# A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations

## Abstract

A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a computational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation process replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and provides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the computation of interface problems separating fluids with different EOS (liquid-gas interface for example) with the Euler equations. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel,more »

- Authors:

- Polytech'Marseille, University Institute of France, Universite de Provence and SMASH Project UMR CNRS 6595 - IUSTI-INRIA, 5 rue E. Fermi, 13453 Marseille Cedex 13 (France). E-mail: Richard.Saurel@polytech.univ-mrs.fr
- Polytech'Marseille, University Institute of France, Universite de Provence and SMASH Project UMR CNRS 6595 - IUSTI-INRIA, 5 rue E. Fermi, 13453 Marseille Cedex 13 (France)

- Publication Date:

- OSTI Identifier:
- 20991578

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 223; Journal Issue: 2; Other Information: DOI: 10.1016/j.jcp.2006.10.004; PII: S0021-9991(06)00476-1; Copyright (c) 2006 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; COMPRESSIBLE FLOW; EFFICIENCY; EQUATIONS OF STATE; EXACT SOLUTIONS; MIXTURES; MULTIPHASE FLOW; OSCILLATIONS; RELAXATION; THERMAL EQUILIBRIUM

### Citation Formats

```
Saurel, Richard, Franquet, Erwin, Daniel, Eric, and Le Metayer, Olivier.
```*A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations*. United States: N. p., 2007.
Web. doi:10.1016/j.jcp.2006.10.004.

```
Saurel, Richard, Franquet, Erwin, Daniel, Eric, & Le Metayer, Olivier.
```*A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations*. United States. doi:10.1016/j.jcp.2006.10.004.

```
Saurel, Richard, Franquet, Erwin, Daniel, Eric, and Le Metayer, Olivier. Tue .
"A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations". United States.
doi:10.1016/j.jcp.2006.10.004.
```

```
@article{osti_20991578,
```

title = {A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations},

author = {Saurel, Richard and Franquet, Erwin and Daniel, Eric and Le Metayer, Olivier},

abstractNote = {A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a computational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation process replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and provides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the computation of interface problems separating fluids with different EOS (liquid-gas interface for example) with the Euler equations. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel, A relaxation-projection method for compressible flows. Part II: computation of interfaces and multiphase mixtures with stiff mechanical relaxation. J. Comput. Phys. (submitted for publication)], the method is extended to the numerical approximation of a non-conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures.},

doi = {10.1016/j.jcp.2006.10.004},

journal = {Journal of Computational Physics},

number = 2,

volume = 223,

place = {United States},

year = {Tue May 01 00:00:00 EDT 2007},

month = {Tue May 01 00:00:00 EDT 2007}

}