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

Abstract

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 well-known Baer-Nunziato model that describes the behavior of granular explosives. In the analysis presented here, we omit source terms representing the exchange of mass, momentum and energy between the phases due to compaction, drag, heat transfer and chemical reaction, but retain the nonconservative nozzling terms that appear naturally in the model. For the Riemann problem the effect of the nozzling terms is confined to the contact discontinuity of the solid phase. Treating the solid contact as a layer of vanishingly small thickness within which the solution is smooth yields jump conditions that connect the states across the contact, as well as a prescription that allows the contribution of the nozzling terms to be computed unambiguously. An iterative method of solution is described for the Riemann problem, that determines the wave structure and the intermediate states of the flow, for given left and right states. A Godunov method based on the solution of the Riemann problem is constructed. It includes 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, demonstrating the accuracy of the numerical method and in particular, the accurate numerical description of the flow in the vicinity of a solid contact where phases couple and 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.