jean-philippe braeunig cea dam Île-de-france, bruyères-le-châtel, lrc cea-ens cachan

36
J.-Ph. Braeunig CEA DAM Ile-de-France Page 1 Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel, LRC CEA-ENS Cachan [email protected] 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.

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Workshop “Numerical Methods for multi-material fluid flows”, Czech Technical University in Prague. A pure eulerian scheme for multi-material fluid flows. Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel, LRC CEA-ENS Cachan [email protected] - PowerPoint PPT Presentation

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Page 1: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 1

Jean-Philippe BraeunigCEA DAM Île-de-France, Bruyères-le-Châtel,

LRC CEA-ENS Cachan

[email protected]

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.

Page 2: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 3: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 4: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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 ,

Page 5: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 6: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 7: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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)

Page 8: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 9: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 10: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 11: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 12: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 13: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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.

Page 14: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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 ?

Page 15: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

pppc

es

2

2

t,coefficienGruneisen 1

(square), velocity sound

energy, internal specific the density, thewith

EOS,given afor ),(P be usLet

e

ep

Page 16: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 17: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 18: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 19: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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.

Page 20: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 20

2D RESULTS

• Perfect shear test

• Density at t=0.25

Page 21: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 21

2D RESULTS

• Perfect shear test

• Pressure at t=0.25

Page 22: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 22

2D RESULTS

• Perfect shear test

• Velocities at t=0.25

Page 23: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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.

Page 24: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 24

2D RESULTS

• Sedov test case• Pressure at t= 2.5 10-9 s

Page 25: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 25

2D RESULTS

• Sedov test case

• Kinetic energy t= 2.5 10-9 s

Page 26: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 27: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 28: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 28

2D RESULTS

• Fall of a volume of water on the ground in air• Video

Page 29: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 29

2D RESULTS

Page 30: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 30

2D RESULTS

Page 31: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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 !

Page 32: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

J.-Ph. Braeunig CEA DAM Ile-de-France Page 32

2D RESULTS

t=0.09 s t=0.132 s

• Pressure field

Page 33: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 34: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 35: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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

Page 36: Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel,  LRC CEA-ENS Cachan

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).