Generalized Riemann Problem Method for the Kapila Model of Compressible Multiphase Flows

Abstract

A second-order accurate and robust numerical scheme is developed for the Kapila model to simulate compressible multiphase flows. The scheme is formulated within the finite volume framework with the generalized Riemann problem (GRP) solver employed as the cornerstone. Besides Riemann solutions, the GRP solver provides time derivatives of flow variables at cell interfaces, achieving second-order accuracy in time within a single stage. The use of the GRP solver enhances the capability of the resulting scheme to handle the stiffness of the Kapila model in two ways. First, the coupled values, i.e., Riemann solutions and time derivatives, give the cell interface values of flow variables at the new time level, yielding an approximation to the cell average of the velocity divergence at the new time level in a computational step. This allows a semi-implicit time discretization to the stiff source term of the volume fraction equation. Second, the effects of source terms are directly included in the numerical flux via the computation of time derivatives. The resulting numerical flux is able to capture the physics of interactions between phases, and the robustness of the scheme is therefore further improved. Several challenging numerical experiments are conducted to demonstrate the good performance of the proposed finite volume scheme. In particular, a test case with a nonlinear smooth solution is designed to verify the numerical accuracy.