1. A COMPRESSIBLE MODEL FOR LOW MACH TWO-PHASE FLOW WITH HEAT AND MASS EXCHANGES N. GRENIER, J.P. VILA & Ph. VILLEDIEU

2. MULTIMAT 2011 - 5-9 septembre 2011 2 CONTEXT AND MOTIVATION Context : COMPERE program from CNES & DLR Research partners : ONERA, ZARM, CNRS, Erlangen university, Air LIquide (Grenoble), Astrium ST (Bremen) Objectives : development of numerical tools for simulating complex fluid behavior inside space launcher tanks: dynamical behavior  sloshing thermal effects  heat and mass exchanges low gravity effects  capillary effects, Marangoni convection … Separated two-phase flow with a free moving interface Gas phase Liquid phase External Heat flux Evaporation Capillary raise

3. MULTIMAT 2011 - 5-9 septembre 2011 3 OUTLINE OF THE PRESENTATION 1. Presentation of the model 2. Numerical method 3. Numerical test cases 4. Conclusion

4. MULTIMAT 2011 - 5-9 septembre 2011 4 OUTLINE OF THE PRESENTATION 1. Presentation of the model 2. Numerical method 3. Numerical test cases 4. Conclusion

5. MULTIMAT 2011 - 5-9 septembre 2011 5 1. Presentation of the model Modeling choices Two fluid model  diffuse interface model Advantage : not necessary to localize (level set method) or reconstruct (VOF method) the interface between the two fluids  easy to implement Drawback : interface diffusion  necessary to define a “mixture” physical model and to use low diffusive numerical scheme Compressible model Advantage : more general, easier to implement into a gas dynamics code (ONERA context) Drawback : ill conditioned for low Mach number flows  low Mach Scheme Same velocity field for both fluids Advantage : hyperbolic model, no closure assumption needed Drawback : impossible to deal with subscale phenomena (subgrid bubbles or droplets …)

6. MULTIMAT 2011 - 5-9 septembre 2011 6 To get a close model, it is now necessary to give a relation between the “mixture” pressure p, the bulk densities, and the mixture specific internal energy e. 1. Presentation of the model Inviscid two-fluid Model with being the mixture total energy per unit volume and the mixture bulk density. Gas bulk density Liquid bulk density

7. MULTIMAT 2011 - 5-9 septembre 2011 7 1. Presentation of the model Extension to non isothermal flows Let T denote the mixture temperature and  the gas volume fraction.  and T are assumed to be the unique solution of the following system : Local mechanical equilibrium Local thermal equilibrium where p = p g (  g,T), p = p l (  l,T) denote the gas and liquid EOS and e=e g (  g,T), e=e l (  l,T) denote the gas and liquid colorific laws. The mixture EOS is then (implicitly) defined by :

8. MULTIMAT 2011 - 5-9 septembre 2011 8 1. Presentation of the model Other interpretation of closure equations (5)-(6) Important consequence : System (1) with pressure law given by (2)-(3) is thermodynamically consistent in the sense that it has a convex entropy in the sense of Lax defined as : where e g, e l are the gas and liquid specific internal energies (implicitly defined by the solution of (2)), s g and s l are specific entropies, and are the real densities. Closure relations (2)-(3) can also be interpreted as a direct consequence of the following modeling assumption for the mixture Gibbs potential : Ideal mixture assumption

9. MULTIMAT 2011 - 5-9 septembre 2011 9 1. Presentation of the model Inclusion of diffusion and capillary effects. with : Viscous stress tensor Capillary stress tensor (body force formulation) Heat flux

10. MULTIMAT 2011 - 5-9 septembre 2011 10 1. Presentation of the model Approximate enthalpy equation for low Mach flows Neglecting viscous and capillary effects, the energy equation is equivalent to : which is the Eulerian formulation of the well-known thermodynamic relation : For low Mach number flow, with imposed pressure on one of the boundaries, one generally has : 1/  dp <<  q, and therefore the energy equation can be replaced by the heat equation :

11. MULTIMAT 2011 - 5-9 septembre 2011 11 1. Presentation of the model Phase change modeling Phase change phenomena can be included in model (7) by just adding a relaxation source term in the r.h.s. : where  (U) is the thermodynamic equilibrium state corresponding to U, defined as the state which maximizes the mixture entropy under the constraints of imposed total volume, total mass and total energy : where This idea was first proposed in : HELLUY P., SEGUIN N., “Relaxation model of phase transition flows”, M2AN, Math. Model. Numer. Anal., vol. 40, num. 2, 2006, p. 331–352. In practice, the thermodynamic equilibrium time scale is assumed to be infinitely small compared to the macroscopic time scale  local thermodynamic equilibrium assumption.

