A five-equation model for the simulation of miscible and viscous compressible fluids

Abstract

Typical multispecies compressible Navier–Stokes computations employ conservative equations for mass fraction transport. Upwind discretisations of these governing equations produce spurious pressure oscillations at diffuse contact surfaces between gases of differing ratio of specific heat capacities which degrade the convergence rate of the algorithm. Adding quasi-conservative equations for volume fraction can solve this error, however this approach has been derived only for immiscible fluids. Here, a five-equation quasi-conservative model is proposed that includes the effects of species diffusion, viscosity and thermal conductivity. The derivation of the model is presented, along with a numerical method to solve the governing equations at second order accuracy in space and time. Formal convergence studies demonstrate the expected order of accuracy is achieved for three benchmark problems, cross-validated against two standard mass fraction models. In these test cases, the new model has between 2 and 10 times lower error for a given grid size. Simulations of a two-dimensional air-SF6 Richtmyer–Meshkov instability demonstrate that the new model converges to the solution with four times fewer points in each direction when compared to the mass fraction model in an identical numerical framework. This represents an ≈40 times lower computational cost for an equivalent error in two-dimensional computations. The proposed model is thus very suitable for Direct Numerical Simulation and Large Eddy Simulation of compressible mixing.