Hyperbolic relaxation models for granular flows

Thierry Gallouët; Philippe Helluy; Jean-Marc Hérard; Julien Nussbaum

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 2, page 371-400
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Gallouët, Thierry, et al. "Hyperbolic relaxation models for granular flows." ESAIM: Mathematical Modelling and Numerical Analysis 44.2 (2010): 371-400. <http://eudml.org/doc/250711>.

@article{Gallouët2010,
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. },
author = {Gallouët, Thierry, Helluy, Philippe, Hérard, Jean-Marc, Nussbaum, Julien},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Two-phase flow; hyperbolicity; relaxation; finite volume; entropy; two-phase flow; entropy},
language = {eng},
month = {3},
number = {2},
pages = {371-400},
publisher = {EDP Sciences},
title = {Hyperbolic relaxation models for granular flows},
url = {http://eudml.org/doc/250711},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Gallouët, Thierry
AU - Helluy, Philippe
AU - Hérard, Jean-Marc
AU - Nussbaum, Julien
TI - Hyperbolic relaxation models for granular flows
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 2
SP - 371
EP - 400
AB - 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.
LA - eng
KW - Two-phase flow; hyperbolicity; relaxation; finite volume; entropy; two-phase flow; entropy
UR - http://eudml.org/doc/250711
ER -

References

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