Numerical Modeling of Two-Phase Flows Using the Two-Fluid Two-Pressure Approach

Abstract

The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity two-pressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity with the definition of Rankine–Hugoniot jump relations. Each field of the convective system is investigated, providing maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two-finite volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to compute two-phase flows.

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

“…So far as the definitions of uI and pI are concerned, we follow [8] and first observe that the characteristic speeds of (1) are always real and given by uI , u k , u k ± c k , k = 1, 2, where c k denotes the speed of sound in phase k. System (1) turns out to be only weakly hyperbolic since there are not enough eigenvectors to span the entire space when uI = u k ± c k for some index k (resonance occurs). When (1) is hyperbolic, one can easily check that similarly to the classical gas dynamics equations, the characteristic fields associated with the eigenvalues u k ± c k are nonlinear while the one associated with u k is linearly degenerate.…”

“…where χ ∈ [0, 1] is a constant (we refer to [8] for the details), which gives a natural definition for the interfacial velocity uI . The usual choices for χ are 0, 1/2 and 1.…”

“…The choice of the coefficient µ is not detailed here (see again [8]) but is related to the ability to provide the system with an entropy balance equation. Indeed, it can be proved that for a specific choice of µ, smooth solutions of (1) verify the conservation law ∂tη + ∂xq = 0, where (η, q) plays the role of a mathematical entropy pair.…”

“…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...).…”