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Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium∗

Published online by Cambridge University Press:  13 February 2012

Gloria Faccanoni ,
Samuel Kokh  and
Grégoire Allaire
Gloria Faccanoni
Affiliation:
IMATH – Université du Sud Toulon-Var, Avenue de l’Université, 83957 La Garde, France. faccanon@univ-tln.fr
Samuel Kokh
Affiliation:
DEN/DANS/DM2S/SFME/LETR, Commissariat à l’Énergie Atomique Saclay, 91191 Gif-sur-Yvette, France; samuel.kokh@cea.fr
Grégoire Allaire
Affiliation:
Conseiller Scientifique du DM2S – Commissariat à l’Énergie Atomique Saclay, 91191 Gif-sur-Yvette, France CMAP, École Polytechnique, CNRS, 91128 Palaiseau, France; allaire@cmap.polytechnique.fr
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Abstract

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.

Keywords

Compressible flowstwo-phase flowshyperbolic systemsphase changerelaxation method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1029 - 1054
Copyright
© EDP Sciences, SMAI, 2012

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References

Allaire, G., Clerc, S. and Kokh, S., A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2002) 577–616. Google Scholar
Allaire, G., Faccanoni, G. and Kokh, S., A strictly hyperbolic equilibrium phase transition model. C. R. Acad. Sci. Paris, Sér. I 344 (2007) 135–140. Google Scholar
K. Annamalai and I.K. Puri, Advanced thermodynamics engineering. CRC Press (2002).
Barberon, Th. and Helluy, Ph., Finite volume simulations of cavitating flows. Comput. Fluids 34 (2005) 832–858. Google Scholar
Benoist, J. and Hiriart-Urruty, J.-B., What is the subdifferential of the closed convex hull of a function? SIAM J. Math. Anal. 27 (1996) 1661–1679. Google Scholar
Benzoni Gavage, S., Stability of multi-dimensional phase transitions in a Van der Waals fluid. Nonlinear Anal. 31 (1998) 243–263. Google Scholar
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).
Brackbill, J.U., Kothe, D.B. and Zemach, C., A continuum method for modeling surface tension. J. Comput. Phys. 100 (1992) 335–354. Google Scholar
H.B. Callen, Thermodynamics and an introduction to thermostatistics. John Wiley & sons, 2nd edition (1985).
F. Caro, Modélisation et simulation numérique des transitions de phase liquide-vapeur. Ph.D. thesis, École Polytechnique (2004). http://www.imprimerie.polytechnique.fr/Theses/Files/caro.pdf.
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). http://www.latp.univ-mrs.fr/IJFVDB/ijfv-caro-coquel-jamet-kokh.pdf.
Chen, G., Levermore, C.D. and Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1992) 787–830. Google Scholar
Coquel, F. and Perthame, B., Relaxation of energy and approximate Riemann solvers for general pressure laws in fluids dynamics. SIAM J. Numer. Anal. 35 (1998) 2223–2249. Google Scholar
J.-M. Delhaye, M. Giot and M.L. Riethmuller, Thermohydraulics of two-phase systems for industrial design and nuclear engineering. Hemisphere Publishing Corporation (1981).
J.-M. Delhaye, M. Giot, L. Mahias, P. Raymond and C. Rénault, Thermohydraulique des réacteurs. EDP Sciences (1998).
Dhir, V.K., Boiling heat transfer. Ann. Rev. Fluid Mech. 30 (1998) 365–401. Google Scholar
Dunn, J.E. and Serrin, J., On the thermomechanics of interstitial working. Arch. Rational Mech. Anal. 88 (1985) 95–133. Google Scholar
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). http://pastel.paristech.org/4785/.
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.
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).
Faccanoni, G., Kokh, S. and Allaire, G., Approximation of liquid-vapor phase transition for compressible fluids with tabulated EOS. C. R. Acad. Sci. Paris Sér. I 348 (2010) 473–478. Google Scholar
Fan, H., One phase Riemann problem and wave interactions in systems of conservation laws of mixed type. SIAM J. Math. Anal. 24 (1993) 840–865. Google Scholar
Fan, H., Traveling waves, Riemann problems and computations of a model of the dynamics of liquid/vapor phase transitions. J. Differ. Equ. 150 (1998) 385–437. Google Scholar
