A positive and entropysatisfying finite volume scheme for the Baer–Nunziato model
Abstract
We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer–Nunziato twophase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in for the isentropic Baer–Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural subcharacteristic condition associated with the relaxation approximation. This last property, which ensures the nonlinear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer–Nunziato model, namely Schwendeman–Wahle–Kapila'smore »
 Authors:
 CMAP, École Polytechnique CNRS, UMR 7641, Route de Saclay, F91128 Palaiseau cedex (France)
 EDFR&D, Département MFEE, 6 Quai Watier, F78401 Chatou Cedex (France)
 Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 bd 11 novembre 1918, F69622 Villeurbanne cedex (France)
 Publication Date:
 OSTI Identifier:
 22622247
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 330; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; CONVERGENCE; DENSITY; ENTROPY; EQUATIONS OF STATE; FLOW MODELS; ISENTROPIC PROCESSES; MATHEMATICAL SOLUTIONS; MULTIPHASE FLOW; NUCLEAR INDUSTRY; NUMERICAL ANALYSIS; PARTIAL DIFFERENTIAL EQUATIONS; RELAXATION; SOUND WAVES; STABILITY; TWOPHASE FLOW
Citation Formats
Coquel, Frédéric, Email: frederic.coquel@cmap.polytechnique.fr, Hérard, JeanMarc, Email: jeanmarc.herard@edf.fr, and Saleh, Khaled, Email: saleh@math.univlyon1.fr. A positive and entropysatisfying finite volume scheme for the Baer–Nunziato model. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2016.11.017.
Coquel, Frédéric, Email: frederic.coquel@cmap.polytechnique.fr, Hérard, JeanMarc, Email: jeanmarc.herard@edf.fr, & Saleh, Khaled, Email: saleh@math.univlyon1.fr. A positive and entropysatisfying finite volume scheme for the Baer–Nunziato model. United States. doi:10.1016/J.JCP.2016.11.017.
Coquel, Frédéric, Email: frederic.coquel@cmap.polytechnique.fr, Hérard, JeanMarc, Email: jeanmarc.herard@edf.fr, and Saleh, Khaled, Email: saleh@math.univlyon1.fr. Wed .
"A positive and entropysatisfying finite volume scheme for the Baer–Nunziato model". United States.
doi:10.1016/J.JCP.2016.11.017.
@article{osti_22622247,
title = {A positive and entropysatisfying finite volume scheme for the Baer–Nunziato model},
author = {Coquel, Frédéric, Email: frederic.coquel@cmap.polytechnique.fr and Hérard, JeanMarc, Email: jeanmarc.herard@edf.fr and Saleh, Khaled, Email: saleh@math.univlyon1.fr},
abstractNote = {We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer–Nunziato twophase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in for the isentropic Baer–Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural subcharacteristic condition associated with the relaxation approximation. This last property, which ensures the nonlinear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer–Nunziato model, namely Schwendeman–Wahle–Kapila's Godunovtype scheme and Tokareva–Toro's HLLC scheme . The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.},
doi = {10.1016/J.JCP.2016.11.017},
journal = {Journal of Computational Physics},
number = ,
volume = 330,
place = {United States},
year = {Wed Feb 01 00:00:00 EST 2017},
month = {Wed Feb 01 00:00:00 EST 2017}
}

Numerical methods for the BaerNunziato (BN) twophase flow model have attracted much attention in recent years. In this paper, we present a new gas kinetic scheme for the BN twophase flow model containing nonconservative terms in the framework of finite volume method. In the view of microscopic aspect, a generalized BhatnagarGrossKrook (BGK) model which matches with the BN model is constructed. Based on the integral solution of the generalized BGK model, we construct the distribution functions at the cell interface. Then numerical fluxes can be obtained by taking moments of the distribution functions, and nonconservative terms are explicitly introduced intomore »

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