Upwind Scheme Using Preconditioned Artificial Dissipation for Unsteady Gas-liquid Two-phase Flow and Its Application to Shock Tube Flow

A stable upwind finite-difference method for unsteady gas-liquid two-phase flows is proposed and applied to shock tube flows. The artificial dissipation terms in the flux difference splitting upwinding scheme are derived using a preconditioned matrix to enhance the stability and convergence of the numerical calculation of mixed compressible and incompressible flows with arbitrary void fractions. A homogeneous gas-liquid two-phase flow model is used. A stable four-stage Runge-Kutta method and the flux difference splitting upwind scheme combined with a third-order MUSCL TVD scheme are employed. Using the proposed method, we compute gas-liquid mixture shock tube problems and compare their results with the exact solution to check the reliability of the proposed method. Shock and expansion wave propagations through the gas-liquid two-phase media are observed in detail. The effect of the preconditioned artificial dissipation on the numerical stability and convergence rate are investigated. We confirm that the proposed method is stable and effective for computations of unsteady two-phase complex flows with arbitrary Mach numbers.