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.
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).
Godlewski, E. and Seguin, N., The Riemann problem for a simple model of phase transition. Commun. Math. Sci. 4 (2006) 227–247. Google Scholar
Gouin, H., 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. CrossRefGoogle Scholar
W. Greiner, L. Neise and H. Stöcker, Thermodynamics and statistical mechanics. Springer (1997).
Ph. Helluy, Quelques exemples de méthodes numériques récentes pour le calcul des écoulements multiphasiques. Mémoire d’habilitation à diriger des recherches (2005).
Ph. Helluy and H. Mathis, Pressure laws and fast Legendre transform. Math. Models Methods Appl. Sci. to appear.
Helluy, Ph. and Seguin, N., Relaxation models of phase transition flows. ESAIM : M2AN 40 (2006) 331–352. Google Scholar
J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis. Grundlehren Text Editions, Springer-Verlag, Berlin (2001).
Jamet, D., Lebaigue, O., Coutris, N. and Delhaye, J.-M., The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change. J. Comput. Phys. 169 (2001) 624–651. Google Scholar
S. Jaouen, Étude mathématique et numérique de stabilité pour des modèles hydrodynamiques avec transition de phase. Ph.D. thesis, Université Paris 6, France (2001).
Jin, S. and Levermore, C.D., Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126 (1996) 449–467. Google Scholar
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).
Korteweg, D.J., 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. Google Scholar
P.G. LeFloch, Hyperbolic systems of conservation laws. Birkhäuser Verlag, Basel (2002).
Le Métayer, O., Massoni, J. and Saurel, R., Elaborating equations of state of a liquid and its vapor for two-phase flow models. Int. J. Thermal Sci. 43 (2004) 265–276. Google Scholar
Le Métayer, O., Massoni, J. and Saurel, R., Modelling evaporation fronts with reactive Riemann solvers. J. Comput. Phys. 205 (2005) 567–610. Google Scholar
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, http://webbook.nist.gov.
R.J. LeVeque, Finite Volume methods for hyperbolic problems. Cambridge University Press, Cambridge. Appl. Math. (2002).
Liu, T.P., Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108 (1987) 153–175. Google Scholar
Matolcsi, T., On the classification of phase transitions. Z. Angew. Math. Phys. 47 (1996) 837–857. Google Scholar
Menikoff, R. and Plohr, B., The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61 (1989) 75–130. Google Scholar
Nukiyama, S., 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). Google Scholar
Petitpas, F., Franquet, E., Saurel, R. and Le Métayer, O., A relaxation-projection method for compressible flows. II. Artificial heat exchanges for multiphase shocks. J. Comput. Phys. 225 (2007) 2214–2248. Google Scholar
P. Ruyer, Modèle de champ de phase pour l’étude de l’ébullition. Ph.D. thesis, École Polytechnique (2006). www.imprimerie.polytechnique.fr/Theses/Files/Ruyer.pdf.
Saurel, R., Cocchi, J.-P. and Butlers, P.-B., Numerical study of cavitation in the wake of a hypervelocity underwater projectile. J. Propuls. Power 15 (1999) 513–522. Google Scholar
Saurel, R., Petitpas, F. and Abgrall, R., Modelling phase transition in metastable liquids : application to cavitating and flashing flows. J. Fluid Mech. 607 (2008) 313–350. Google Scholar
Shearer, M., Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type. Proc. R. Soc. Edinb. 93 (1983) 133–244. CrossRefGoogle Scholar
Slemrod, M., Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 81 (1983) 301–315. Google Scholar
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).
P. Van Carey, Liquid-vapor phase-change phenomena. Taylor and Francis (1992).
A. Voß, Exact Riemann solution for the Euler equations with nonconvex and nonsmooth equation of State. Ph.D. thesis, RWTH-Aachen (2004). http://www.it-voss.com/papers/thesis-voss-030205-128-final.pdf.