HLLC-type and path-conservative schemes for a single-velocity six-equation two-phase flow model: A comparative study

The present article deals with the numerical integration of a six-equation single-velocity two-phase flow model with stiff mechanical relaxation. This model can be employed to approximate efficiently the well known single-velocity single-pressure five-equation model of Kapila et al. (2001). Work in the literature has shown the efficiency of the six-equation model in simulating complex two-phase flows involving cavitation and evaporation processes. The aim of this work is to present and discuss various numerical schemes for this two-phase model focusing on the integration of the nonconservative terms appearing in the phasic energy equations. In fact, previous work has suggested that the choice of the discretization method for the nonconservative terms often does not play a significant role. Two new methods are proposed: a path-conservative HLLC-type scheme that is based on the Dal Maso–LeFloch–Murat theory, and a generalized HLLC-type scheme that is based on a Suliciu's Riemann solver. The latter scheme has the important property of preserving the positivity of the intermediate states of the conserved quantities. Moreover, we also approximate solutions of the six-equation model by applying two path-conservative schemes recently proposed in the literature, which have been derived from the Osher and HLLEM Riemann solvers. We show comparisons of the different numerical schemes for several test cases, including cavitation problems and shock tubes. An efficiency study for first and second order schemes is also presented. Numerical results show that different methods corresponding to different numerical treatments of the nonconservative terms give analogous results and they are all able to produce accurate approximations of solutions of the Kapila's five-equation model, except, as expected, for shocks in two-phase mixtures with very high pressure ratios.