The shock-wave/turbulent-boundary-layer interaction (STBLI) problem is an important aerodynamic phenomenon which can be found in many practical applications. The ability to accurately predict this particular flow is therefore of a practical concern. However, it has been impeded by lacks of an accurate numerical method and an adequate turbulence closure model. Therefore, in the first part of the study, a comparative study is made of three different centered schemes (Beam-Warming, SIAF and LU-SGS) and an upwind scheme in computing a compression corner. The Baldwin-Lomax mixing length model is adopted to close the turbulent shear stress in the mass averaged, two dimensional compressible Navier-Stokes equations. Computations are performed for a Mach number of 2.90 with the Reynolds number $Re_δ$ (based on the incoming boundary layer thickness) of $1.6×10^6$ for α=16° ramp. The upwind scheme is found to yield oscillation-free solutions around the shock while all of the centered schemes give oscillatory solutions. The present study leads us to believe that any centered difference scheme using scalar artificial dissipation should be suspect in predicting the shock-wave/turbulent-boundary-layer interaction problem.
In the second part, a comparative study is made on the performance of several low-Reynolds-number κ-ω models and the κ-ω model in predicting the shock-wave / turbulent-boundary-layer interaction over supersonic compression ramps of 16°, 20° and 24° at Mach numbers of 2.85, 2.79 and 2.84, respectively.
The model equations are numerically solved by a higher order upwind scheme with the 3rd order MUSCL type TVD. The computational results reveal that all of the low-Reynolds-number κ-ω models, particularly those employing $y^+$ in their damping functions give erroneously large skin friction in the redeveloping region. It is also interesting to note that the κ-ω models adjusted elaborately based on DNS data do not perform better as expected than the conventional low-Reynolds number κ-ω models. On the other hand, the κ-ω model which does not adopt any low-Reynolds-number modification yields reasonably correct skin friction, but with somewhat late onset of pressure rise. By recasting the ω equation into the general form of the e equation, it is inferred that turbulent cross diffusion term between κ and ω is critical to guarantee better performance of the κ-ω model for the skin friction prediction in the redeveloping region. Finally, an asymptotic analysis of a fully developed incompressible channel flow with the κ-ω and the κ-ω models revealed that the cross diffusion mechanism inherent in the κ-ω model contributes to the better performance of the κ-ω model. Therefore, in order to improve the prediction capability of the κ-ω model, the cross-diffusion term should be included in the e equation. The new coefficients of the cross diffusion modified κ-ω model is determined based on the near wall asymptotic approach and a consistency condition. It is found that the proposed model with the new constants generally performs very well as compared to others in the prediction of various STBLI problems.
