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Relaxation schemes for the multicomponent Euler system

Published online by Cambridge University Press:  15 November 2003

Stéphane Dellacherie
Stéphane Dellacherie*
Affiliation:
Commissariat à l'Énergie Atomique, 91191 Gif-sur-Yvette, France. [email protected].
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Abstract

We show that it is possible to construct a class of entropicschemes for the multicomponent Euler system describing a gas or fluidhomogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. Afirst order Chapman–Enskog expansion shows that the relaxed systemformally converges when the relaxation frequencies go to the infinitytoward a multicomponent Navier–Stokes system with the classical Fick andNewton laws, with a thermal diffusion which can be assimilated to a Soret effect in the case of a fluid mixture,and with also a pressure diffusion or a density diffusion respectively for a gas or fluid mixture. We also discuss on the link between the convexity of the entropies of each species and the existence of the Chapman–Enskog expansion.

Keywords

Multicomponent Euler systemrelaxation schemeentropic schemeChapman–Enskog expansion.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 6 , November 2003 , pp. 909 - 936
Copyright
© EDP Sciences, SMAI, 2003

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References

Arun, In, Numerical evaluation of an energy relaxation method for inviscid real fluids. SIAM J. Sci. Comput. 21 (1999) 340–365.
Abgrall, R. and Saurel, R., A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425–467.
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
P. Andries, Étude d'équations modèles pour la simulation découlements raréfiés. Ph.D. thesis, Pierre and Marie Curie University, Paris VI, France (2000).
R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, §18.4. John Wiley edition (1960).
G. Chanteperdrix, P. Villedieu and J.P. Vila, Un modèle bifluide compressible pour la simulation numérique d'écoulements diphasiques à phases séparées. Preprint, ONERA report (2002).
Coquel, F. and Perthame, B., Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal. 35 (1998) 2223–2249. CrossRef
S. Chapman and T.G. Cowling, The Mathematical theory of non uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge University Press (1970, reedited in 1990).
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, Pierre and Marie Curie University, Paris VI, France (1990). Cf. also the technical note ONERA No. 1991–3.
Dellacherie, S., Sur le caractère entropique des schémas de relaxation appliqués à une équation d'état non classique. C. R. Acad. Sci. Paris Sér. I 332 (2001) 765–770. CrossRef
Dellacherie, S., On the Wang Chang-Uhlenbeck equations. Discrete Contin. Dynam. Systems-Series B 3 (2003) 229–253.
S. Dellacherie and N. Rency, Relations de fermeture pour le système des équations d'Euler multi-espèces. Construction et étude des schémas de relaxation en multi-espèces et en multi-constituants. CEA report R-5999 (2001).
Després, B., Inégalité entropique pour un solveur conservatif du système de la dynamique des gaz en coordonnées de Lagrange. C. R. Acad. Sci. Paris Sér. I 324 (1997) 1301–1306. CrossRef
E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, Appl. Math. Sci. 118 (1996).
K.E. Grew and T.L. Ibbs, Thermal diffusion in gases. Cambridge University Press (1952).
F. Lagoutière, Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants. Ph.D. thesis, Pierre and Marie Curie University, Paris VI, France (2000).
Montarnal, P. and Shu, C.W., Real gas computation using an energy relaxation method and high-order WENO schemes. J. Comput. Phys. 148 (1999) 59–80. CrossRef
Mulder, W., Osher, S. and Sethian, J.A., Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100 (1992) 209–228. CrossRef
Perthame, B., Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27 (1990) 1405–1421. CrossRef
Villedieu, P. and Mazet, P.A., Schémas cinétiques pour les équations d'Euler hors équilibre thermochimique. Édition Gauthier-Villars, La Recherche Aérospatiale 2 (1995) 85–102.
F. De Vuyst, Schémas non conservatifs et schémas cinétiques pour la simulation numérique d'écoulements hypersoniques non visqueux en déséquilibre thermochimique. Ph.D. thesis, Pierre and Marie Curie University, Paris VI, France (1994).
C.S. Wang Chang and G.E. Uhlenbeck, Transport phenomena in polyatomic gases. Report no. CM-681, University of Michigan Research Report, July (1951).