In this work, we propose a general framework for the construction of pressure law for phase transition. These equations of state are particularly suitable for a use in a relaxation finite volume scheme. The approach is based on a constrained convex optimization problem on the mixture entropy. It is valid for both miscible and immiscible mixtures. We also propose a rough pressure law for modelling a super-critical fluid.
Mots-clés : finite volume, entropy optimization, relaxation, phase transition, reactive flows, critical point
@article{M2AN_2006__40_2_331_0,
author = {Helluy, Philippe and Seguin, Nicolas},
title = {Relaxation models of phase transition flows},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {331--352},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
year = {2006},
doi = {10.1051/m2an:2006015},
mrnumber = {2241826},
zbl = {1108.76078},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2006015/}
}
TY - JOUR AU - Helluy, Philippe AU - Seguin, Nicolas TI - Relaxation models of phase transition flows JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 331 EP - 352 VL - 40 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2006015/ DO - 10.1051/m2an:2006015 LA - en ID - M2AN_2006__40_2_331_0 ER -
%0 Journal Article %A Helluy, Philippe %A Seguin, Nicolas %T Relaxation models of phase transition flows %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 331-352 %V 40 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2006015/ %R 10.1051/m2an:2006015 %G en %F M2AN_2006__40_2_331_0
Helluy, Philippe; Seguin, Nicolas. Relaxation models of phase transition flows. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 2, pp. 331-352. doi : 10.1051/m2an:2006015. http://www.numdam.org/articles/10.1051/m2an:2006015/
[1] , and, A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2002) 577-616.
[2] and, Finite volume simulations of cavitating flows. In Finite volumes for complex applications, III (Porquerolles, 2002), Lab. Anal. Topol. Probab. CNRS, Marseille (2002) 441-448 (electronic).
[3] and, Finite volume simulation of cavitating flows. Comput. Fluids 34 (2005) 832-858. | Zbl
[4] , and, Practical computation of axisymmetrical multifluid flows. Int. J. on Finite Volumes 1 (2003) 1-34. http://averoes.math.univ-paris13.fr/IJFV
[5] , A reduced stability condition for nonlinear relaxation to conservation laws. J. Hyper. Diff. Eqns 1 (2004) 149-170. | Zbl
[6] , Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal. 21 (1984) 1013-1037. | Zbl
[7] , Un algorithme rapide pour le calcul de transformées de Legendre-Fenchel discrètes. C.R. Acad. Sci. Paris Sér. I Math. 308 (1989) 587-589. | Zbl
[8] , Thermodynamics and an introduction to thermostatistics, second edition. Wiley and Sons (1985). | Zbl
[9] , Modélisation et simulation numérique des transitions de phase liquide-vapeur. Ph.D. thesis, École Polytechnique, Paris, France (November 2004).
[10] ,,, A compressible model for separated two-phase flows computations. In ASME Fluids Engineering Division Summer Meeting. ASME, Montreal, Canada (July 2002).
[11] , and, Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1994) 787-830. | Zbl
[12] , Contribution à l'étude théorique et à l'approximation par éléments finis du système hyperbolique de la dynamique des gaz multidimensionnelle et multiespèces. Ph.D. thesis, Université Paris VI, France (1991).
[13] , Relaxation schemes for the multicomponent Euler system. ESAIM: M2AN 37 (2003) 909-936. | Numdam | Zbl
[14] , Entropy and partial differential equations | MR
[15] , and, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61. | Zbl
[16] and, Nonclassical shocks and kinetic relations: strictly hyperbolic systems. SIAM J. Math. Anal. 31 (2000) 941-991 (electronic). | Zbl
[17] , Optimisation et analyse convexe. Mathématiques, Presses Universitaires de France, Paris (1998). | MR | Zbl
[18] and, Fundamentals of convex analysis. Grundlehren Text Editions, Springer-Verlag, Berlin (2001). | MR | Zbl
[19] , Étude mathématique et numérique de stabilité pour des modèles hydrodynamiques avec transition de phase. Ph.D. thesis, Université Paris VI (November 2001).
[20] and, Physique statistique. Physique théorique, Ellipses, Paris (1994).
[21] , Hyperbolic systems of conservation laws and the mathematical theory of shock waves, in CBMS Regional Conf. Ser. In Appl. Math. 11, Philadelphia, SIAM (1972). | MR | Zbl
[22] and, High-order schemes, entropy inequalities, and nonclassical shocks. SIAM J. Numer. Anal. 37 (2000) 2023-2060. | Zbl
[23] and, A class of approximate Riemann solvers and their relation to relaxation schemes. J. Comput. Phys. 172 (2001) 572-591. | Zbl
[24] , The Riemann problem for general systems of conservation laws. J. Differ. Equations 56 (1975) 218-234. | Zbl
[25] , A fast computational algorithm for the Legendre-Fenchel transform. Comput. Optim. Appl. 6 (1996) 27-57. | Zbl
[26] , Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16 (1998) 171-185. | Zbl
[27] and, Multidimensional case of an entropic variational formulation of conservative hyperbolic systems. Rech. Aérospatiale 5 (1984) 369-378. | Zbl
[28] and, The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61 (1989) 75-130. | Zbl
[29] , Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27 (1990) 1405-1421. | Zbl
[30] and, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425-467. | Zbl
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