A simple HLLC-type Riemann solver for compressible non-equilibrium two-phase flows

Abstract

A simple, robust and accurate HLLC-type Riemann solver for two-phase 7-equation type models is built. It involves 4 waves per phase, i.e. the three conventional right- and left-facing and contact waves, augmented by an extra “interfacial” wave. Inspired by the Discrete Equations Method (Abgrall and Saurel, 2003), this wave speed (uI) is assumed function only of the piecewise constant initial data. Therefore it is computed easily from these initial states. The same is done for the interfacial pressure PI. Interfacial variables uI and PI are thus local constants in the Riemann problem. Thanks to this property there is no difficulty to express the non-conservative system of partial differential equations in local conservative form. With the conventional HLLC wave speed estimates and the extra interfacial speed uI, the four-waves Riemann problem for each phase is solved following the same strategy as in Toro et al. (1994) for the Euler equations. As uI and PI are functions only of the Riemann problem initial data, the two-phase Riemann problem consists in two independent Riemann problems with 4 waves only. Moreover, it is shown that these solvers are entropy producing. The method is easy to code and very robust. Its accuracy is validated against exact solutions as well as experimental data.