J.-Ph. Braeunig CEA DAM Ile-de-France Page 1
Jean-Philippe BraeunigCEA DAM Île-de-France, Bruyères-le-Châtel,
LRC CEA-ENS Cachan
with Prof. J.-M. Ghidaglia and B.Desjardins
September 10-14, 2007
A pure eulerian scheme for multi-material fluid flows
Workshop “Numerical Methods for multi-material fluid flows”,
Czech Technical University in Prague.
J.-Ph. Braeunig CEA DAM Ile-de-France Page 2
IMPROVEMENT EXPECTED
• GOALS :• Second order scheme in 2D-3D• Cartesian structured meshes, plane or axi-symmetric• Robust scheme• Fully conservative method• Multi-material computations with real fluid EOS• Interface capturing in 2D-3D• Enable materials to slip by each others within a cell
J.-Ph. Braeunig CEA DAM Ile-de-France Page 3
SINGLE-MATERIAL : FINITE VOLUME FVCF SCHEME
0 2
1
1
d
i
iinn
Adt
VVVol
• VFFC : Finite Volume with Characteristic Flux (Ghidaglia et al. [1])
up
rightleftV
down
leftArightA
upA
downA
VolThe finite volume scheme reads :
0
ΓΩ dΓn F V dΩ
t
The vector of quantities V per volume unit is cell centered.
E
uV
NpnuV
nupnuE
pnnuu
nu
nFnFd
ii
i )(
)( )(
)(
)(
1
with
EOS.given afor pressure the),(
energy, totalspecific the21
energy, internal specific the
,in velocity the density, the
,),,0(
2
d
ep
u/eE
e
u
nunN t
P
J.-Ph. Braeunig CEA DAM Ile-de-France Page 4
SINGLE-MATERIAL : FINITE VOLUME FVCF SCHEME
LRRL
LRLRRL
LRRL nFF
nVJsignnFF
nVV
2
),( 2
),,(
LR )( ),( and isigndiagnVJsign
RL ),( )(with nVJdiag i
• The FVCF flux through the face between CL and CR is obtained by upwinding the normal fluxes in the diagonal space:
CL CR
LRn
V
LL FV , RR FV ,
J.-Ph. Braeunig CEA DAM Ile-de-France Page 5
multi-material : INTERFACE CAPTURING NIP
• NIP : Natural Interface Positioning• 1D interface capturing, two main difficulties :
• How to manage the time step CFL condition with very small volumes and how goes the interface through a cell face ?
• How to define the flux at xint between two different materials,
with different EOS ?
pure cell, only material 1
pure cell,only material 2
mixed cell, both materials
1 and 2
1 1 2 2
xL xRxint x
int
J.-Ph. Braeunig CEA DAM Ile-de-France Page 6
multi-material : INTERFACE CAPTURING NIP
• How to manage the time step CFL condition with very small volumes and how goes the interface through a cell boundary ?
pure cell mixed cellpure cell
tn
tn*
tn+1
1intnx
1intnxnxint
tn+1*
1intnx
pure cell mixed cell pure cell
mixed cellpure cell pure cell
J.-Ph. Braeunig CEA DAM Ile-de-France Page 7
multi-material : INTERFACE CAPTURING NIP
• How to define the flux at xint between two different materials, with different EOS ?
