1. 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 IUSTI

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

3. 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

4. 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 ?

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

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

8. 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)

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
Uranium-Molybdene mixture
Epoxy-Periclase mixture

11. 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

12. 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

13. 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.

14. 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.

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

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

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

18. 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

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

20. 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

21. 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)

23. Thank you for your attention