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Hyperbolic relaxation models for granular flows

Published online by Cambridge University Press:  27 January 2010

Thierry Gallouët ,
Philippe Helluy ,
Jean-Marc Hérard  and
Julien Nussbaum
Thierry Gallouët
Affiliation:
IRMA, 7 rue Descartes, 67084 Strasbourg Cedex, France. helluy@math.u-strasbg.fr
Philippe Helluy
Affiliation:
IRMA, 7 rue Descartes, 67084 Strasbourg Cedex, France. helluy@math.u-strasbg.fr
Jean-Marc Hérard
Affiliation:
IRMA, 7 rue Descartes, 67084 Strasbourg Cedex, France. helluy@math.u-strasbg.fr
Julien Nussbaum
Affiliation:
IRMA, 7 rue Descartes, 67084 Strasbourg Cedex, France. helluy@math.u-strasbg.fr
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Abstract

In this work we describe an efficient model for the simulation of a two-phase flow made of a gas and a granular solid. The starting point is the two-velocity two-pressure model of Baer and Nunziato [Int. J. Multiph. Flow16 (1986) 861–889]. The model is supplemented by a relaxation source term in order to take into account the pressure equilibrium between the two phases and the granular stress in the solid phase. We show that the relaxation process can be made thermodynamically coherent with an adequate choice of the granular stress. We then propose a numerical scheme based on a splitting approach. Each step of the time marching algorithm is made of two stages. In the first stage, the homogeneous convection equations are solved by a standard finite volume Rusanov scheme. In the second stage, the volume fraction is updated in order to take into account the equilibrium source term. The whole procedure is entropy dissipative. For simplified pressure laws (stiffened gas laws) we are able to prove that the approximated volume fraction stays within its natural bounds.

Keywords

Two-phase flowhyperbolicityrelaxationfinite volumeentropy
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 2 , March 2010 , pp. 371 - 400
Copyright
© EDP Sciences, SMAI, 2010

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References

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