• In dimension one, we integrate the eulerian system on a moving volume :
0
dΩx
F
t
VΩ(t)
dΓ)nuV( dΩ Vt
dΩt
VΓ(t) ΓΓΩ(t)Ω(t)
dΓn F dΩx
FΓΓ(t)Ω(t)
tΓΓΓΓ
)Γ(u ΓΓ)Γ(u ΓΩ(t)
nu,n,N
dΓNp dΓnF dΩ Vt ΓΓ
0with
000
It comes:
The eulerian normal flux reads : ΓΓΓΓΓ Np)nuV (nF
Ω(t)
J.-Ph. Braeunig CEA DAM Ile-de-France Page 8
multi-material : INTERFACE CAPTURING NIP
• We write the conservation laws on partial volumes left (L) and right (R)
• are calculated by the FVCF scheme
nuint
LNp intintRNp intint
L
RLR
and RL
) ,,0( with 0
) ,,0( with 0
intint
intintintint
11
intint
intintintint
11
RRRRR
nR
nR
nR
nR
LLLLL
nL
nL
nL
nL
nunNNpAdt
VVolVVol
nunNNpAdt
VVolVVol
intRn int
Ln
intint111
111
int111
1
)4(
)3(
)2(
)1(
upm
AdtE
m
mE
m
Adtv
m
mv
pm
Adtu
m
mu
Adtmm
LnL
nLn
L
nLn
L
LnL
nLn
L
nLn
L
LnL
nLn
L
nLn
L
LnL
nL
intint111
111
int111
1
)4(
)3(
)2(
)1(
and
upm
AdtE
m
mE
m
Adtv
m
mv
pm
Adtu
m
mu
Adtmm
RnR
nRn
R
nRn
R
RnR
nRn
R
nRn
R
RnR
nRn
R
nRn
R
RnR
nR
J.-Ph. Braeunig CEA DAM Ile-de-France Page 9
2D EXTENSION
• The 1D approach will be extended in 2D/3D by directional splitting
• Single-material:
• Directional splitting does not change the scheme if fluxes on each directions are computed with values at the same time tn
0)()()( nRear,nFront,nDown,nUp,nLeft,nRight,1
zzzyyyxxx
nn
AAAdt
VVVol
nup,
nright ,nleft ,
ndown,
xAyA
Vol
J.-Ph. Braeunig CEA DAM Ile-de-France Page 10
Capture d’interface VFFC-NIP
• Extension to the multi-material case in 2D/3D• Phase x
11 1
11
1
2
2
22
Condensate
J.-Ph. Braeunig CEA DAM Ile-de-France Page 11
2D EXTENSION
• Multi-material: • Directions are splitted, using the same 1D algorithm in each direction• At the beginning of the computation, you know the order of the
materials within a mixed cell in each direction. This order is kept by the method.
• Condensation of mixed cells if one have to deal with neighbouring mixed cells.
Condensate
Flux at interfaces
1D conservative projection
FVCF
11 1
11 1
222
2
22
1 1 1 1 1 1 222222
1 1 1 1 2222
11 1
11 1
222
2
22 1
tn
tn,x
FVCF
Remap andinterface positioning
J.-Ph. Braeunig CEA DAM Ile-de-France Page 12
2D EXTENSION
ucpx
ct
ucpx
ct
with ,0
with ,0
• Interface pressure and velocity in 1D:• FVCF is written in lagrangian coordinates • Calculation of two local Riemann invariants with equations:
• Two conditions to obtain pressure and velocity :
• The solution is :rrrrrrr
lllllll
ucpucp
ucpucp
intintint
intintint
011
,min~with
~~~~
~~
~~~~
~~
~~
11
int
int
ni
ni
ni
ni
nii
iii
rrll
rl
rrll
rrrlll
rrll
rlrrll
rrll
rlllrr
peeTdsdt
dxcc
cc
pp
cc
ucucu
cc
uucc
cc
pcpcp
J.-Ph. Braeunig CEA DAM Ile-de-France Page 13
2D EXTENSION
~~~~
~~
~~~~
~~
~~
int
int
rrll
rl
rrll
rrrlll
rrll
rlrrll
rrll
rlllrr
cc
ppn
cc
ucucnu
ncc
uucc
cc
pcpcpnp
n
• 2D interface pressure gradient and velocity are obtained by writing 1D equations in direction of the interface normal vector, in order to impose perfect sliding:
cl
cr
• In phase x of the directional splitting, we only consider the x component of pressure gradient and velocity :
x
rrll
rl
rrll
xrr
xllx
x
rrll
rlrrll
rrll
rlllrrx
ncc
pp
cc
ucucu
nncc
uucc
cc
pcpcp
rl ~~~~
~~
~~~~
~~
~~
int
int
• Unfortunately, we did not find 2D/3D expressions of pint and uint that set positive entropy dissipation for each phase of the directional splitting.
