Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) no. 5, pp. 1029-1054.

In the present work we investigate the numerical simulation of liquid-vapor phase change in compressible flows. Each phase is modeled as a compressible fluid equipped with its own equation of state (EOS). We suppose that inter-phase equilibrium processes in the medium operate at a short time-scale compared to the other physical phenomena such as convection or thermal diffusion. This assumption provides an implicit definition of an equilibrium EOS for the two-phase medium. Within this framework, mass transfer is the result of local and instantaneous equilibria between both phases. The overall model is strictly hyperbolic. We examine properties of the equilibrium EOS and we propose a discretization strategy based on a finite-volume relaxation method. This method allows to cope with the implicit definition of the equilibrium EOS, even when the model involves complex EOS's for the pure phases. We present two-dimensional numerical simulations that shows that the model is able to reproduce mechanism such as phase disappearance and nucleation.

Classification : 76T10,  76N10,  65M08
Mots clés : compressible flows, two-phase flows, hyperbolic systems, phase change, relaxation method
     author = {Faccanoni, Gloria and Kokh, Samuel and Allaire, Gr\'egoire},
     title = {Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {1029--1054},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {5},
     year = {2012},
     doi = {10.1051/m2an/2011069},
     zbl = {1267.76110},
     mrnumber = {2916371},
     language = {en},
     url = {}
Faccanoni, Gloria; Kokh, Samuel; Allaire, Grégoire. Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) no. 5, pp. 1029-1054. doi : 10.1051/m2an/2011069.

[1] G. Allaire, S. Clerc and S. Kokh, A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2002) 577-616. | MR 1927402 | Zbl 1169.76407

[2] G. Allaire, G. Faccanoni and S. Kokh, A strictly hyperbolic equilibrium phase transition model. C. R. Acad. Sci. Paris, Sér. I 344 (2007) 135-140. | MR 2288604 | Zbl 1109.35066

[3] K. Annamalai and I.K. Puri, Advanced thermodynamics engineering. CRC Press (2002). | Zbl 1003.80001

[4] Th. Barberon and Ph. Helluy, Finite volume simulations of cavitating flows. Comput. Fluids 34 (2005) 832-858. | Zbl 1134.76392

[5] J. Benoist and J.-B. Hiriart-Urruty, What is the subdifferential of the closed convex hull of a function? SIAM J. Math. Anal. 27 (1996) 1661-1679. | MR 1416513 | Zbl 0876.49018

[6] S. Benzoni Gavage, Stability of multi-dimensional phase transitions in a Van der Waals fluid. Nonlinear Anal. 31 (1998) 243-263. | MR 1487544 | Zbl 0928.76015

[7] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004). | MR 2128209 | Zbl 1086.65091

[8] J.U. Brackbill, D.B. Kothe and C. Zemach, A continuum method for modeling surface tension. J. Comput. Phys. 100 (1992) 335-354. | MR 1167749 | Zbl 0775.76110

[9] H.B. Callen, Thermodynamics and an introduction to thermostatistics. John Wiley & sons, 2nd edition (1985). | Zbl 0095.23301

[10] F. Caro, Modélisation et simulation numérique des transitions de phase liquide-vapeur. Ph.D. thesis, École Polytechnique (2004).

[11] F. Caro, F. Coquel, D. Jamet and S. Kokh, A simple finite-volume method for compressible isothermal two-phase flows simulation. International Journal on Finite Volumes (2006). | MR 2465464

[12] G. Chen, C.D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1992) 787-830. | MR 1280989 | Zbl 0806.35112

[13] F. Coquel and B. Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluids dynamics. SIAM J. Numer. Anal. 35 (1998) 2223-2249. | MR 1655844 | Zbl 0960.76051

[14] J.-M. Delhaye, M. Giot and M.L. Riethmuller, Thermohydraulics of two-phase systems for industrial design and nuclear engineering. Hemisphere Publishing Corporation (1981).

[15] J.-M. Delhaye, M. Giot, L. Mahias, P. Raymond and C. Rénault, Thermohydraulique des réacteurs. EDP Sciences (1998).

