Robust numerical coupling of pressure and pressureless gas dynamics equations for Eulerian spray DNS and LES

Large eddy simulation (LES) and direct numerical simulation (DNS) of polydisperse evaporating sprays with Eulerian models are very promising tools for high performance computing of combustion applications since they are able to predict the turbulent dispersion and evaporation. However, the spray system of conservation equations has a convective part which is similar either to gas dynamics Euler equations with a real gas type state law or to the pressureless gas dynamics (PGD), depending on the local flow regime and droplet Stokes number; so they usually involve singularities due to model closure assumptions and require dedicated numerical schemes. The present contribution introduces a new generation of numerical methods based on relaxation schemes which are able to treat both PGD and general gas dynamics as well as to cope in a robust manner with vacuum zones and natural singularities of the resulting system of conservation equations. The approach relies on the coupling between a relaxed model for PGD and a relaxed model for gas dynamics using an energy threshold. The proposed hybrid relaxation scheme and algorithms are validated through comparisons with analytical solutions and other numerical strategies on one-dimensional (1D) and two-dimensional (2D) configurations. They exhibit a very robust behavior and are a very promising candidate for more complex applications since they provide solutions to key numerical issues of the actual Eulerian spray DNS and LES models. Though the energy is considered here as isotropic, the method can be extended to nonequilibrium gas dynamics to describe the spray dynamics with higher accuracy.