Relaxation and numerical approximation of a two-fluid two-pressure diphasic model

Abstract

Abstract. This paper is concerned with the numerical approximation of the solutions of a two-fluid two-pressure model used in the modelling of two-phase flows. We present a relaxation strategy for easily dealing with both the nonlinearities associated with the pressure laws and the nonconservative terms that are inherently present in the set of convective equations and that couple the two phases. In particular, the proposed approximate Riemann solver is given by explicit formulas, preserves the natural phase space, and exactly captures the coupling waves between the two phases. Numerical evidences are given to corroborate the validity of our approach.Mathematics Subject Classification. 76T10, 35L60, 76M12.

A Godunov-type method for the seven-equation model of compressible two-phase flow

“…The objective of this section is to mention some asymptotic properties satisfied by the solutions of (1). For that, we assume that the time-scales involved in the pressure relaxation and drag force coefficients are much smaller than the other characteristic time-scales of the system.…”

“…Then, the model (and related ones, see among others Stewart & Wendroff [41], Abgrall & Saurel [37], [38]...) has gained interest for the modelling and computation of two phase flows. See for instance, in a non-exhaustive way, Kapila et al [29], Glimm et al [23], Abgrall & Saurel [37], Gavrilyuk & Saurel [22], Gallouët, Hérard & Seguin [21], Coquel, Gallouët, Hérard & Seguin [13], and more recently Ambroso, Chalons, Coquel & Galié [1], Tokareva & Toro [42], and the references therein. One of the main features of this model is to involve two distinct velocities u 1 and u 2 and two pressures p 1 and p 2 associated with the two phases.…”

Well-Balanced Roe-Type Numerical Scheme for a Model of Two-Phase Compressible Flows

“…The reader is referred to [1,11,16,20,24,25] for numerical schemes for multi-phase multi-pressure models. In [33,34], well-balanced numerical schemes for two-phase flow models were built by using stationary waves to track the source terms.…”

Large Time-Step Numerical Scheme for the Seven-Equation Model of Compressible Two-Phase Flows

“…It was first proposed in Baer & Nunziato [4] and has since aroused more and more interest, see for instance Embid & Baer [7], Stewart & Wendroff [13], Abgrall & Saurel [11], Gallouët, Hérard & Seguin [8], Andrianov & Warnecke [3], Karni et al [9] Schwendeman, Wahle & Kapila [12], Munkejord [10], Tokareva & Toro [14], Ambroso, Chalons, Coquel & Galié [1], Ambroso, Chalons & Raviart [2], and the references therein. One of the main features of this model is that it is hyperbolic, at least in the context of subsonic flows.…”

“…This property will be used in the numerical strategy. Numerous papers are devoted to the numerical study of two-fluid two-pressure models, see again for instance [8], [3], [9], [12], [10], [14], [1], [2] and the references therein. Many of the proposed methods are based on time-explicit, exact or approximate, Godunov-type methods (Roe or Roe-like scheme, HLL or HLLC scheme...).…”