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

1. 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 • jean-philippe.braeunig@cea.fr • with Prof. J.-M. Ghidaglia and B.Desjardins September 10-14, 2007

2. 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 IMPROVEMENT EXPECTED

3. SINGLE-MATERIAL : FINITE VOLUME FVCF SCHEME • VFFC : Finite Volume with Characteristic Flux (Ghidaglia et al. [1]) The vector of quantities V per volume unitis cell centered. with The finite volumescheme reads :

4. SINGLE-MATERIAL : FINITE VOLUME FVCF SCHEME • The FVCF flux through the face between CL and CRis obtained by upwinding the normal fluxes in the diagonal space: CR CL

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 xintbetween two different materials, with different EOS ? x xR xint xL 1 1 2 2 pure cell, only material 2 mixed cell, both materials 1 and 2 pure cell, only material 1

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 tn tn* pure cell pure cell mixed cell tn+1* tn+1 pure cell pure cell mixed cell

7. multi-material : INTERFACE CAPTURING NIP • How to define the flux at xintbetween 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. 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. 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

10. Capture d’interface VFFC-NIP • Extension to the multi-material case in 2D/3D • Phase x 2 2 2 Condensate 1 1 2 1 1 1 1

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. 2 2 2 2 tn 1 2 2 1 1 1 1 1 1D conservative projection 1 1 2 2 2 1 2 1 2 2 1 1 Flux at interfaces Condensate 1 2 2 2 1 1 2 1 2 1 Remap and interface positioning 2 2 2 tn,x 1 2 2 1 1 1 1 1

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. 2D EXTENSION • 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 n cr • 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 pintand uint that set positive entropy dissipation for each phase of the directional splitting.

14. 2D EXTENSION 1 2 2 2 1 1 2 1 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. 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. 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 2 2 1 2 1 Layer i (u)i (p)i

17. At the end of each 1D calculation of the directional splitting, interfaces normal vectors are updated this way: 2D EXTENSION Interface positioning with respect to volume fraction (VOF) following D. L. Youngs [5] • 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. Blastwave test case of Woodward & Collela [2] using 400 cells Comparison by Liska&Wendroff [3] of density profile at time t=0.038 1D RESULT interfaces VFFC-NIP

19. 2D RESULTS • Perfect shear test • Velocities at t=0 s Ux=0.4 Uy=1. Ux=-0.4 Uy=-1.

20. 2D RESULTS • Perfect shear test • Density at t=0.25

21. 2D RESULTS • Perfect shear test • Pressure at t=0.25

22. 2D RESULTS • Perfect shear test • Velocities at t=0.25

23. 2D RESULTS • Sedov test case • Perfect gas EOS with specific heat ratio 1.66 • Mesh is 100x100 cells P=1.5 Rho=1. Ux=Uy=0. P=5. 1019 Rho=1. Ux=Uy=0.

24. 2D RESULTS • Sedov test case • Pressure at t= 2.5 10-9 s

25. 2D RESULTS • Sedov test case • Kinetic energy t= 2.5 10-9 s

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. 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

28. 2D RESULTS • Fall of a volume of water on the ground in air • Video

29. 2D RESULTS

30. 2D RESULTS

31. 2D RESULTS Air is ejected by sliding between water and wall even when there is only one mixed cell left !

32. 2D RESULTS • Pressure field t=0.132 s t=0.09 s

33. 2D RESULTS • Shock wave in water interaction with a bubble of air • Shock wave at Mach 6 • 200x100 cells Water Water Air

34. 2D RESULTS • Geometry T=0 T=2.6µs T=6.8µs T=8.0µs T=8.3µs T=9.3µs

35. 2D RESULTS • Pressure T=0 T=2.6µs T=6.8µs T=8.0µs T=8.3µs T=9.3µs

36. 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. CONCLUSION [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).