12. MULTIMAT 2011 - 5-9 septembre 2011 12 OUTLINE OF THE PRESENTATION 1. Presentation of the model 2. Numerical method 3. Numerical test cases 4. Conclusion

13. MULTIMAT 2011 - 5-9 septembre 2011 13 2. Numerical scheme A finite volume relaxation scheme Each time step is divided in two stages : Transport step  Eulerian finite volume scheme Numerical flux on edge e K KeKe n e,K e Relaxation step  local thermodynamic equilibrium Note that, by construction, the second step is entropy diminishing.

14. MULTIMAT 2011 - 5-9 septembre 2011 14 2. Numerical scheme Expression of the hyperbolic numerical flux (isothermal case  no energy equation) Which expression choosing for p e and v e ? Remark : a similar idea has been proposed by Liou (AUSM+up scheme, JCP 2006) and by Li & Gu (all Mach Roe type scheme, JCP 2008) for the compressible gas dynamics system. Low Mach Scheme Centered scheme for pressure (see Dellacherie (2011) recent work on low Mach number schemes) Centered expression + stabilizing pressure term. Expression of the positive parameter  e will be given later. nene ULUL URUR e

15. MULTIMAT 2011 - 5-9 septembre 2011 15 2. Numerical scheme Semi-implicit version of the scheme (isothermal case) To avoid a restrictive stability condition based on the sound celerity, mass conservation equations are solved with a implicit scheme. An explicit scheme is used to compute the new velocity from the momentum equation : with and Newton algorithm

16. MULTIMAT 2011 - 5-9 septembre 2011 16 2. Numerical scheme Formal justification of the stabilizing role of “-  (p R - p L ) “ Let us consider the following modified system for isothermal flows Modified convective velocity Remark : the same property holds for the non isothermal case but with the entropy instead of the free energy. Proposition : the term has a stabilizing effect in the sense for that any smooth solution of (1’) one has the following free energy balance equation : where denotes the free energy of the mixture. Dissipative source term if  v is proportional to – grad(p)

17. MULTIMAT 2011 - 5-9 septembre 2011 17 2. Numerical scheme the semi-implicit scheme is entropic (in the sense of Lax). Stability theoretical result : Under the two following conditions (i) (ii) In practice, we take : How to choose the value of  e ? with cfl much larger than 1 for low mach number flows.

18. MULTIMAT 2011 - 5-9 septembre 2011 18 Where, respectively, denote the gas, respectively the liquid, mass numerical flux. In practice, two variants of the scheme can be used : an explicit scheme with respect to the fluid temperature, a fully implicit scheme with respect to all thermodynamic variables 2. Numerical scheme Discretization of the enthalpy equation To respect the maximum principle on the temperature, we use the following upwind scheme based on the sign of the mass fluxes :

19. MULTIMAT 2011 - 5-9 septembre 2011 19 2. Numerical scheme Relaxation step U *  U n+1 If both phases can coexist (gas – liquid thermodynamic equilibrium) , v and h are left unchanged during this step. We thus have : System of 3 equations and 3 unknowns else only one phase can be present in the cell at the end of the time step  Remark : in practice, for numerical purpose, a minimal lower value is imposed for gas and liquid mass fractions.

20. MULTIMAT 2011 - 5-9 septembre 2011 20 OUTLINE OF THE PRESENTATION 1. Presentation of the model 2. Numerical method 3. Numerical test cases 4. Conclusion

21. MULTIMAT 2011 - 5-9 septembre 2011 21 3. Numerical test cases Linear oscillations in a 2D rectangular tank ρ 1 =1 kg.m -3 ; c 1 =300 m.s -1 ρ 2 =1000 kg.m -3 ; c 2 =1200 m.s -1 Transverse acceleration : a 0 = 0.01 g Coarse cartesian grid : 40 X 20 Ma = 2 10 -5 Possibility to compute an analytical solution as a série expansion by potential flow theory. (see for example Landau & Lifschitz T6, fluid Mechanics) 22

