A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver

Compressible multi-phase flows are found in a variety of scientific and engineering problems. The development of accurate and efficient numerical algorithms for multi-phase flow simulations remains one of the challenging issues in computational fluid dynamics. A main difficulty of numerical methods for multi-phase flows is that the model equations cannot always be written in conservative form, though they may be hyperbolic and derived from physical conservation principles. In this work, assuming a hyperbolic model, a path-conservative method is developed to deal with the non-conservative character of the equations. The method is applied to solve the five-equation model of Saurel and Abgrall for two-phase flow. As another contribution of the work, a simplified HLLC-type approximate Riemann solver is proposed to compute the Godunov state to be incorporated into the Godunov-type path-conservative method. A second order, semi-discrete version of the method is then constructed via a MUSCL reconstruction with Runge–Kutta time stepping. Moreover, the method is then extended to the two-dimensional case by directional splitting. The method is systematically assessed via a series of test problems with exact solutions, finding satisfactory results.