Compressible Two-Phase Flows: Two-Pressure Models and Numerical Methods

A conservative hyperbolic model for compressible two-phase two-fluid model is studied and numerical methods for its approximate solution are proposed. The derivation of the governing equations of the model is based on the principles of extended thermodynamics. The field equations form a hyperbolic system of balance equations in conservative form, which guarantees the well-posedness of the initialvalue problem (its solvability, at least locally in time). The system of governing equations consists of well-known conservation laws for the mixture mass, momentum, and energy, which are completed by the additional balance laws for the relative velocity of phases and for the volume concentration of one phase. The closure constitutive relation for the model is the equation of state for the mixture, which can be derived from known equations of state for each phase. The eigenstructure analysis of the one-dimensional case shows the existence of six real eigenvalues, four of which are connected with two speeds of sound in the pure phases, and two correspond to the mixture flow velocity. A corresponding, complete set of linearly independent eigenvectors is given explicitly and the nature of the associated characteristic fields is studied. For the case of isentropic flow it is shown that in terms of the individual phase parameters the originally conservative system of governing equations can be transformed to the well-known non-conservative model of Baer-Nunziato-type. In this situation our model differs from the latter by the definition of interfacial pressure and by extra terms in the momentum equations related to lift forces. Finite volume shock-capturing methods for solving numerically the governing equations are studied, test problems are proposed and numerical results are presented and discussed.