A relaxation scheme for a hyperbolic multiphase flow model

Abstract

This article is the first of two in which we develop a relaxation finite volume scheme for the convective part of the multiphase flow models introduced in the series of papers [17,16,6]. In the present article we focus on barotropic flows where in each phase the pressure is a given function of the density. The case of general equations of state will be the purpose of the second article. We show how it is possible to extend the relaxation scheme designed in [12] for the barotropic Baer-Nunziato two phase flow model to the multiphase flow model with N -where N is arbitrarily large -phases. The obtained scheme inherits the main properties of the relaxation scheme designed for the Baer-Nunziato two phase flow model. It applies to general barotropic equations of state. It is able to cope with arbitrarily small values of the statistical phase fractions. The approximated phase fractions and phase densities are proven to remain positive and a fully discrete energy inequality is also proven under a classical CFL condition. For N = 3, the relaxation scheme is compared with Rusanov's scheme, which is the only numerical scheme presently available for the three phase flow model (see [6]). For the same level of refinement, the relaxation scheme is shown to be much more accurate than Rusanov's scheme, and for a given level of approximation error, the relaxation scheme is shown to perform much better in terms of computational cost than Rusanov's scheme. Moreover, contrary to Rusanov's scheme which develops strong oscillations when approximating vanishing phase solutions, the numerical results show that the relaxation scheme remains stable in such regimes.

A four-field three-phase flow model with both miscible and immiscible components

“…Actually, while restricting to two-phase flow models, we recall that the numerical scheme, initially developped in [32,50] for the two-phase flow case with immiscible components, has been indeed much improved, both in terms of accuracy and stability, using the relaxation scheme [51,52]; a detailed comparison of the latter with other schemes, namely the approximate Godunov solver [53], and the HLLC scheme [54], confirmed its advantages and strong potentialities. Moreover, a recent accurate and efficient relaxation scheme has been proposed in [55] for the barotropic immiscible three-phase flow model [9], which is precisely an extension of the one developped in [51,52] for two-phase flows with immiscible components.…”

Simulation and preliminary validation of a three-phase flow model with energy

“…• There is a need for more accurate and robust Riemann solvers (see for instance [1,2,10,36] in that direction).…”

Simulation of a homogeneous relaxation model for a three-phase mixture with miscible phases

“…Loss of coolant accident in pressurized water reactors and vapor explosions are two typical examples of compressible multiphase flows in this framework [11,12]. The Date: June 1, 2022. literature about immiscible multiphase configurations is huge; the reader may refer to recent publications [16,13] for the modelling and analysis and to [28,31] for numerical approximations of such immiscible models. Recently models of compressible flows with miscible phases have been proposed [27,22,23,18,17], see also [1].…”