# A positive and entropy-satisfying 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 two-phase 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 sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear 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, F-91128 Palaiseau cedex (France)
- EDF-R&D, Département MFEE, 6 Quai Watier, F-78401 Chatou Cedex (France)
- Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 bd 11 novembre 1918, F-69622 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; TWO-PHASE FLOW

### Citation Formats

```
Coquel, Frédéric, E-mail: frederic.coquel@cmap.polytechnique.fr, Hérard, Jean-Marc, E-mail: jean-marc.herard@edf.fr, and Saleh, Khaled, E-mail: saleh@math.univ-lyon1.fr.
```*A positive and entropy-satisfying finite volume scheme for the Baer–Nunziato model*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2016.11.017.

```
Coquel, Frédéric, E-mail: frederic.coquel@cmap.polytechnique.fr, Hérard, Jean-Marc, E-mail: jean-marc.herard@edf.fr, & Saleh, Khaled, E-mail: saleh@math.univ-lyon1.fr.
```*A positive and entropy-satisfying finite volume scheme for the Baer–Nunziato model*. United States. doi:10.1016/J.JCP.2016.11.017.

```
Coquel, Frédéric, E-mail: frederic.coquel@cmap.polytechnique.fr, Hérard, Jean-Marc, E-mail: jean-marc.herard@edf.fr, and Saleh, Khaled, E-mail: saleh@math.univ-lyon1.fr. Wed .
"A positive and entropy-satisfying 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 entropy-satisfying finite volume scheme for the Baer–Nunziato model},

author = {Coquel, Frédéric, E-mail: frederic.coquel@cmap.polytechnique.fr and Hérard, Jean-Marc, E-mail: jean-marc.herard@edf.fr and Saleh, Khaled, E-mail: saleh@math.univ-lyon1.fr},

abstractNote = {We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer–Nunziato two-phase 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 sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear 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 Godunov-type 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}

}