Exact and numerical solutions of the Riemann problem for a conservative model of compressible two-phase flows

TL;DR

This paper analyzes the Riemann problem for a conservative two-phase flow model, deriving explicit solutions, studying wave interactions, and comparing with non-conservative models and numerical results.

Contribution

It provides explicit solutions and wave analysis for the conservative SHTC two-phase flow model, highlighting differences from non-conservative models and deriving the Kapila limit system.

Findings

Overlapping rarefaction waves are possible.

Shocks can occur inside rarefaction waves.

Conservative form offers advantages over non-conservative models.

Abstract

In this work we study the solution of the Riemann problem for the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) two-phase flow model introduced in \cite{Romenski2007,Romenski2009}. All characteristic fields are carefully studied and explicit expressions are derived for the Riemann invariants and the Rankine-Hugoniot conditions. Due to the presence of multiple characteristics in the system under consideration, non-standard wave phenomena can occur. Therefore we briefly review admissibility conditions for discontinuities and then discuss possible wave interactions. In particular we will show that overlapping rarefaction waves are possible and moreover we may have shocks that lie inside a rarefaction wave. In contrast to nonconservative two phase flow models, such as the Baer-Nunziato system, we can use the advantage of the conservative form of the model under consideration. Furthermore, we show the relation between the considered conservative SHTC system and the corresponding barotropic version of the nonconservative Baer-Nunziato model. Additionally, we derive the reduced four equation Kapila system for the case of instantaneous relaxation, which is the common limit system of both, the conservative SHTC model and the non-conservative Baer-Nunziato model. Finally, we compare exact solutions of the Riemann problem with numerical results obtained for the conservative two-phase flow model under consideration, for the non-conservative Baer-Nunziato system and for the Kapila limit. The examples underline the previous analysis of the different wave phenomena, as well as differences and similarities of the three systems.