The intrinsic phenomena of cavitation and concentration in Riemann solutions for the isentropic two-phase model with the logarithmic equation of state

Abstract

The Riemann solutions for a simplified two-phase flow model with the logarithmic equation of state are obtained in fully explicit forms for all four possible different structures. The intrinsic phenomena of cavitation and concentration are identified and analyzed in the Riemann solutions by using the vanishing pressure limit in the mixture momentum equation. It is shown that the Riemann solution consisting of 1-rarefaction wave, 2-contact discontinuity, and 3-rarefaction wave converges to a solution made up of three different vacuum states together with the left and right states separated successively by four contact discontinuities as the perturbed parameter tends to zero. By comparison, the Riemann solution composed of 1-shock wave, 2-contact discontinuity, and 3-shock wave also tends to a solution consisting of a single delta shock wave in the vanishing pressure limit. In addition, it should be pointed out that Dirac delta measures are developed simultaneously in the densities of liquid and gas in the limiting situation.