22. MULTIMAT 2011 - 5-9 septembre 2011 22 3. Numerical test cases Linear oscillations in a 2D rectangular tank Second order low Mach scheme Second order Godunov type scheme Exact solution Numerical Scheme Godunov scheme Low Mach scheme Time step 0.00050.05 Total CPU time 201

23. MULTIMAT 2011 - 5-9 septembre 2011 23 3. Numerical test cases Dynamical test case : bubble rise inside a liquid : Sussman et al test case (Sussman, M. and Smereka, P. and Osher, S., A Level Set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics, 114, 146-159, 1994) Explicit Godunov type scheme with real EOS Cartesian mesh 140 X 233 Semi-implicit low Mach scheme with real EOS Cartesian mesh 140 X 233 Explicit Godunov type scheme with modified EOS Cartesian mesh 140 X 233

24. MULTIMAT 2011 - 5-9 septembre 2011 24 3. Numerical test cases Bubble rise inside a liquid : Sussman et al test case (Sussman, M. and Smereka, P. and Osher, S., A Level Set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics, 114, 146-159, 1994) Sussman et al Solution with Level Set method and incompressible model Usual Godunov type scheme with real EOS Semi-implicit low Mach scheme with real EOS

25. MULTIMAT 2011 - 5-9 septembre 2011 25 3. Numerical test cases Rayleigh-Bénard instability Critical Rayleigh number for instability : Wall with imposed temperature Liquid phase Gas phase Periodic boundary conditions g with

26. MULTIMAT 2011 - 5-9 septembre 2011 26 3. Numerical test cases Rayleigh-Bénard instability Stable Unstable

27. MULTIMAT 2011 - 5-9 septembre 2011 27 3. Numerical test cases Marangoni convection test case No gravity. Static contact angle :  = 90° Liquid phase Adiabatic Wall Wall with imposed temperature T = T 0 Adiabatic Wall Gas phase Wall with imposed temperature T = T 1 <T 0

28. MULTIMAT 2011 - 5-9 septembre 2011 28 3. Numerical test cases Marangoni convection test case Volume fraction field Temperature field Coarse grid Medium grid Fine grid

29. MULTIMAT 2011 - 5-9 septembre 2011 29 3. Numerical test cases 1D Evaporation test case Outlet with imposed pressure : p = p 0 Gas phase Wall with imposed heat flux q w Evaporation front  =  l, p = p 0, T = T sat (p 0 ), u = u I  =  v, p= p 0, u= 0, T = f(x) Approximate theoretical solution Liquid phase

30. MULTIMAT 2011 - 5-9 septembre 2011 30 3. Numerical test cases 1D Evaporation test case Interface position vs time for several values of L v and q w.

31. MULTIMAT 2011 - 5-9 septembre 2011 31 3. Numerical test cases 1D Evaporation test case

32. MULTIMAT 2011 - 5-9 septembre 2011 32 OUTLINE OF THE PRESENTATION 1. Presentation of the model 2. Numerical method 3. Numerical test cases 4. Conclusion

33. MULTIMAT 2011 - 5-9 septembre 2011 33 CONCLUSIONS AND FUTURE PROSPECTS An Eulerian two-fluid model with diffuse interface has been applied to the simulation of low Mach separated two-phase flows with heat and mass transfers. Using formal arguments, a simple semi-implicit low Mach scheme has been proposed for this model. For isothermal flows, this scheme has been proved to be entropy diminishing under a CFL condition which do not depend on the sound celerity. This methodology can be very easily implemented in existing industrial compressible CFD codes for multi-physics applications (work in progress at ONERA). It is a very interesting alternative to classical approaches based on one-fluid incompressible model with VOF or Level Set methods.

34. MULTIMAT 2011 - 5-9 septembre 2011 34 CONCLUSIONS AND FUTURE PROSPECTS This two-fluid approach has been successfully applied to several academic problems for low Mach two-phase flows. Future works will be devoted to the assessment of the method for more complex phase change problems. extension of the model to more complex physical problems : multi- component gas phase with an incondensable specie, 3 phases problems … parallelization of the code for 3D applications

35. MULTIMAT 2011 - 5-9 septembre 2011 35 Thank you for your attention …..

36. MULTIMAT 2011 - 5-9 septembre 2011 36 Back – up

37. MULTIMAT 2011 - 5-9 septembre 2011 37 with the mixture bulk density. 1. Presentation of the model Purely Dynamical model (inviscid Isothermal flow) To get a close model, it is necessary to give a relation between the “mixture” pressure p and the bulk gas density and the bulk liquid density. Remark: The gas volume fraction  is not explicitly transported in this model.

