The Application of Linear and Non-linear Two-Equation Turbulence Models in Hypersonic Flows

A non-linear two-equation k-ε model has been implemented in the OpenFOAM framework and has been assessed for the computation of hypersonic flows with shock boundary layer interactions (SWBLI), together with two other linear, low-Reynolds number models, namely the Launder-Sharma k-ε model and the k-ω SST model. Supersonic and hypersonic flow computations resulting from the use of these three models have been compared with experimental benchmark cases over a range of conditions. The three original models tested do generally return reasonable predictions of wall pressure in most cases, with the non-linear model resulting in the best capability among the three in predicting flow separation. The wall heat flux in the interaction region, however, is overpredicted, in most cases by all three models (sometimes showing quite a dramatic overprediction). While the overall wall heat flux predictions of the non-linear model, are closer to the measured values, it is nevertheless evident that there is a need for further improvement. To address this need, the inclusion of a new source term in the dissipation rate equation is proposed, which aims to restrict the turbulent length scales in the shock wave boundary layer interaction region. This is inactive in incompressible flows and exerts only a minor influence in supersonic SWBLI cases.Computations of a range of supersonic and hypersonic flows with SWBLI show that this inclusion of the proposed source term in the dissipation rate equation, of both the non-linear and the linear k-ε models, has significant effects only in hypersonic flows. These effects are mainly confined to the thermal predictions of these k-ε models. In the non-linear model predictions, the over-estimation of the wall heat flux in the SWBLI region is largely eliminated, while in the corresponding predictions of the linear model the over-estimation is substantially reduced. The cubic non-linear k-ε model tested, with the proposed new source term to the dissipation rate equation, is thus shown to be a very reliable and cost-effective tool for the RANS modeling of supersonic and hypersonic flows.