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A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems

Published online by Cambridge University Press:  15 January 2003

Thierry Gallouët ,
Jean-Marc Hérard  and
Nicolas Seguin
Thierry Gallouët
Affiliation:
LATP-UMR CNRS 6632, C.M.I., Université de Provence, 13453 Marseille Cedex 13, France. gallouet@cmi.univ-mrs.fr., herard@cmi.univ-mrs.fr.
Jean-Marc Hérard
Affiliation:
LATP-UMR CNRS 6632, C.M.I., Université de Provence, 13453 Marseille Cedex 13, France. gallouet@cmi.univ-mrs.fr., herard@cmi.univ-mrs.fr. Département MFTT, Électricité de France - R&D, 78401 Chatou Cedex, France. herard@chi80bk.der.edf.fr., seguin@chi80bk.der.edf.fr.
Nicolas Seguin
Affiliation:
LATP-UMR CNRS 6632, C.M.I., Université de Provence, 13453 Marseille Cedex 13, France. gallouet@cmi.univ-mrs.fr., herard@cmi.univ-mrs.fr. Département MFTT, Électricité de France - R&D, 78401 Chatou Cedex, France. herard@chi80bk.der.edf.fr., seguin@chi80bk.der.edf.fr.
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Abstract

The present paper is devoted to the computation of single phase or two phase flows using the single-fluid approach. Governing equations rely on Euler equations which may be supplemented by conservation laws for mass species. Emphasis is given on numerical modelling with help of Godunov scheme or an approximate form of Godunov scheme called VFRoe-ncv based on velocity and pressure variables. Three distinct classes of closure laws to express the internal energy in terms of pressure, density and additional variables are exhibited. It is shown first that a standard conservative formulation of above mentioned schemes enables to predict “perfectly” unsteady contact discontinuities on coarse meshes, when the equation of state (EOS) belongs to the first class. On the basis of previous work issuing from literature, an almost conservative though modified version of the scheme is proposed to deal with EOS in the second or third class. Numerical evidence shows that the accuracy of approximations of discontinuous solutions of standard Riemann problems is strengthened on coarse meshes, but that convergence towards the right shock solution may be lost in some cases involving complex EOS in the third class. Hence, a blend scheme is eventually proposed to benefit from both properties (“perfect” representation of contact discontinuities on coarse meshes, and correct convergence on finer meshes). Computational results based on an approximate Godunov scheme are provided and discussed.

Keywords

Godunov schemeEuler systemcontact discontinuitiesthermodynamicsconservative schemes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 6 , November 2002 , pp. 1133 - 1159
Copyright
© EDP Sciences, SMAI, 2002

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