Multiphase shock relations and extra physics 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

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

Reduction to a 5 equations model 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) Not conservative

To deal with realistic applications shock 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 ?

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 ?

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

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

Consequences 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)

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

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

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 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

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

Relaxation system 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.

Comparison with conventional methods Conventional Godunov average supposes a single pressure, velocity and temperature 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.

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

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

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

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

Shock tube problem with mixtures of materials (2)

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

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

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