A relaxation method for two-phase flow models with hydrodynamic closure law

Abstract

This paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two-phase flows in pipelines. The PDEs are closed by two highly nonlinear algebraic relations, namely, a pressure law and a hydrodynamic one. The severe nonlinearities encoded in these laws make the classical approximate Riemann solvers virtually intractable at a reasonable cost of evaluation. We propose a strategy for relaxing solely these two nonlinearities. The relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate. Such a property not only makes it trivial to solve the Riemann problem but also enables us to enforce some further stability requirements, in addition to those coming from a Chapman-Enskog analysis. The new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions. Extensive numerical evidences are provided.

An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment

“…Several solutions can exist and several criteria of selection have already been proposed, see [12][13][14]. We refer to [1,7,8] for some other examples. In this paper we shall not be able to impose any condition (except the entropy inequality, see below) in order to define weak solutions to the system so that the solutions may differ from the ones computed with other numerical schemes.…”

The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model

“…The first test case is a Riemann problem constructed by Baudin et al [2], and whose solution is a pure rarefaction. Baudin et al took the liquid to have a constant density.…”

“…As has been pointed out by several researchers [2,3,7,8,10,23], the complexity of these laws severely restricts the possibilities for constructing a Roe solver by purely algebraic manipulations. Nevertheless, Roe-type schemes have been proposed for this model.…”

“…Baudin et al [2,3] suggested a relaxation scheme of the type proposed by Jin and Xin [14]. This is somewhat related to the Roe scheme in that one needs only to solve a linear Riemann problem at each cell interface.…”

A method to couple HEM and HRM two-phase flow models

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