The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow

This paper considers the Riemann problem and an associated Godunov method for a model of compressible two-phase flow. The model is a reduced form of the wellknown Baer-Nunziato model which describes the behavior of granular explosives. In the analysis presented here, we focus on the effect of non-conservative nozzling terms in the two-phase model and omit source terms representing the exchange of mass, momentum and energy between the phases due to compaction, drag, heat transfer and chemical reaction. For the Riemann problem, the effect of the nozzling terms is confined to the contact discontinuity of the solid phase. A thin-layer analysis of the solid contact is presented resulting in jump conditions that connect the states of the flow on either side and an equation that is used to evaluate an integral of the nozzling term in the derivation of a Godunov method. An iterative method of solution is described for the Riemann problem that determines the wave structure and the intermediate states for given left and right states of the flow. A Godunov method is constructed based on the solution of the Riemann problem, and involves non-conservative flux contributions derived from an integral of the nozzling terms over a grid cell. The Godunov method is extended to second-order accuracy using a method of slope limiting, and an adaptive Riemann solver is described and used for computational efficiency. Numerical results are presented and these demonstrate the accuracy of the numerical method, and, in particular, the accurate numerical description of the flow in the vicinity of a solid contact where the phases couple and the nozzling terms are important. The numerical method is compared with other methods available in the literature and found to give more accurate results for the problems considered.