The generalized Riemann problem scheme for a laminar two-phase flow model with two-velocities

In this paper, we propose a generalized Riemann problem (GRP) scheme for a laminar two-phase flow model. The model takes into account the distinctions between different densities and velocities, and is obtained by averaging vertical velocities across each layer for the two-phase flows. The rarefaction wave and the shock wave are analytically resolved by using the Riemann invariants and Rankine-Hugoniot condition, respectively. The source term is incorporated into the resolution of the GRP method. We further extend the GRP method to the two-dimensional (2-D) system, which is non-conservative. The Strang splitting method is applied, but it still can not provide explicit Riemann invariants and shock relations, which prevent us to apply the GRP method directly. Another splitting technique is also applied to the 2-D case, such that each split subsystem contains only one family of waves. Numerical experiments on some typical problems show that the proposed method achieves good performance.