Relaxation models of phase transition flows
ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 2, pp. 331-352.

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.

DOI : 10.1051/m2an:2006015
Classification : 76M12, 65M12
Mots clés : finite volume, entropy optimization, relaxation, phase transition, reactive flows, critical point
Helluy, Philippe  ; Seguin, Nicolas 1

1 Laboratoire J.-L. Lions, Université Paris VI, France.
@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] 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.

[2] 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).

[3] T. Barberon and P. Helluy, Finite volume simulation of cavitating flows. Comput. Fluids 34 (2005) 832-858. | Zbl

[4] T. Barberon, P. Helluy and S. Rouy, Practical computation of axisymmetrical multifluid flows. Int. J. on Finite Volumes 1 (2003) 1-34. http://averoes.math.univ-paris13.fr/IJFV

[5] F. Bouchut, A reduced stability condition for nonlinear relaxation to conservation laws. J. Hyper. Diff. Eqns 1 (2004) 149-170. | Zbl

[6] Y. Brenier, Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal. 21 (1984) 1013-1037. | Zbl

[7] Y. Brenier, 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] H.B. Callen, Thermodynamics and an introduction to thermostatistics, second edition. Wiley and Sons (1985). | Zbl

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

[10] 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).

[11] G.Q. Chen, C. David Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1994) 787-830. | Zbl

[12] 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).

[13] S. Dellacherie, Relaxation schemes for the multicomponent Euler system. ESAIM: M2AN 37 (2003) 909-936. | Numdam | Zbl

[14] L.C. Evans, Entropy and partial differential equations | MR

[15] A. Harten, P.D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61. | Zbl

[16] B.T. Hayes and P.G. Lefloch, Nonclassical shocks and kinetic relations: strictly hyperbolic systems. SIAM J. Math. Anal. 31 (2000) 941-991 (electronic). | Zbl

[17] J.-B. Hiriart-Urruty, Optimisation et analyse convexe. Mathématiques, Presses Universitaires de France, Paris (1998). | MR | Zbl

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

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

[20] L. Landau and E. Lifchitz, Physique statistique. Physique théorique, Ellipses, Paris (1994).

[21] 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). | MR | Zbl

[22] P.G. Lefloch and C. Rohde, High-order schemes, entropy inequalities, and nonclassical shocks. SIAM J. Numer. Anal. 37 (2000) 2023-2060. | Zbl

[23] R.J. Leveque and M. Pelanti, A class of approximate Riemann solvers and their relation to relaxation schemes. J. Comput. Phys. 172 (2001) 572-591. | Zbl

[24] T.P. Liu, The Riemann problem for general systems of conservation laws. J. Differ. Equations 56 (1975) 218-234. | Zbl

[25] Y. Lucet, A fast computational algorithm for the Legendre-Fenchel transform. Comput. Optim. Appl. 6 (1996) 27-57. | Zbl

[26] Y. Lucet, Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16 (1998) 171-185. | Zbl

[27] P.-A. Mazet and F. Bourdel, Multidimensional case of an entropic variational formulation of conservative hyperbolic systems. Rech. Aérospatiale 5 (1984) 369-378. | Zbl

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

[29] B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27 (1990) 1405-1421. | Zbl

[30] R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425-467. | Zbl

Cité par Sources :