Relaxation schemes for the multicomponent Euler system

Stéphane Dellacherie

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 6, page 909-936
  • ISSN: 0764-583X

Abstract

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We show that it is possible to construct a class of entropic schemes for the multicomponent Euler system describing a gas or fluid homogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. A first order Chapman–Enskog expansion shows that the relaxed system formally converges when the relaxation frequencies go to the infinity toward a multicomponent Navier–Stokes system with the classical Fick and Newton 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.

How to cite

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Dellacherie, Stéphane. "Relaxation schemes for the multicomponent Euler system." ESAIM: Mathematical Modelling and Numerical Analysis 37.6 (2010): 909-936. <http://eudml.org/doc/194198>.

@article{Dellacherie2010,
abstract = { We show that it is possible to construct a class of entropic schemes for the multicomponent Euler system describing a gas or fluid homogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. A first order Chapman–Enskog expansion shows that the relaxed system formally converges when the relaxation frequencies go to the infinity toward a multicomponent Navier–Stokes system with the classical Fick and Newton 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. },
author = {Dellacherie, Stéphane},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Multicomponent Euler system; relaxation scheme; entropic scheme; Chapman–Enskog expansion.; entropic schemes; first-order Chapman-Enskog expansion; hyperbolicity},
language = {eng},
month = {3},
number = {6},
pages = {909-936},
publisher = {EDP Sciences},
title = {Relaxation schemes for the multicomponent Euler system},
url = {http://eudml.org/doc/194198},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Dellacherie, Stéphane
TI - Relaxation schemes for the multicomponent Euler system
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 6
SP - 909
EP - 936
AB - We show that it is possible to construct a class of entropic schemes for the multicomponent Euler system describing a gas or fluid homogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. A first order Chapman–Enskog expansion shows that the relaxed system formally converges when the relaxation frequencies go to the infinity toward a multicomponent Navier–Stokes system with the classical Fick and Newton 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.
LA - eng
KW - Multicomponent Euler system; relaxation scheme; entropic scheme; Chapman–Enskog expansion.; entropic schemes; first-order Chapman-Enskog expansion; hyperbolicity
UR - http://eudml.org/doc/194198
ER -

References

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  1. Arun In, Numerical evaluation of an energy relaxation method for inviscid real fluids. SIAM J. Sci. Comput.21 (1999) 340–365.  
  2. R. Abgrall and R. Saurel, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys.150 (1999) 425–467.  
  3. 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.  
  4. P. Andries, Étude d'équations modèles pour la simulation découlements raréfiés. Ph.D. thesis, Pierre and Marie Curie University, Paris VI, France (2000).  
  5. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, §18.4. John Wiley edition (1960).  
  6. 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).  
  7. F. Coquel and B. Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal.35 (1998) 2223–2249.  
  8. 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).  
  9. 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.  
  10. S. Dellacherie, Sur le caractère entropique des schémas de relaxation appliqués à une équation d'état non classique. C. R. Acad. Sci. Paris Sér. I332 (2001) 765–770.  
  11. S. Dellacherie, On the Wang Chang-Uhlenbeck equations. Discrete Contin. Dynam. Systems-Series B3 (2003) 229–253.  
  12. 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).  
  13. B. Després, Inégalité entropique pour un solveur conservatif du système de la dynamique des gaz en coordonnées de Lagrange. C. R. Acad. Sci. Paris Sér. I324 (1997) 1301–1306.  
  14. E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, Appl. Math. Sci. 118 (1996).  
  15. K.E. Grew and T.L. Ibbs, Thermal diffusion in gases. Cambridge University Press (1952).  
  16. 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).  
  17. P. Montarnal and C.W. Shu, Real gas computation using an energy relaxation method and high-order WENO schemes. J. Comput. Phys.148 (1999) 59–80.  
  18. W. Mulder, S. Osher and J.A. Sethian, Computing interface motion in compressible gas dynamics. J. Comput. Phys.100 (1992) 209–228.  
  19. B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal.27 (1990) 1405–1421.  
  20. P. Villedieu and P.A. Mazet, Schémas cinétiques pour les équations d'Euler hors équilibre thermochimique. Édition Gauthier-Villars, La Recherche Aérospatiale2 (1995) 85–102.  
  21. 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).  
  22. C.S. Wang Chang and G.E. Uhlenbeck, Transport phenomena in polyatomic gases. Report no. CM-681, University of Michigan Research Report, July (1951).  

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