Simulation numériquedes écoulements complexes

écoulement moyen et viscosité sous-maille

en collaboration avec

- Pierre Borgnat , Hatem Touil (LP, ENSL)- Adrien Cahuzac, Jérôme Boudet, Joëlle Caro

Liang Shao, Jean-Pierre Bertoglio (LMFA, ECL)- Guillaume Balarac (LEGI, Grenoble)- Federico Toschi (TU Eindhoven)

Inhomogénéité - phénoménologie

!

" = #u2

: cisaillement

turbulence homogène et isotropeturbulence cisaillée

écoulement moyen + fluctuations

tourbillon

l 3/23/1)(/ lll!

"" #$uteddy!

"1

sheart

l

!"!

'uL

Kolmogorov’s theory

!"#2'u$

taux de dissipation d’énergie

!

u = u + " u : décomposition de Reynolds

écoulement autour d’un cylindre àReD=47000 (régime turbulent sous-critique)

Simulation numérique desécoulements turbulents

goal= get a physically-sound numerical representation of the flow

-Direct Numerical Simulation (all scales of motion) is usually prohibitive

- alternatives = partial representation of the flow : - Reynolds-Averaged Navier-Stokes : mean flow

- Large-Eddy Simulation : large-scale dynamics of the flow

Does it make sense (from a physical viewpoint) ? What are the governing equations?

subgridgridgridt UU !" div)~(NS

~+=

Ssubgridsubgridsubgrid

~2I)(tr !"" #=#

SGS viscosity = property of the flow, not of the fluid… (very) bad news !

?)etc.,~

,~,( gridgridsubgrid UU !"#

4

shearuniformlocally:j

j

i

iix

x

uuu !+"=

#

#

i

22

2dS),(),(),(

4

1)(T ! "

"

#

$

%%

&

'()+()(=

lB

j

j

iisgs lxul

x

ulxulxu

ll

*

+*

*++

,

!

" # sgs $ S = 2 % sgs(&) $ S 2

' Tsgs(&) & with & = grid spacing

- after some (exact) calculations from the Navier-Stokes equations…

!

" # u ($) % # S & $ and "U($) % S & $ ' ( sgs($) = (Cs$)2& S ) S ( )

- in the context of LES :

Effet du cisaillement en équations

Lévêque et al., J. Fluid Mech. (2007)Shear-Improved Smagorinsky Model

5

If : Smagorinsky model for HI turbulence (Cs =0.17)

If : Shear-dominated SGS viscosity

SS~~

>>!

SS~~!>>

( ) ( ) SCu

C sssgs

~)( 22!"#$

#

#!"#$%

&

( )S

SCssgs ~

~ 2

2!

"#$%

MINUS :PLUS :→ physically-sound, no ad-hoc parameter→ low computational cost→ easy to parallelize→ convenient for boundary conditions

→ requires estimation of mean-flowas the simulation runs!

!

" sgs(#) = (Cs#)2$ S % S ( )

signal processing

Application to complex-geometryunsteady turbulent flows (real-world flows)

How to extract (or reconstruct) the mean flow as the simulation runs ?

- physically sound : smoothing in time mean flow = low-frequency component of the flow

-manageable from computational viewpoint : recursive filter-adaptive : Kalman filtering

Cahuzac et al., in press Phys. Fluids (2010)

grid cut-offsmoothingcut-off (fc)

SGS

very-largescales resolved

scales

SISM

f

Euu(f)in

stab

ilitie

s

turb

ulen

ce cascade

Exponentially-weighted moving averageexponential smoothing

!

cexp =2" fc #t

3$ 3.628 fc #t

conceptually simple local in space low memory storage

MINUS :PLUS :

uniform physically-sound cut-off frequency ? unavoidable phase delay :

!

"t / cexp

)1(

exp

)(

exp

)1()1( ++

!+!"=nnn

ucucu

-cexp : smoothing factor- equivalent tp first-order low pass filter with cut-off frequency fc :

at each grid point

Low-pass Kalman filter adaptive exponential smoothing

)1()()1( +++=

nnn

uuu !

control increment = random noise with fixed variance

!

"# u

=2$ fc %t

3& u

*

)1()1()1( ++++=

nnnuuu !

deviation from the estimated mean = random noise with variance

: observation

: model

K(n+1) optimal Kalman gain (given by theory)!

"#u

(n+1)=max u

*$ u

(n+1)% u

(n+1), 0.1$ u

*2[ ]

periodsstationaryfor/ .expcuu=!!

""

)1()1()()1()1()1( ++++

!+!"=nnnnn

uKuKu

why ?

predict and update :

10!/*ufc =

!

u*

= uwall

3.106 mesh-pointsΔr+=1, RΔθ+=20, Δz+=25

resolution: 49x89x41Δx+=52, Δz+=31, Δy+=0.5…24

!

St = fs "D /U#

!

fc = 2 f s

Strouhal numberfriction velocity

Practical evaluation of the methodTurb’Flow solver (Jérôme Boudet, LMFA):

- Finite volume- Convective fluxes by centered schemes on four points- Diffusive fluxes by centered scheme on two points- SGS model = Shear-Improved Smagorinsky Model (SISM)

plane-channel Flow at Rew = 395 flow past a circular cylinder at ReD = 47,000

Cahuzac et al. Phys. Fluids (2010)

11

… spatial average (along x-z)

… exponential smoothing compared tospatial average (along x)

… Kalman filtering

Velocity probes at y+=10 in the plane-channel flow (Rew=395)

12

streamwise velocity profile turbulent intensity profiles

Plane-channel flow

comparisons with DNS data

friction velocity:

1.59.0!

= smuDNS

w

1.exp.54.0!

= smuw

within the 10% level of accuracyreviewed for standard SGS models

→

13

spanwise spectraof streamwise velocity

plane-channel flow

comparisons with DNS data

two-point velocity correlationfor spanwise separation

14

plane-channel flow

comparisons with DNS data

two-point correlation functions… promising for aero-acoustics

streamwise

spanwise

!

Cp =p " p#1

2$U#

2

!

Cf ="w

1

2#U$

2

% Re

2

2

1

!

="

U

pC rmsp

#

flow past a circular cylinder

Friction and pressure coefficients

flow past a circular cylinder

mean flow andturbulence intensities

Work in progress...

flow past a circular cylinder

comparisons with experimental dataspectra in frequency

(a)

(a)(b)

(b)

(c)

(c)

1.8 D

… promising for aero-acoustics

fc

« Adaptivity » of the Kalman filter

flow past a circular cylinder

a drawback of the smoothing: phase delay

KalmanKt/!

10 <<KalmanK

… make KKalman maximum inseparation zone

Conclusion

from a physical viewpoint :- take into account mean-flow inhomogeneity + unsteadiness :

SISM (no ad-hoc parameter) + (adaptive) temporal smoothing

from a numerical viewpoint :- inconditionnally stable- low cost algorithms- easy to parallelize + convenient for boundary conditions

To-Do list:- address realistic turbo-machinery flow / aero-acoustic features

Flocon project (LMFA) : Adrien Cahuzac + Jérôme Boudet- reduce phase delay :

- improve Kalman filter- consider Lagrangian filtering (along fluid trajectory)

Plane channel flow (academic flow)Lévêque et al., J. Fluid Mech. (2007)Boudet et al., J. of Therm. Science. (2008)Cahuzac et al. Phys. Fluids (2010)

22

flow past a circular cylinder

comparisons with experimental data

flow characteristics

Mesh resolution