Viscous Regularization for the Non-equilibrium Seven-Equation Two-Phase Flow Model

In this paper, a viscous regularization is derived for the non-equilibrium seven-equation two-phase flow model (SEM). This regularization, based on an entropy condition, is an artificial viscosity stabilization technique that selects a weak solution satisfying an entropy-minimum principle. The viscous regularization ensures nonnegativity of the entropy residual, is consistent with the viscous regularization for Euler equations when one phase disappears, does not depend on the spatial discretization scheme chosen, and is compatible with the generalized Harten entropies. We investigate the behavior of the proposed viscous regularization for two important limit-cases. First, a Chapman–Enskog expansion is performed for the regularized SEM and we show that the five-equation flow model of Kapila is recovered with a well-scaled viscous regularization. Second, a low-Mach asymptotic limit of the regularized seven-equation flow model is carried out whereby the scaling of the non-dimensional numbers associated with the viscous terms is determined such that an incompressible two-phase flow model, with a properly scaled regularization, is recovered. Both limit-cases are illustrated with one-dimensional numerical results, including two-phase flow shock tube tests and steady-state two-phase flows in converging-diverging nozzles. A continuous finite element discretization is employed for all numerical simulations.