference ∆T0
as a function of Qh
Feedback loop switched on
Wel and ∆Wel = Wel minus Wel0
η = Wel ldquo
Qh + WLS
rdquo
and ηη0
∆T and ∆T minus ∆T0
as a function of Qh φ and G
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 79 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(1)
Feedback loop switched off (G=0)
η0(Qh) (bull) ∆T0(Qh) ()
60 80 100 120 140 1600
02
04
06
Qh [W]
η0[
]
60 80 100 120 140 160400
425
450
475
∆T0[K
]
Feedback loop switched on (G 6= 0)
Qh=75 W ηη0(G φ)
0 45 90 135 180 225 270 315 3600
05
1
φ [deg]
ηη0
G = 22
G = 46
G = 10
rArr The electroacoustic feedback loop impacts the operation of the TA engine
Possibility to ր η (ie to have η gt η0)
Here φopt asymp 270
Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 80 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the device
⋆ The device (a ldquoStirling typerdquo Thermo-acousto-electric engine working with 5 bars air)
CHX1
CHX2
Regenerator
HHX (Qh)
TBT
Coupling Duct
Alternator (Wel)
0 20 cm
Auxiliary source S (WLS)
Microphone micro
φ
G
TAC
TAC
Auxiliary source Acoustominuselectric conversion(electrodynamic alternator)
⋆ The control parameters
The heat input Qh The phase shift φ
The power supplied to the auxiliary source WLS (controlled via the voltage gain G)⋆ The output
Feedback loop switched off
The output power Wel0
The efficiency η0 = Wel0Qh
The temperature difference ∆T0
as a function of Qh
Feedback loop switched on
Wel and ∆Wel = Wel minus Wel0
η = Wel ldquo
Qh + WLS
rdquo
and ηη0
∆T and ∆T minus ∆T0
as a function of Qh φ and G
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 79 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(1)
Feedback loop switched off (G=0)
η0(Qh) (bull) ∆T0(Qh) ()
60 80 100 120 140 1600
02
04
06
Qh [W]
η0[
]
60 80 100 120 140 160400
425
450
475
∆T0[K
]
Feedback loop switched on (G 6= 0)
Qh=75 W ηη0(G φ)
0 45 90 135 180 225 270 315 3600
05
1
φ [deg]
ηη0
G = 22
G = 46
G = 10
rArr The electroacoustic feedback loop impacts the operation of the TA engine
Possibility to ր η (ie to have η gt η0)
Here φopt asymp 270
Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 80 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(1)
Feedback loop switched off (G=0)
η0(Qh) (bull) ∆T0(Qh) ()
60 80 100 120 140 1600
02
04
06
Qh [W]
η0[
]
60 80 100 120 140 160400
425
450
475
∆T0[K
]
Feedback loop switched on (G 6= 0)
Qh=75 W ηη0(G φ)
0 45 90 135 180 225 270 315 3600
05
1
φ [deg]
ηη0
G = 22
G = 46
G = 10
rArr The electroacoustic feedback loop impacts the operation of the TA engine
Possibility to ր η (ie to have η gt η0)
Here φopt asymp 270
Possibility to destroy TA oscillations (here for φ equiv 50 minus 60 and G ge 10)
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 80 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(2)
The variation of η(Φ Qh) ∆T (Φ Qh) with a fixed voltage gain G (here G = 32)
0 45 90 135 180 225 270 315 3600
05
1
(a)
ηη0
70W
71W
80W
0 45 90 135 180 225 270 315 360minus10
0
10
20
(b)
φ [deg]
∆Tminus
∆T0[K
]
rArr η gt η0 while ∆T gt ∆T0
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 81 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Active Control of a TAO the results(3)
The net output power gain ∆Wel compared to the power provided to the auxiliary sourceWLS with an assigned phase shift (here φ = 270)
0 05 1 15 210minus4
10minus3
10minus2
10minus1
100
gain G
WLS∆W
el[W
]
∆Wel () and WLS (bull) as a function of G for Qh = 70W
∆Wel () and WLS (N) as a function of G for Qh = 150W
rArr Possibility to have ∆Wel gt WLS (indisputable proof of concept )
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 82 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Plan
Preacuteambule de quelle thermoacoustique parle-t-on
Theacuteorie lineacuteaire de la thermoacoustique
Exemple le tube de Sondhauss
Effets de saturation non lineacuteaires
Dynamique des auto-oscillateurs thermoacoustiques
Forccedilage des auto-oscillateurs thermoacoustique
Conclusion
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 83 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
Conclusion
⋆ Conclusion of part 1-2 (linear theory design tools Sondhauss tube)
