A volume-fraction based algorithm for hybrid barotropic and non-barotropic two-fluid flow problems

The aim of this paper is to describe a simple Eulerian interface-capturing approach for the efficient numerical resolution of a hybrid barotropic and non-barotropic two-fluid flow problem in more than one space dimension. We use the compressible Euler equations as a model system with the thermodynamic property of each of the barotropic and non-barotropic fluid components characterized by the Tait and Noble–Abel equations of state, respectively. The algorithm is based on a volume fraction formulation of the equations together with an extended equation of state that is devised to give an approximate treatment for the mixture of more than one fluid component within a grid cell. A standard high-resolution wave propagation method is employed to solve the proposed two-fluid model with the dimensional-splitting technique incorporated in the method for multidimensional problems. Several numerical results are presented in one and two space dimensions that show the feasibility of the algorithm as applied to a reasonable class of practical problems without the occurrence of any spurious oscillation in the pressure near the smeared material interfaces. This includes, in particular, solutions for a study on the variation of the jet velocity with the incident shock pressure arising from the collapse of an air cavity in water under a shock wave.