The formulation of the RANS equations for hypersonic turbulent flows

Abstract

Accurate prediction of hypersonic turbulent flows is essential to the design of high-speed aerospace vehicles. Such flows are mainly predicted using the Reynolds-Averaged-Navier-Stokes (RANS) approach in general and in particular turbulence models using the effective viscosity approximation. Several terms involving the turbulent kinetic energy (TKE) appear explicitly in the RANS equations through the modelling of the Reynolds stresses and mean total energy and the molecular and turbulent diffusion terms. Some of these terms are often ignored in low, or even supersonic, speed simulations with zero-equation models, as well as some one-or twoequation models. The omission of these terms may not be suitable under hypersonic conditions, but there are nevertheless codes and software packages that still make such approximations, even for very high-speed turbulent flow simulations. To clarify the impact of ignoring the TKE terms in the RANS equations, two linear two-equation models and one nonlinear two-equation model are applied to the computation of two hypersonic benchmark cases, a 2D zero-pressure gradient flat plate case and an axisymmetric shock wave boundary layer interaction (SWBLI) case. The predicted surface friction coefficients and velocity profiles with different combinations of TKE terms showed little differences in the zero-pressure gradient case. However, in the SWBLI case, comparisons show that the flow separation would be delayed or accelerated with different combinations of TKE compared to the one with all the TKE terms included. Therefore, it is highly recommended to include all the TKE terms in the mean flow equations when dealing with simulations of hypersonic turbulent flows, especially for flows with shock wave boundary layer interactions. As a further consequence, since the TKE terms may not be obtained explicitly in zero-equation, or some one-equation, models, it is debatable whether these models are suitable for simulations of hypersonic turbulent flows with SWBLIs.