[16] V.K. Dhir, Boiling heat transfer. Ann. Rev. Fluid Mech. 30 (1998) 365-401.

[17] J.E. Dunn and J. Serrin, On the thermomechanics of interstitial working. Arch. Rational Mech. Anal. 88 (1985) 95-133. | MR 775366 | Zbl 0582.73004

[18] G. Faccanoni, Étude d'un modèle fin de changement de phase liquide-vapeur. Contribution à l'étude de la crise d'ébullition. Ph.D. thesis, École Polytechnique, France (2008).

[19] G. Faccanoni, S. Kokh and G. Allaire, Numerical simulation with finite volume of dynamic liquid-vapor phase transition, Finite Volumes for Complex Applications V. ISTE and Wiley (2008) 391-398. | MR 2451432

[20] G. Faccanoni, G. Allaire and S. Kokh, Modelling and numerical simulation of liquid-vapor phase transition, in Conf. Proc. of EUROTHERM-84, Seminar on Thermodynamics of Phase Changes, Namur (2009).

[21] G. Faccanoni, S. Kokh and G. Allaire, Approximation of liquid-vapor phase transition for compressible fluids with tabulated EOS. C. R. Acad. Sci. Paris Sér. I 348 (2010) 473-478. | MR 2607042 | Zbl 1258.76162

[22] H. Fan, One phase Riemann problem and wave interactions in systems of conservation laws of mixed type. SIAM J. Math. Anal. 24 (1993) 840-865. | MR 1226854 | Zbl 0804.35074

[23] H. Fan, Traveling waves, Riemann problems and computations of a model of the dynamics of liquid/vapor phase transitions. J. Differ. Equ. 150 (1998) 385-437. | MR 1658672 | Zbl 0927.76045

[24] H. Fan and M. Slemrod, The Riemann problem for systems of conservation laws of mixed type, in Conf. Proc. on Shock Induced Transitions and Phase Structure in General Media Institute of Mathematics and its Applications. Minneapolis (1990) 61-91. | MR 1240333 | Zbl 0807.76031

[25] C. Fouillet, Généralisation à des mélanges binaires de la méthode du second gradient et application à la simulation numérique directe de l'ébullition nuclée. Ph.D. thesis, Université Paris 6 (2003).

[26] E. Godlewski and N. Seguin, The Riemann problem for a simple model of phase transition. Commun. Math. Sci. 4 (2006) 227-247. | MR 2204085 | Zbl 1103.35070

[27] H. Gouin, Utilization of the second gradient theory in continuum mechanics to study the motion and thermodynamics of liquid-vapor interfaces. Physicochemical Hydrodynamics - Interfacial Phenomena B 174 (1987) 667-682.

[28] W. Greiner, L. Neise and H. Stöcker, Thermodynamics and statistical mechanics. Springer (1997). | Zbl 0823.73001

[29] Ph. Helluy, Quelques exemples de méthodes numériques récentes pour le calcul des écoulements multiphasiques. Mémoire d'habilitation à diriger des recherches (2005).

[30] Ph. Helluy and H. Mathis, Pressure laws and fast Legendre transform. Math. Models Methods Appl. Sci. to appear. | MR 2795505 | Zbl 1268.76054

[31] Ph. Helluy and N. Seguin, Relaxation models of phase transition flows. ESAIM : M2AN 40 (2006) 331-352. | Numdam | MR 2241826 | Zbl 1108.76078

[32] J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis. Grundlehren Text Editions, Springer-Verlag, Berlin (2001). | MR 1865628 | Zbl 0998.49001

[33] D. Jamet, O. Lebaigue, N. Coutris and J.-M. Delhaye, The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change. J. Comput. Phys. 169 (2001) 624-651. | MR 1836527 | Zbl 1047.76098

[34] S. Jaouen, Étude mathématique et numérique de stabilité pour des modèles hydrodynamiques avec transition de phase. Ph.D. thesis, Université Paris 6, France (2001).

