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

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

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 … External Heat flux Gas phase Capillary raise Evaporation Liquid phase Separated two-phase flow with a free moving interface

OUTLINE OF THE PRESENTATION • 1.Presentation of the model • 2. Numerical method • 3. Numerical test cases • 4. Conclusion

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 …)

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

1.Presentation of the model Extension to non isothermal flows Let Tdenote the mixture temperature andathe gas volume fraction. a and Tare assumed to be the unique solution of the following system : Local mechanical equilibrium Local thermal equilibrium wherep = pg(rg,T), p = pl(rl,T)denote the gas and liquid EOS ande=eg(rg,T), e=el(rl,T)denote the gas and liquid colorific laws. The mixture EOS is then (implicitly) defined by :

1.Presentation of the model Other interpretation of closure equations (5)-(6) 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 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 eg, el are the gas and liquid specific internal energies (implicitly defined by the solution of (2)), sg and sl are specific entropies, and are the real densities.

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

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/r dp << dq, and therefore the energy equation can be replaced by the heat equation :

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 P(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 In practice, the thermodynamic equilibrium time scale is assumed to be infinitely small compared to the macroscopic time scale  local thermodynamic equilibrium assumption. • 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.

2. Numerical scheme K Transport step Eulerian finite volume scheme e ne,K Ke Numerical flux on edgee Relaxation step  local thermodynamic equilibrium Note that, by construction, the second step is entropy diminishing. A finite volume relaxation scheme Each time step is divided in two stages :

2. Numerical scheme ne e UR UL Centered scheme for pressure (see Dellacherie (2011) recent work on low Mach number schemes) Centered expression + stabilizing pressure term. Expression of the positive parameter ge will be given later. Expression of the hyperbolic numerical flux (isothermal case  no energy equation) Low Mach Scheme Which expression choosing for pe and ve ? 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.

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

2. Numerical scheme 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. Formal justification of the stabilizing role of “- g (pR - pL) “ Let us consider the following modified system for isothermal flows Modified convective velocity Dissipative source term if dv is proportional to – grad(p) Remark : the same property holds for the non isothermal case but with the entropy instead of the free energy.

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

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 : • 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 toall thermodynamic variables

2. Numerical scheme Relaxation step U* Un+1 r, v and h are left unchanged during this step. We thus have : If both phases can coexist (gas – liquid thermodynamic equilibrium) 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.

3. Numerical test cases Linear oscillations in a 2D rectangular tank • ρ1=1 kg.m-3 ; c1=300 m.s-1 • ρ2=1000 kg.m-3 ; c2=1200 m.s-1 • Transverse acceleration : a0 = 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) r2

3. Numerical test cases Linear oscillations in a 2D rectangular tank Exact solution Second order low Mach scheme Second order Godunov type scheme

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) 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 Explicit Godunov type scheme with real EOS Cartesian mesh 140 X 233

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) Semi-implicit low Mach scheme with real EOS Sussman et al Solution with Level Set method and incompressible model Usual Godunov type scheme with real EOS

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

3. Numerical test cases Rayleigh-Bénard instability Unstable Stable Unstable Unstable Stable Stable

3. Numerical test cases Marangoni convection test case Adiabatic Wall Gas phase Wall with imposed temperature T = T0 Wall with imposed temperature T = T1<T0 Liquid phase Adiabatic Wall No gravity. Static contact angle : q = 90°

3. Numerical test cases Coarse grid Fine grid Medium grid Marangoni convection test case Temperature field Volume fraction field

3. Numerical test cases 1D Evaporation test case Evaporation front Outlet with imposed pressure : p = p0 Wall with imposed heat flux qw Liquid phase Gas phase r=rv , p= p0 , u= 0, T = f(x) r=rl , p = p0 , T = Tsat(p0), u = uI Approximate theoretical solution

3. Numerical test cases 1D Evaporation test case Interface position vs time for several values of Lv and qw.

3. Numerical test cases 1D Evaporation test case

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.

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

Thank you for your attention …..

Back – up

1.Presentation of the model Purely Dynamical model (inviscid Isothermal flow) with the mixture bulk density. To get a close model, it is necessary to give a relation betweenthe “mixture” pressure pand the bulk gas density and the bulk liquid density . Remark: The gas volume fraction a is not explicitly transported in this model.

1.Presentation of the model Purely dynamical model • Let adenote the gas volume fraction : a is defined as the solution of Local pressure equilibrium between the two non miscible fluids where p = pg(rg) and p = pl(rl)denote the gas and liquid equation of state. Mixture EOS Remark : if the expressions of pg and pl are complex, p is only implicitly defined in function of the bulk densities.

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

1.Presentation of the model Other interpretation of closure equations (5)-(6) 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 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 s is an arbitrary concave function, eg, el are the specific internal energies, implicitly defined by the solution of (5), sg and sl are the specific entropies, and are the real fluid densities.

1.Presentation of the model Purely dynamical model (3/3) Proposition : If pg and pl are strictly non decreasing functions, model (1)-(2)-(3) is hyperbolic and has a convex entropy in the sense of Lax defined as : Lax entropy (convex function of the conservative variables) Entropy flux wherefgand flare thefree energy of the gas and liquid phases and are defined as :

4. APPLICATIONS Linear oscillations in a 2D rectangular tank • ρ1=1 kg.m-3 ; c1=300 m.s-1 • ρ2=1000 kg.m-3 ; c2=1200 m.s-1 • Transverse acceleration : a0 = 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) r2

4. APPLICATIONS Linear oscillations in a 2D rectangular tank Exact solution Second order low Mach scheme Second order Godunov type scheme

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

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