Modeling shock unsteadiness in shock/turbulence interaction

Presence of shock waves can drastically alter the characteristics of a turbulent boundary layer. Shock/turbulence interactions can cause flow separation and high heating rates, both of which are critical to vehicle design. Commonly studied flow configurations include compression ramps, cylinder-flare combinations, double cones, single or double fins on a plate, oblique shocks impinging on a boundary layer and transonic airfoils.

Engineering prediction of shock/turbulence interaction relies on Reynolds averaged Navier-Stokes (RANS) simulations. Although the predictions are satisfactory for small ramp angles (up to 16 deg.), there is noticeable disagreement with experimental data for higher deflections. RANS turbulence models cannot predict the size of the separation region, the peak heat transfer rate at reattachment, and the mean velocity profiles on the ramp.

We study the modeling of homogeneous isotropic turbulence interacting with a normal shock. The standard k-epsilon model grossly over-predicts the amplification of the turbulent kinetic energy across the shock, because the underlying eddy viscosity assumption breaks down in a rapidly distorting mean flow. Modifications based on the realizability constraint reduce the eddy viscosity, and thus yield a lower amplification of turbulence. However, it is shown that eddy viscosity corrections are not enough to match results from linear theory and direct numerical simulation (DNS). This is because the existing models do not account for some of the key physical processes involved in these interactions, e.g. the unsteady motion of the shock. The coupling of the unsteady shock motion with the incoming turbulent fluctuations is found to reduce the amplification of the turbulent kinetic energy. We modify the k-equation to incorporate the shock unsteadiness mechanism and model it using linear analysis. The resulting equation yields a significant improvement over the existing models. The equation for the solenoidal dissipation rate is also modified so that it predicts the correct turbulence decay rate behind the shock.

The figures below show the amplification of turbulent kinetic energy in the interaction of homogeneous isotropic turbulence with a normal shock at Mach numbers of 1.29, 2.0 and 3.0. Predictions of the standard k-epsilon model (green), a realizable model (blue), the new model that includes shock-unsteadiness effects (red), and the k-eps model with zero eddy viscosity in the shock (purple) are compared with DNS data (black symbols). It can be seen that the new k-epsilon model (red line) reproduces DNS data of shock/isotropic turbulence interaction well.