1. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 1 Jean-Philippe Braeunig CEA DAM Île-de-France, Bruyères-le-Châtel, LRC CEA-ENS Cachan jean-philippe.braeunig@cea.fr 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.

2. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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

3. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 3 SINGLE-MATERIAL : FINITE VOLUME FVCF SCHEME VFFC : Finite Volume with Characteristic Flux (Ghidaglia et al. [1]) The finite volume scheme reads : The vector of quantities V per volume unit is cell centered. with

4. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 4 SINGLE-MATERIAL : FINITE VOLUME FVCF SCHEME The FVCF flux through the face between C L and C R is obtained by upwinding the normal fluxes in the diagonal space: CLCL CRCR

5. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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 x int 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 xLxL xRxR x int x

6. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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 cell pure cell tntn t n* t n+1 t n+1* pure cell mixed cell pure cell mixed cell pure cell

7. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 7 multi-material : INTERFACE CAPTURING NIP How to define the flux at x int between two different materials, with different EOS ? In dimension one, we integrate the eulerian system on a moving volume : It comes: The eulerian normal flux reads :

8. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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 L R

9. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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 t n

10. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 10 Capture d’interface VFFC-NIP Extension to the multi-material case in 2D/3D Phase x 1 1 1 1 1 1 2 2 2 2 Condensate

11. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2 1 tntn t n,x Remap and interface positioning

12. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 12 2D EXTENSION 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 :

13. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 13 2D EXTENSION 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: clcl crcr In phase x of the directional splitting, we only consider the x component of pressure gradient and velocity : Unfortunately, we did not find 2D/3D expressions of p int and u int that set positive entropy dissipation for each phase of the directional splitting.

14. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 14 2D EXTENSION 1 1 1 1 2 2 2 2 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: What is small variations and for which quantities ?

15. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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:

16. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 16 2D EXTENSION Variation of p in layer i is a function of variations of density and internal energy of the form: Some analyses of the scheme give: 1 1 1 2 2 2 Layer i  (u) i  (p) i

17. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 17 2D EXTENSION Interface positioning with respect to volume fraction (VOF) following D. L. Youngs [5] 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.

18. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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 interfaces

19. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 19 2D RESULTS Perfect shear test Velocities at t=0 s Ux=0.4 Uy=1. Ux=-0.4 Uy=-1.

20. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 20 2D RESULTS Perfect shear test Density at t=0.25

21. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 21 2D RESULTS Perfect shear test Pressure at t=0.25

22. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 22 2D RESULTS Perfect shear test Velocities at t=0.25

23. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 23 2D RESULTS Sedov test case Perfect gas EOS with specific heat ratio 1.66 Mesh is 100x100 cells P=5. 10 19 Rho=1. Ux=Uy=0. P=1.5 Rho=1. Ux=Uy=0.

24. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 24 2D RESULTS Sedov test case Pressure at t= 2.5 10 -9 s

25. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 25 2D RESULTS Sedov test case Kinetic energy t= 2.5 10 -9 s

26. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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:

27. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 27 2D RESULTS Fall of a volume of water on the ground in air Initial state of air is v=0, density = 1 kg/m 3, pressure = 10 5 Pa Initial state of water is v x =0 m/s, v y =-15 m/s, density = 1 kg/m 3, pressure = 10 5 Pa Gravity is g=-9.81 m/s 2 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

28. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 28 2D RESULTS Fall of a volume of water on the ground in air Video

29. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 29 2D RESULTS

30. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 30 2D RESULTS

31. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 31 2D RESULTS Air is ejected by sliding between water and wall even when there is only one mixed cell left !

32. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 32 2D RESULTS t=0.09 s t=0.132 s Pressure field

33. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 33 Shock wave in water interaction with a bubble of air Shock wave at Mach 6 200x100 cells Water Air 2D RESULTS

34. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 34 Geometry T=0 T=6.8µs T=2.6µs T=8.0µs T=8.3µsT=9.3µs 2D RESULTS

35. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 35 Pressure T=0 T=6.8µs T=2.6µs T=8.0µs T=8.3µsT=9.3µs 2D RESULTS

36. J.-Ph. Braeunig CEA DAM Ile-de-FrancePage 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).www.math.ntnu.no/conservation/ [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).