Numerical Simulation of Homogeneous Equilibrium Two-Phase Flows with Shock-Stable Schemes

In this paper, we introduce the two-phase versions of shock-stable schemes, RoeM and AUSMPW+, for the computation of compressible gas-liquid two-phase mixture flows. From the mixture EOS, we derived and evaluated a new shock discontinuity sensing term, which is commonly used in RoeM and AUSMPW+ for the stable numerical flux calculation. And the developed two-phase RoeM and AUSMPW+ are preconditioned for the simulation of all Mach number flows, which are generally of interest for many gas-liquid two-phase applications due to the large variation in a speed of sound. Developed schemes are applied on several liquid-gas, large property discrepancy two-phase test problems from highly compressible to nearly incompressible flow conditions, and the numerical results show their accurate and robust behavior for all speed two-phase flows. I. Introduction ECENTLY, there have been many researches on the computation of compressible gas-liquid two-phase flows, because of its wide application areas such as cavitation phenomenon in hydraulic machine, high speed underwater projectile, explosion in water, liquid shock-gas bubble interaction, and so on. Among the several levels to describe two different phases 1 , homogeneous equilibrium model (HEM) is considered in this paper for its practical engineering capability with relative efficiency. Under the assumption of homogeneous equilibrium, our first interest lies in the extending of RoeM 2 and AUSMPW+ 3 schemes to compressible gas-liquid two-phase flows for the robust simulation. Both schemes are originally developed for the high resolution simulation of high-speed gas dynamics. The RoeM scheme, based on the Roe’s flux difference splitting (FDS), is a shock-stable scheme without any tunable parameters while maintaining the accuracy of the original Roe scheme. From the linear perturbation analysis on the odd-even decoupling problem, RoeM introduces Mach-number-based functions near the shock discontinuity. The AUSMPW+ scheme is the improved version of the AUSMPW scheme. By the use of pressure-based weighting functions, AUSMPW+ can reflect both properties across a cell interface adequately, and its numerical results show the successful elimination of the overshoots behind shocks and/or the oscillations near a wall. Both RoeM and AUSMPW+ schemes are akin to the recently developed advanced schemes for the gas dynamics, and they possess the similar shock discontinuity sensing term (SDST). In order to maintain the merits of both schemes even in twophase flows, we introduce a new SDST derived from the mixture equation of state (EOS). And due to the advection property of AUSM-type schemes, we scale the control functions of the AUMSPW+ scheme to prevent numerical instabilities near the large-density-ratio phase interface. The speed of sound of liquid phase is about 1400-1500m/s, while that of gas phase is about 300-400m/s, and that of mixture region is about O(2)m/s. Because of the large variation in speed of sound, there exist compressible and incompressible regions simultaneously for many engineering gas-liquid two-phase problems. From the viewpoint of the compressible governing equation system, the condition number – ratio of largest to smallest eigenvalue –