Construction of a low Mach finite volume scheme for the isentropic Euler system with porosity

¹ Hydro-Québec – TransÉnergie et Équipement, DCMÉ, Prévisions de contrôle du réseau, Édifice Jean-Lesage, 75 boulevard René-Lévesque Ouest, Montréal, QC H2Z 1A4, Canada
² Université de Pau et des Pays de l’Adour, E2S UPPA, CNRS, INRIA, équipe CAGIRE, LMAP, Pau, France
³ Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides, 91191 Gif-sur-Yvette, France
⁴ Université Sorbonne Paris Nord, LAGA, CNRS UMR 7539, Institut Galilée, 99 Av. J.-B. Clément, 93430 Villetaneuse, France

^* Corresponding author: pascal.omnes@cea.fr

Received: 14 January 2021
Accepted: 2 April 2021

Abstract

Classical finite volume schemes for the Euler system are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the linear wave equation taking into account porosity. We explain the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergence-free velocity, while this is not the case in the Cartesian case. On Cartesian meshes, a fix is proposed and accuracy at low Mach number is proved to be recovered. Based on the linear study, a numerical scheme and a low Mach fix for the non-linear system, with a non-conservative source term due to the porosity variations, is proposed and tested.