J.-Ph. Braeunig CEA DAM Ile-de-France Page 14
2D EXTENSION
1 1 1 1 2222 FVCFFVCF
In small layers in the condensate,the CFL condition is not fulfilled !
• What is CFL condition ?• In the case of linear advection, using a basic upwind
scheme:
udt
dxa
u
uu
dt
dxa
u
uu
dt
dxau
dt
dxa
dt
dxauu
adx
uua
dt
uu
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
ni
of s variationsmall 1,
0 if
tymonotonici 1,
0 if
1
0 with 0
11
11
11
• What is small variations and for which quantities ?
J.-Ph. Braeunig CEA DAM Ile-de-France Page 15
2D EXTENSION
• What is small variations and for which quantities ?• We choose to control variations of pressure p.• Variation of p is a function of variations of density and
internal energy, given by:
dp
cΓdedp
e
pΓ
pppc
es
2
2
t,coefficienGruneisen 1
(square), velocity sound
energy, internal specific the density, thewith
EOS,given afor ),(P be usLet
e
ep
J.-Ph. Braeunig CEA DAM Ile-de-France Page 16
2
2
11
12
11
ρ
Γ
p
)(c
ερρ and
p
Γρ
εee
ερρρ
Γ
p
)(cee
p
Γρ
p
pp
n
n
n
n
pnn
n
nn
pnn
pnn
n
n
n
nnn
n
nn
ni
nni
i
i
i
i
ii
i
ii
ii
ii
i
i
i
i
ii
i
iii
2D EXTENSION
• Variation of p in layer i is a function of variations of density and internal energy of the form:
2n
ini
nip
ni
c
pu
i
22
2nn
ini
nip
nni
i
iiΓ
pp
• Some analyses of the scheme give:
interfacesat quantities denotes 2/1 )( , with 2/12/1 ixxxdt
dxiii
nin
i
FVCF1 1 1 222
FVCF
Layer i(u)i
(p)i
J.-Ph. Braeunig CEA DAM Ile-de-France Page 17
2D EXTENSION
Interface positioning with respect to volume fraction (VOF)
following D. L. Youngs [5]
2f
1f 1f 1f
1f
1f 1f1f
2f1f
2f)f,F(f i2
i1n
• At the end of each 1D calculation of the directional splitting, interfaces normal vectors are updated this way:
• Notice that the explicit position of the interface is not needed in this method. Only the interface normal vector and materials ordering are needed.
i
ii f
fn
J.-Ph. Braeunig CEA DAM Ile-de-France Page 18
1D RESULT
VFFC-NIP
• Blastwave test case of Woodward & Collela [2] using 400 cells
• Comparison by Liska&Wendroff [3] of density profile at time t=0.038
0
1
1000
u
p
0
1
100
u
p
0
1
01.0
u
p
interfaces
J.-Ph. Braeunig CEA DAM Ile-de-France Page 19
2D RESULTS
• Perfect shear test
• Velocities at t=0 s
Ux=0.4Uy=1.
Ux=-0.4Uy=-1.
J.-Ph. Braeunig CEA DAM Ile-de-France Page 20
2D RESULTS
• Perfect shear test
• Density at t=0.25
J.-Ph. Braeunig CEA DAM Ile-de-France Page 21
2D RESULTS
• Perfect shear test
• Pressure at t=0.25
J.-Ph. Braeunig CEA DAM Ile-de-France Page 22
2D RESULTS
• Perfect shear test
• Velocities at t=0.25
J.-Ph. Braeunig CEA DAM Ile-de-France Page 23
2D RESULTS
• Sedov test case•Perfect gas EOS with specific heat ratio 1.66•Mesh is 100x100 cells
P=5. 1019
Rho=1.Ux=Uy=0.
P=1.5Rho=1.Ux=Uy=0.
