Kinetic-energy- and pressure-equilibrium-preserving schemes for real-gas turbulence in the transcritical regime

Abstract

Numerical simulations of compressible turbulent flows governed by real-gas equations of state, such as high-pressure transcritical flows, are strongly susceptible to instabilities. In addition to the inherent multi-scale nature of the flow, the presence of a pseudo-interface can generate spurious pressure oscillations when conventional schemes are utilized. This study proposes a general framework to derive and analyze discretization methods that are able to preserve kinetic energy by convection, and simultaneously maintain pressure equilibrium in discontinuity-free compressible real-gas flows. The formal analysis reveals that the discrete pressure-equilibrium condition can be fulfilled at most to second-order accuracy, as it requires the spatial differential operator to satisfy a discrete chain rule when total, or internal energy, are directly discretized. A novel class of schemes based on the solution of a pressure equation is thus proposed, which preserves mass, momentum, kinetic energy and pressure equilibrium, but not total energy. Extensive numerical tests of increasing complexity confirm the theoretical predictions, and show that the proposed scheme is capable of providing non-dissipative, stable and oscillation-free simulations, unlike existing methods tailored for the transcritical regime.