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Relaxation models of phase transition flows

Published online by Cambridge University Press:  21 June 2006

Philippe Helluy  and
Nicolas Seguin
Philippe Helluy
Affiliation:
ISITV/MNC, BP 56, 83162 La Valette Cedex, France. helluy@univ-tln.fr
Nicolas Seguin
Affiliation:
Laboratoire J.-L. Lions, Université Paris VI, Boite courrier 187, 75252 Paris Cedex 05, France.
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Abstract

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 optimizationproblem 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.

Keywords

Finite volumeentropy optimizationrelaxationphase transitionreactive flowscritical point.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 2 , March 2006 , pp. 331 - 352
Copyright
© EDP Sciences, SMAI, 2006

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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. CrossRef
T. Barberon and P. Helluy, 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).
Barberon, T. and Helluy, P., Finite volume simulation of cavitating flows. Comput. Fluids 34 (2005) 832–858. CrossRef
Barberon, T., Helluy, P. and Rouy, S., Practical computation of axisymmetrical multifluid flows. Int. J. on Finite Volumes 1 (2003) 1–34. http://averoes.math.univ-paris13.fr/IJFV
Bouchut, F., A reduced stability condition for nonlinear relaxation to conservation laws. J. Hyper. Diff. Eqns 1 (2004) 149–170. CrossRef
Brenier, Y., Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal. 21 (1984) 1013–1037. CrossRef
Brenier, Y., Un algorithme rapide pour le calcul de transformées de Legendre-Fenchel discrètes. C.R. Acad. Sci. Paris Sér. I Math. 308 (1989) 587–589.
H.B. Callen, Thermodynamics and an introduction to thermostatistics, second edition. Wiley and Sons (1985).
F. Caro, Modélisation et simulation numérique des transitions de phase liquide-vapeur. Ph.D. thesis, École Polytechnique, Paris, France (November 2004).
G. Chanteperdrix, P. Villedieu, J.-P. Vila, A compressible model for separated two-phase flows computations. In ASME Fluids Engineering Division Summer Meeting. ASME, Montreal, Canada (July 2002).
Chen, G.Q., David Le, C.vermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1994) 787–830. CrossRef
J.-P. Croisille, 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).
Dellacherie, S., Relaxation schemes for the multicomponent Euler system. ESAIM: M2AN 37 (2003) 909–936. CrossRef
L.C. Evans, Entropy and partial differential equations (2004). http://math.berkeley.edu/~evans/entropy.and.PDE.pdf
Harten, A., Lax, P.D. and Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35–61. CrossRef
Hayes, B.T. and LeFloch, P.G., Nonclassical shocks and kinetic relations: strictly hyperbolic systems. SIAM J. Math. Anal. 31 (2000) 941–991 (electronic). CrossRef
J.-B. Hiriart-Urruty, Optimisation et analyse convexe. Mathématiques, Presses Universitaires de France, Paris (1998).
J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis. Grundlehren Text Editions, Springer-Verlag, Berlin (2001).
S. Jaouen, Étude mathématique et numérique de stabilité pour des modèles hydrodynamiques avec transition de phase. Ph.D. thesis, Université Paris VI (November 2001).
L. Landau and E. Lifchitz, Physique statistique. Physique théorique, Ellipses, Paris (1994).
P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, in CBMS Regional Conf. Ser. In Appl. Math. 11, Philadelphia, SIAM (1972).
LeFloch, P.G. and Rohde, C., High-order schemes, entropy inequalities, and nonclassical shocks. SIAM J. Numer. Anal. 37 (2000) 2023–2060. CrossRef
LeVeque, R.J. and Pelanti, M., A class of approximate Riemann solvers and their relation to relaxation schemes. J. Comput. Phys. 172 (2001) 572–591. CrossRef
Liu, T.P., The Riemann problem for general systems of conservation laws. J. Differ. Equations 56 (1975) 218–234. CrossRef
Lucet, Y., A fast computational algorithm for the Legendre-Fenchel transform. Comput. Optim. Appl. 6 (1996) 27–57. CrossRef
Lucet, Y., Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16 (1998) 171–185. CrossRef
Mazet, P.-A. and Bourdel, F., Multidimensional case of an entropic variational formulation of conservative hyperbolic systems. Rech. Aérospatiale 5 (1984) 369–378.
Menikoff, R. and Plohr, B.J., The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61 (1989) 75–130. CrossRef
Perthame, B., Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27 (1990) 1405–1421. CrossRef
Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425–467. CrossRef