Design tools are available but- they are based on the linearization to 1st order of the governing equations- they are based on a 1-D description both acoustics and heat transport
⋆ Conclusion of part 3 (nonlinear processes)NL process Academic understanding Appropriate modeling Impact on TA Engines
TA Heat pumping significant
NL acoustics not much
Streaming significant
Edge effects significant
Turbulence
⋆ Conclusion of part 4 (dynamics of TA Oscillators) Even the simplest TAO exhibit complicated behaviors which are not very well unders-tood The study of the transient regime may provide a deeper physical insight on the me-chanisms responsible for sound saturation and may therefore provide new opportunities to increase the enginersquos efficiencies
⋆ Conclusion of part 5 (External forcing of TA oscillators) The forcing of a TAO allows studying synchronization phenomena But it may also be used to increase the efficiency of TA engines
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 84 85
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-
Avant-propos Theacuteorie lineacuteaire Sondhauss tube Effets Non lineacuteaires Dynamique des TAO Forccedilage des TAO Conclusion
RemerciementsLrsquoOR thermoacoustique du LAUM
Pierrick LOTTON Guillaume PENELET Gaeumllle POIGNAND Cocircme OLIVIER(DR CNRS) (MCF) (IR) (Doc)
Les anciens doctorants
Cyril DESJOUY Matthieu GUEDRA Flavio BANNWART
Autres collegravegues du LAUM
M BRUNEAU JP DALMONT J GILBERT V GUSEV M LECLERCQ P PICART
ou drsquoailleurs
T BIWA (Tohoku Univ Japan)
merci de mrsquoavoir eacutecouteacute
G Penelet Ecole theacutematique Acoustique Non Lineacuteaire et Milieux Complexes Oleacuteron 1-6 Juin 2014 85 85
- Preacuteambule de quelle thermoacoustique parle-t-on
-
- Quelle thermoacoustique
- Tomographie Thermoacoustique
- Haut-parleur TA
- Effet piston
- Instabiliteacutes de combustion
- Machines thermoacoustiques
- The Rayleigh criterium
- The TA Laser a prototypical example
- Auto-oscillateur thermo-acoustique contexte et probleacutematique
-
- Theacuteorie lineacuteaire de la thermoacoustique
-
- Basic principles
- Basic equations
- Le point de vue thermodynamique
- Sound amplification
- Heat transport by sound
- Amplification of a standing wave
- How to build an intrinsically reversible Thermoacoustic engine
- Examples
- Design of thermoacoustic engines
-
- Exemple le tube de Sondhauss
-
- Le systegraveme consideacutereacute
- Analogies eacutelectroacoustique
- Scheacutema eacutelectrique eacutequivalent
- Approximation par une DDE
- Reacutesolution numeacuterique
- Reacutesolution analytiquemeacutethode des temps multiples
- Reacutesolution analytiquemeacutethode des temps multiples (2)
- Saturation par pompage thermoacoustique
- Autres meacutecanismes de saturation
-
- Effets de saturation non lineacuteaires
-
- Thermoacoustic heat transport by sound
- Higher harmonics generation
- Higher harmonics generation shock waves
- Entrance effects
- Acoustic streaming
- Turbulence
- Preliminary conclusion
-
- Dynamique des auto-oscillateurs thermoacoustiques
-
- Pourquoi eacutetudier le transitoire
- Description of the TAO in the time domain
- ThermoAcoustic wave amplification
- Unsteady Heat Transfer
- Summary
- Comparison with experiments (1)
- Comparison with experiments (2)
- Interfeacuteromeacutetrie
- Principe de la mesure
- Monteacutee en teacutempeacuterature
- Deacuteclenchement simple masse volumique instantaneacutee
- Deacuteclenchement simple analyse spectrale
- Deacuteclenchement simple moyenne glissante
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement masse volumique instantaneacutee
- Cycle de deacuteclenchementsarrecirct premier deacuteclenchement moyenne glissante
- Bilan
-
- Forccedilage des auto-oscillateurs thermoacoustique
-
- The ThermoAcoustic Oscillator as an interesting dynamical system
- Synchonization Phenomena
- Synchronization of a TAo by a loudspeaker experiment
- Synchronization phase locking phase trapping beating death hellip
- Synchronization the obtained Arnold Tongues
- External forcing of relaxational oscillations
- Active control of a TAOthe idea
- Active control of a TAOthe device
- Active control of a TAO the results(1)
- Active control of a TAO the results(2)
- Active control of a TAO the results(3)
-
- Conclusion
-