Discrete Equation Method (DEM) for the simulation of viscous, compressible, two-phase flows

• A scheme for the discretization of a viscous seven-equations model is presented.
• An asymptotic analysis at a discrete level of the seven equation model is performed.
• Resetting the relaxation terms, a five discretized equations system is obtained.
• The proposed formulation is validated by performing several test cases.
• The influence of viscosity on the shape of shock transition is illustrated.

A numerical approximation of a five equations model for the simulation of two-phase flows, including viscous effects is illustrated. A DEM approach is developed obtaining a clear formulation of viscous fluxes and an accurate resolution method for all terms. Starting from a system composed by the Navier–Stokes equations for each phase, a final 1D semi-discrete equation for the discretization of pressure and velocity equilibrium model is obtained.First, the influence of viscous terms is studied comparing viscous and inviscid solutions in two-phase configurations. This analysis allows to identify the thermodynamic quantities, such as pressure, velocity, phase densities and volume fraction that are influenced by the viscous effects. Then, several gas/liquid mixtures, obtained modifying the gas/liquid viscosity ratio, are compared in order to study the influence of viscous effects in terms of phase volume fraction, density and entropy. Finally, the influence of mesh on the viscosity, in terms of minimum mesh size needed to evaluate the viscous effects in the simulation, is shown.