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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 IUSTI

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

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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 Not conservative Kapila & al (2001)

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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 ?

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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 ?

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W1*W1* W10W10 W2*W2* W20W20 W20W20 W10W10 σ1σ1 W10W10 W1LW1L W20W20 W2LW2L (u+c) 2 Dispersion mechanism

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W1*W1* W10W10 W2*W2* W20W20 σ1σ1 W10W10 W1LW1L W20W20 W2LW2L W2RW2R W20W20 W1RW1R σ 1 Dispersion mechanism

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Consequences The two-phase shock is smooth shock = succession of equilibrium states (P 1 =P 2 and u 1 =u 2 ) 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 P 1 =P 2 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)

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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

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

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+ 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 The reduced model (with 5 equations) is now closed

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A new projection method Saurel & al (2005) Volume fractions definition t x x i-1/2 x i+1/2 t n+1 u* i-1/2 u* i+1/2 S + i-1/2 S - i+1/2 Euler equations context u 1, P 1, e 1 u 2, P 2, e 2 u 3, P 3, e 3 V1V1 V2V2 V3V3

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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.

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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.

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Propagation of a density discontinuity in an uniform flow with JWL EOS P = P CJ = Pa u = 1000 m/s ρ = ρ CJ = 2182 kg/m 3 P = P CJ = Pa u = 1000 m/s ρ = 100 kg/m 3 00,51

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Shock tube problem in extreme conditions Euler equations and JWL EOS P = P CJ = Pa ρ = ρ CJ = 2182 kg/m 3 P = Pa ρ = 100 kg/m 3 00,51 The Godunov method fails in these conditions

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Shock tube problem with almost pure fluids Liquid-Gas interface with the 5 equations model 00,81 P = 10 5 Pa α air = P = 10 9 Pa α water = ρ water = 1000 kg/m 3 ρ air = 50 kg/m 3 Stiffened Gas EOS

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Shock tube problem with mixtures of materials Epoxy-Spinel Mixture 00,61 P = 10 5 Pa α epoxy = 0,5954 P = Pa α epoxy = 0,5954 ρ epoxy = 1185 kg/m 3 ρ spinel = 3622 kg/m 3 Stiffened Gas EOS

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Shock tube problem with mixtures of materials (2)

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Mixture Hugoniot tests Comparison with experiments and the 7 equations model New method 7 equations model P = 10 5 Pa α epoxy = 0,5954 ρ epoxy = 1185 kg/m 3 ρ spinel = 3622 kg/m 3 Up Piston Epoxy-Spinel Mixture

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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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