Numerical relaxation techniques for mass transfer in three-phase liquid-vapor-gas flows

We describe liquid-vapor-gas flows by a hyperbolic single-velocity three-phase compressible flow model with instantaneous pressure relaxation that we studied in previous work. The model includes thermal relaxation terms to account for heat transfer, and chemical relaxation terms to describe mass transfer between the liquid and vapor phases. To numerically solve the model system we use a fractional step method where we alternate between the solution of the homogeneous system via finite volume HLLC-type schemes and the solution of systems of ordinary differential equations that take into account the relaxation source terms. In this work we propose a novel numerical procedure for chemical relaxation that can efficiently describe arbitrary-rate mass transfer, both slow finite-rate processes and stiff instantaneous ones. The main idea consists in describing the relaxation process by a system of ordinary differential equations that admits an analytical semi-exact exponential solution. This relaxation system is built by employing the relaxed models that can be derived analytically from the parent three-phase flow model in the limit of instantaneous mechanical and thermal relaxation processes, in order to guarantee the constraints of pressure and temperature equilibrium during phase transition. Some numerical experiments in one and two dimensions are presented to show the effectiveness of the proposed method.