[35] S. Jin and C.D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126 (1996) 449-467. | MR 1404381 | Zbl 0860.65089

[36] S. Kokh, Aspects numériques et théoriques de la modélisation des écoulements diphasiques compressibles par des méthodes de capture d'interfaces. Ph.D. thesis, Université Paris 6 (2001).

[37] D.J. Korteweg, Sur la forme que prennent les équations des mouvements des fluides si l'on tient compte des forces capillaires par des variations de densité. Arch. Néer. Sci. Exactes Sér. II 6 (1901) 1-24. | JFM 32.0756.02

[38] P.G. Lefloch, Hyperbolic systems of conservation laws. Birkhäuser Verlag, Basel (2002). | MR 1927887 | Zbl 1019.35001

[39] O. Le Métayer, J. Massoni and R. Saurel, Elaborating equations of state of a liquid and its vapor for two-phase flow models. Int. J. Thermal Sci. 43 (2004) 265-276.

[40] O. Le Métayer, J. Massoni and R. Saurel, Modelling evaporation fronts with reactive Riemann solvers. J. Comput. Phys. 205 (2005) 567-610. | MR 2134994 | Zbl 1088.76051

[41] E.W. Lemmon, M.O. Mclinden and D.G. Friend, Thermophysical properties of fluid systems, in WebBook de Chimie NIST, Base de Données Standard de Référence NIST Numéro 69, National Institute of Standards and Technology, edited by P.J. Linstrom and W.G. Mallard. Gaithersburg MD, 20899,

[42] R.J. Leveque, Finite Volume methods for hyperbolic problems. Cambridge University Press, Cambridge. Appl. Math. (2002). | MR 1925043 | Zbl 1010.65040

[43] T.P. Liu, Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108 (1987) 153-175. | MR 872145 | Zbl 0633.35049

[44] T. Matolcsi, On the classification of phase transitions. Z. Angew. Math. Phys. 47 (1996) 837-857. | MR 1424031 | Zbl 0881.73008

[45] R. Menikoff and B. Plohr, The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61 (1989) 75-130. | MR 977944 | Zbl 1129.35439

[46] S. Nukiyama, The maximum and minimum values of the heat Q transmitted from metal to boiling water under atmospheric pressure. Int. J. Heat Mass Transfer 9 (1966) 1419-1433. (English translation of the original paper published in J. Jpn Soc. Mech. Eng. 37 (1934) 367-374).

[47] F. Petitpas, E. Franquet, R. Saurel and O. Le Métayer, A relaxation-projection method for compressible flows. II. Artificial heat exchanges for multiphase shocks. J. Comput. Phys. 225 (2007) 2214-2248. | MR 2349701 | Zbl 1183.76831

[48] P. Ruyer, Modèle de champ de phase pour l'étude de l'ébullition. Ph.D. thesis, École Polytechnique (2006).

[49] R. Saurel, J.-P. Cocchi and P.-B. Butlers, Numerical study of cavitation in the wake of a hypervelocity underwater projectile. J. Propuls. Power 15 (1999) 513-522.

[50] R. Saurel, F. Petitpas and R. Abgrall, Modelling phase transition in metastable liquids : application to cavitating and flashing flows. J. Fluid Mech. 607 (2008) 313-350. | MR 2436919 | Zbl 1147.76060

[51] M. Shearer, Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type. Proc. R. Soc. Edinb. 93 (1983) 133-244. | MR 688788 | Zbl 0511.35059

[52] M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 81 (1983) 301-315. | MR 683192 | Zbl 0505.76082

[53] L. Truskinovsky, Kinks versus shocks, in Shock induced transitions and phase structures in general media, edited by R. Fosdick et al. Springer Verlag, Berlin (1991). | MR 1240340 | Zbl 0818.76036

[54] P. Van Carey, Liquid-vapor phase-change phenomena. Taylor and Francis (1992).

[55] A. Voß, Exact Riemann solution for the Euler equations with nonconvex and nonsmooth equation of State. Ph.D. thesis, RWTH-Aachen (2004).