A Discrete Equation Method for Two-Dimensional Calculations of Two-Phase Compressible Mixtures

At present time there are different numerical schemes to solve the hyperbolic equations system for two-phase mixture. Those schemes, mainly, rely on a second order Godunov-type scheme (Godunov, 1969), with approximate Riemann solver for the resolution of conservation equations, and set of nonconservative equations. In this paper we applied the Discrete Equation Method (DEM) processed in Saurel & Abgrall (2003) for two-phase compressible mixtures calculation using exact Riemann solver (Leonov & Chudanov, 2008). Thanks to a deeper analysis of the model, a class of schemes, those are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities, was proposed. Such analysis provides a more accurate estimate of closure terms and an accurate resolution method for the conservative fluxes as well as non-conservative terms even for situations involving discontinuous solutions. The relaxation parameters are determined also, so the resulting model is free of input parameters. The second order accuracy numerical scheme was obtained using an extension of the conventional MUSCL approach. Such method allows extending for multidimensions using splitting procedures. Some common set of 2D numerical tests, calculated with previous issues of two-phase model, is produced. Here a two-phase shock tube with tangential velocity discontinuity, the advection of a square gas bubble in uniform liquid flow, the shock wave interactions with density discontinuities are presented.