38. MULTIMAT 2011 - 5-9 septembre 2011 38 1. Presentation of the model Purely dynamical model Let  denote the gas volume fraction : Local pressure equilibrium between the two non miscible fluids where p = p g (  g ) and p = p l (  l ) denote the gas and liquid equation of state. Mixture EOS Remark : if the expressions of p g and p l are complex, p is only implicitly defined in function of the bulk densities.  is defined as the solution of

39. MULTIMAT 2011 - 5-9 septembre 2011 39 1. Presentation of the model Example : stiffened gas model for both fluids. Expression of the Gibbs potential for each fluid : Fluid i Equation of state Fluid i calorific law With these notations, system (5)-(6) is equivalent to : Mixture specific volume

40. MULTIMAT 2011 - 5-9 septembre 2011 40 1. Presentation of the model Other interpretation of closure equations (5)-(6) Important property : System (4) with pressure law given by (5)-(6) is thermodynamically consistent in the sense that it has a infinite set of convex entropies in the sense of Lax defined as : where  is an arbitrary concave function, e g, e l are the specific internal energies, implicitly defined by the solution of (5), s g and s l are the specific entropies, and are the real fluid densities. Closure relations (5)-(6) can also be interpreted as a direct consequence of the following modeling assumption for the mixture Gibbs potential : Ideal mixture assumption

41. MULTIMAT 2011 - 5-9 septembre 2011 41 Proposition : If p g and p l are strictly non decreasing functions, model (1)-(2)-(3) is hyperbolic and has a convex entropy in the sense of Lax defined as : 1. Presentation of the model where f g and f l are the free energy of the gas and liquid phases and are defined as : Lax entropy (convex function of the conservative variables) Entropy flux Purely dynamical model (3/3)

42. MULTIMAT 2011 - 5-9 septembre 2011 42 4. APPLICATIONS Linear oscillations in a 2D rectangular tank ρ 1 =1 kg.m -3 ; c 1 =300 m.s -1 ρ 2 =1000 kg.m -3 ; c 2 =1200 m.s -1 Transverse acceleration : a 0 = 0.01 g Coarse cartesian grid : 40 X 20 Ma = 2 10 -5 Possibility to compute an analytical solution as a série expansion by potential flow theory. (see for example Landau & Lifschitz T6, fluid Mechanics) 22

43. MULTIMAT 2011 - 5-9 septembre 2011 43 4. APPLICATIONS Linear oscillations in a 2D rectangular tank Second order low Mach scheme Second order Godunov type scheme Exact solution Numerical Scheme Godunov scheme Low Mach scheme Time step 0.00050.05 Total CPU time 201

44. MULTIMAT 2011 - 5-9 septembre 2011 44 1. Presentation of the model References R. ABGRALL, R. SAUREL. A simple method for compressible multifuid flows, SIAM J. Sci. Comput. 21 (3) : 1115-1145, (1999). 66 G. ALLAIRE, G. FACCANONI et S. KOKH, A strictly hyperbolic equilibrium phase transition model. C. R. Acad. Sci. Paris Sér. I, 344 pp. 135–140, 2007. CARO F., COQUEL F., JAMET D., KOKH S., “A Simple Finite-Volume Method for Compressible Isothermal Two-Phase Flows Simulation”, Int. J. on Finite Volumes, vol. 3, num. 1, 2006, p. 1–37. HELLUY P., SEGUIN N., “Relaxation model of phase transition flows”, M2AN, Math. Model. Numer. Anal., vol. 40, num. 2, 2006, p. 331–352. LE METAYER O., MASSONI J., SAUREL R., “Elaborating equations of state of a liquid and its vapor for two-phase flow models”, Int. J. of Th. Sci., vol. 43, num. 3, 2004, p. 265–276. G. CHANTEPERDRIX, JP VILA, P. VILLEDIEU, A compressible model for separated two-phase flow computations, FEDSM02, 14-18 July, Montreal, Quebec, Canada, 2002