Modelling of interfaces separating compressible fluids and mixtures of materials

1. Modelling of interfaces separating compressible fluids and mixtures of materials Multiphase shock relations and extra physics Erwin FRANQUET and Richard SAUREL Polytech Marseille, UMR CNRS 6595 - IUSTI

2. Baer & Nunziato (1986) and Saurel & al. (2003) Chinnayya & al (2004) A Multiphase model with 7 equationsFor solving interfaces problems and shocks into mixtures

3. When dealing with interfaces and mixtures with stiff mechanical relaxation the 7 equations model can be reduced to a 5 equations model Infinite drag coefficient Infinite pressure relaxation parameter • Kapila & al (2001) Reduction to a 5 equations model Not conservative

4. To deal with realistic applicationsshock relations are mandatory • 7 unknowns : α1, Y1, ρ, u, P, e, σ • 4 conservation laws : • Mixture EOS : • One of the variable behind the shock is given (often σ or P) An extra relation is needed : jump of volume fraction or any other thermodynamic variable How to determine it ?

5. Informations from the resolution of the 7 equations model Impact of an epoxy-spinel mixture by a piston at several velocities Fully dispersed waves Why are the waves dispersed in the mixture ?

6. W1* W10 σ1 W1L W10 (u+c)2 W2* W2L W20 W20 Dispersion mechanism W10 W20

7. W1* W10 σ1 ’ σ1 W2* W2R W20 W20 Dispersion mechanism W1R W1L W10 W2L W20

8. The two-phase shock is smooth • shock = succession of equilibrium states (P1=P2 and u1=u2) • We can use the 7 equations model in the limit : • That is easier to integrate between pre and post shock states. • In that case, the energy equations reduce to : • And can be integrated as : and As P1=P2 at each point, we have : The mixture energy jump is known without ambiguity : To fulfill the mixture energy jump each phase must obey : Saurel & al (2005) Consequences

9. Some properties • Identifies with the Hugoniot adiabat of each phase • Symmetric and conservative formulation • Entropy inequality is fulfilled • Single phase limit is recovered • Validated for weak and strong shocks for more than 100 experimental data • The mixture Hugoniot curve is tangent to mixture isentrope

10. Shock relations validation Epoxy-Spinel mixture Paraffine-Enstatite mixture Epoxy-Periclase mixture Uranium-Molybdene mixture

11. The second difficulty comes from the average of the volume fraction inside computational cells : • It is not a conservative variable • It necessitates the building of a new numerical method The reduced model (with 5 equations) is now closed + Mixture EOS + Rankine-Hugoniot relations • Consequences : • A Riemann solver can be built • This one can be used to solve numerically the 5 equations model

12. t u*i-1/2 S+i-1/2 S-i+1/2 u*i+1/2 t n+1 Volume fractions definition Euler equations context xi-1/2 xi+1/2 x V1 V2 V3 u1, P1, e1 u2, P2, e2 u3, P3, e3 A new projection methodSaurel & al (2005)

13. Conservation and entropy inequality are preserved if : This ODE system is solved in each computational cell so as to reach the mechanical equilibrium asymptotic state ( ) It can be written as an algebraic system solved with the Newton method. Relaxation system

14. Comparison with conventional methods • Conventional Godunov average supposes a single pressure, velocity andtemperature in the cell. In the new method, we assume only mechanical equilibrium and not temperature equilibrium. • It guarantees conservation and volume fraction positivity • The method does not use any flux and is valid for non conservative equations • In the case of the ideal gas and the stiffened gas EOS with the Euler equations both methods are equivalent. Results are different for more complicated EOS (Mie-Grüneisen for example) • The new method gives a cure to anomalous computation of some basic problems: - Sliding lines - Propagation of a density discontinuity in an uniform flow with Mie Grüneisen EOS. • It can be used in Lagrange or Lagrange + remap codes.

15. P = PCJ = 2 1010 Pa u = 1000 m/s ρ = ρCJ = 2182 kg/m3 P = PCJ = 2 1010 Pa u = 1000 m/s ρ = 100 kg/m3 0 0,5 1 Propagation of a density discontinuity in an uniform flow with JWL EOS

16. P = PCJ = 2 1010 Pa ρ = ρCJ = 2182 kg/m3 P = 2 108 Pa ρ = 100 kg/m3 0 0,5 1 • The Godunov method fails in these conditions Shock tube problem in extreme conditionsEuler equations and JWL EOS

17. P = 109 Pa αwater = 1-10-8 P = 105 Pa αair = 1-10-8 0 0,8 1 ρwater = 1000 kg/m3 ρair = 50 kg/m3 Stiffened Gas EOS Shock tube problem with almost pure fluidsLiquid-Gas interface with the 5 equations model

18. P = 1010 Pa αepoxy = 0,5954 P = 105 Pa αepoxy = 0,5954 0 0,6 1 ρepoxy = 1185 kg/m3 ρspinel = 3622 kg/m3 Stiffened Gas EOS Shock tube problem with mixtures of materialsEpoxy-Spinel Mixture

19. Shock tube problem with mixtures of materials (2)

20. Piston Epoxy-Spinel Mixture P = 105 Pa αepoxy = 0,5954 ρepoxy = 1185 kg/m3 ρspinel = 3622 kg/m3 Up New method 7 equations model Mixture Hugoniot testsComparison with experiments and the 7 equations model

21. Air (Ideal Gas EOS) Copper (Stiffened Gas EOS) U = 5000 m/s TNT (JWL EOS) RDX (Mie-Grüneisen EOS) 2D impact of a piston on a solid stucture

23. Thank you for your attention