1. NUMERICAL MODELING OF COMPRESSIBLE TWO-PHASE FLOWS Hervé Guillard INRIA Sophia-antipolis, Pumas Team, B.P. 93, 06902 Sophia-Antipolis Cedex, France, Herve.Guillard@sophia.inria.fr Thanks to Fabien Duval, Mathieu Labois, Angelo Murrone, Roxanna Panescu, Vincent Perrier

2. Crossing the “wall “ of sound Some examples of two-phase flows Granular medium : HMX

3. Some examples of two-phase flows : Steam generator in a nuclear power plant

4. Multi-scale phenomena Need for macro-scale description and averaged models

5. Example of Interface problems : Shock-bubble interaction Non structured tet mesh : 18M nodes 128 processors 3h30 h

7. model suitable for two fluid studies : no general agreement large # of different models : homogeneous, mixture models, two-fluid models, drift-flux models number of variables, definition of the unknowns number of equations large # of different approximations conservative, non-conservative, incompressible vs incompressible techniques, TWO-PHASE MODELS

8. -Construction of a general 2-phase model - Non-equilibrium thermodynamics of two phase non-miscible mixtures - Equilibria in two phase mixture -Reduced “hyperbolic” models for equilibrium situations - Technical tool : Chapman-Enskog expansion - A hierarchy of models - Some examples - Reduced “parabolic” models - First-order Chapman-Enskog expansion - Iso-pressure, iso-velocity model - Traveling waves and the structure of two-phase shock OVERVIEW OF THIS TALK

9. HOMOGENIZED MODELS Reference textbooks : Ishii (1984), Drew-Passman (1998) Let us consider 2 unmiscible fluids described by the Euler eq Introduce averaging operators Let X_k be the characteristic function of the fluid region k where σ is the speed of the interface

10. Let f be any regular enough function Define averaged quantities : etc Multiply the eq by X_k and apply Gauss and Leibnitz rules

12. THE TWO FLUID MODEL Models for :

13. How to construct these models ? Use the entropy equation :

14. Assume : Then first line : One important remark ( Coquel, Gallouet,Herard, Seguin, CRAS 2002 ) : The two-fluid system + volume fraction equation is (always) hyperbolic but the field associated with the eigenvalue is linearly degenerate if and only if

15. Final form of the entropy equation :

16. Simplest form ensuring positive entropy production :

17. Summary - Two fluid system + volume fraction eq = hyperbolic system the entropy production terms are positive - This system evolves to a state characterized by - pressure equality - velocity equality - temperature equality - chemical potential equality Deduce from this system, several reduced systems characterized by instantaneous equilibrium between - pressure - pressure + velocity - pressure + velocity + temperature -.............

18. One example : Bubble column : AMOVI MOCK UP (CEA Saclay ) Pressure relaxation time Velocity relaxation time Temperature relaxation time Bubble transit time

19. Construction of reduced models : Technical tool : The Chapmann-Enskog expansion What is a Chapman-Enskog asymptotic expansion ? - technique introduced by Chapmann and Enskog to compute the transport coefficients of the Navier-Stokes equations from the Boltzmann equations -technique used in the Chen-Levermore paper on hyperbolic relaxation problems

20. CHAPMAN-ENSKOG EXPANSION

22. # # #

23. Some examples : Assume pressure equilibrium : “classical” two-fluid model (Neptune) eos : solve p1 = p2 for the volume fraction Non-hyperbolic !

25. Some examples : Assume : - pressure equilibrium - velocity equilibrium one-pressure, one velocity model (Stewart-Wendroff 1984, Murrone-Guillard, 2005)

26. one-pressure, one velocity model (Stewart-Wendroff 1984, Murrone-Guillard, 2005) Hyperbolic system u-c, u+c gnl, u,u ld Entropy

27. Some examples : Assume - pressure equilibrium - velocity equilibrium - temperature equilibrium Multi-component Euler equations : eos : solve : p1 = p2, T1=T2

28. A Small summary : Model # eqs complexity hyperbolic conservative contact respect total non equilibrium 7 +++ yes no yes pressure equilibrium 6 +++ no no ? pressure and velocity equilibrium 5 ++ yes no yes pressure and velocity and temperature 4 + yes yes no equilibrium

29. Why the 4 equation conservative model cannot compute a contact 1 u p Ti 0 u p Ti+1 1 u p Ti YupTYupT 1 u p Ti 1 u p Ti 0 u p Ti+1 Not possible at constant pressure keeping constant the conservative variables R. Abgrall, How to prevent pressure oscillations in multi-component flow calculation: a quasi-conservative approach, JCP, 1996

30. “Parabolic” reduced system Goal : Introduce some effects related to non-equlibrium

31. One example of “parabolic” two-phase flow model Is a relative velocity (drift – flux models)

32. Mathematical properties of the model : First-order part : hyperbolic Second-order part : dissipative

33. Comparison of non-equilibrium model (7 eqs) Vs Equilibrium model (5 eqs) with dissipative Terms (air-water shock tube pb)

35. Sedimentation test-case (Stiffened gas state law) Note : velocities of air and water are of opposite sign

36. Sedimentation test-case (Perfect gas state law) Note : velocities of air and water are of opposite sign 5 eqs dissipative model Non-equilibrium model

38. Non equilibrium Model (7 eqs) Equilibrium Model (5 eqs)

39. Two-phase flows models have non-conservative form Non-conservative models : Definition of shock solution Traveling waves

40. Weak point of the model : Non conservative form Shock solutions are not defined One answer : LeFloch, Raviart-Sainsaulieu change into Define the shock solutions as limits of travelling waves solution of the regularized dissipative system for Drawback of the approach : the limit solution depends on the viscosity tensor

41. How to be sure that the viscosity tensor encode the right physical informations ? The dissipative tensor retains physical informations coming from the non-equilibrium modell

42. Convergence of travelling waves solutions of the 5eqs dissipative model toward shock solutions

43. ISOTHERMAL MODEL with DARCY-LIKE DRIFT LAW

44. Rankine-Hugoniot Relations :

45. NUMERICAL TESTS Infinite drag term (gas and liquid velocities are equal)

46. TRAVELLING WAVE SOLUTIONS

47. TRAVELLING WAVES II If TW exists, they are characterized by a differential system of Degree 2 Isothermal case : This ODE has two equilibrium point Stable one unstable one

48. PressurevelocityGas Mass fraction Drag Coeff 10000 kg/m3/s Drag Coeff 5000 kg/m3/s

50. CONCLUSIONS - Hierarchy of two-fluid models characterized by stronger and stronger assumptions on the equilibriums realized in the two fluid system - on-going work to define shock solutions for two-phase model as limit of TW of a dissipative system characterized by a viscosity tensor that retain physical informations on disequilibrium