J.-Ph. Braeunig CEA DAM Ile-de-France Page 24
2D RESULTS
• Sedov test case• Pressure at t= 2.5 10-9 s
J.-Ph. Braeunig CEA DAM Ile-de-France Page 25
2D RESULTS
• Sedov test case
• Kinetic energy t= 2.5 10-9 s
J.-Ph. Braeunig CEA DAM Ile-de-France Page 26
2D RESULTS
• Fall of a volume of water on the ground in air• EOS for air is assumed to be perfect gas:
• EOS for water is assumed to be stiffened gas:
1 and
4.1 with )1(
air2
airair
air
airair
Γp
c
ep
1 and
1021 ,7 with )1(
water2
8
waterwater
water
waterwater
Γp
c
Paep
J.-Ph. Braeunig CEA DAM Ile-de-France Page 27
2D RESULTS
• Fall of a volume of water on the ground in air• Initial state of air is v=0, density = 1 kg/m3, pressure = 105 Pa
• Initial state of water is vx=0 m/s, vy=-15 m/s, density = 1 kg/m3, pressure = 105 Pa
• Gravity is g=-9.81 m/s2
• Boundary conditions are wall with perfect sliding• Box is 20 m x 15 m• Water is 10 m x 8 m• Mesh is 100x75 cells
J.-Ph. Braeunig CEA DAM Ile-de-France Page 28
2D RESULTS
• Fall of a volume of water on the ground in air• Video
J.-Ph. Braeunig CEA DAM Ile-de-France Page 29
2D RESULTS
J.-Ph. Braeunig CEA DAM Ile-de-France Page 30
2D RESULTS
J.-Ph. Braeunig CEA DAM Ile-de-France Page 31
2D RESULTS
Air is ejected by sliding between water and wall even when there is only one mixed cell left !
J.-Ph. Braeunig CEA DAM Ile-de-France Page 32
2D RESULTS
t=0.09 s t=0.132 s
• Pressure field
J.-Ph. Braeunig CEA DAM Ile-de-France Page 33
• Shock wave in water interaction with a bubble of air
• Shock wave at Mach 6
• 200x100 cells
Water Water
Air
smu
Pap
mkg
/0
10
/10005
3
smu
Pap
mkg
/68.2113
108381.1
/1.132110
3
smu
Pap
mkg
/0
10
/15
3
2D RESULTS
J.-Ph. Braeunig CEA DAM Ile-de-France Page 34
• Geometry
T=0
T=6.8µs
T=2.6µs
T=8.0µs
T=8.3µs T=9.3µs
2D RESULTS
J.-Ph. Braeunig CEA DAM Ile-de-France Page 35
• Pressure
T=0
T=6.8µs
T=2.6µs
T=8.0µs
T=8.3µs T=9.3µs
2D RESULTS
J.-Ph. Braeunig CEA DAM Ile-de-France Page 36
CONCLUSION
• The method is locally conservative in mass, momentum and total energy, and allow sliding of materials by each others.
• Next studies on NIP aimed to improve accuracy in the mixed cells and a second order interface capturing.
• More physics, as surface tension or turbulence, will be added to this modelling.
[1] J.-M. Ghidaglia, A. Kumbaro, G. Le Coq, On the numerical solution to two fluid models via a cell centered finite volume method. Eur. J. Mech. B Fluids 20 (6) (2001) 841-867.[2] P. R. Woodward and P. Collela, The numerical simulation of 2D fluid flow with strong shock. Journal of Computational Physics, 1984, pp.115-173[3] R. Liska and B. Wendroff, Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. Conservation Laws Preprint Server, www.math.ntnu.no/conservation/, (2001).[4] J.-P. Braeunig, B. Desjardins, J.-M. Ghidaglia, A pure eulerian scheme for multi-material fluid flows, Eur. J. Mech. B Fluids, submitted.[5] D. L. Youngs, Time-Dependent multi-material flow with large fluid
distortion, Numerical Methods for Fluid Dynamics, edited by K.W. Morton and M. J. Baines, pp. 273-285, (1982).