A discretization of the phase mass balance in fractional step algorithms for the drift-flux model

We address in this paper a parabolic equation used to model the phase mass balance in two-phase flows, which differs from the mass balance for chemical species in compressible multicomponent flows by the addition of a nonlinear term of the form ∇·ρφ(y)ur, where y is the unknown mass fraction, ρ stands for the density, φ is a regular function such that φ(0) = φ(1) = 0 and ur is a (not necessarily divergence free) velocity field. We propose a finite-volume scheme for the numerical approximation of this equation, with a discretization of the nonlinear term based on monotone flux functions. Under the classical assumption that the discretization of the convection operator must be such that it vanishes for a constant y, we prove the existence and uniqueness of the solution, together with the fact that it remains within its physical bounds, i.e., within the interval [0, 1]. Then this scheme is combined with a pressure correction method to obtain a semi-implicit fractional step scheme for the so-called drift-flux model. To satisfy the above-mentioned assumption a specific time stepping algorithm with particular approximations for the density terms is developed. Numerical tests are performed to assess the convergence and stability properties of this scheme.