Compressible unsteady Gortler vortices subject to free-stream vortical disturbances

The perturbations triggered by free-stream vortical disturbances in compressible boundary layers developing over concave walls are studied numerically and through asymptotic methods. We employ an asymptotic framework based on the limit of high Gortler number, the scaled parameter defining the centrifugal effects; we use an eigenvalue formulation where the free-stream forcing is neglected; and we solve the receptivity problem by integrating the compressible boundary-region equations complemented by appropriate initial and boundary conditions that synthesize the influence of the free-stream vortical flow. Near the leading edge, the boundary-layer perturbations develop as thermal Klebanoff modes and, when centrifugal effects become influential, these modes turn into thermal Gortler vortices, i.e. streamwise rolls characterized by intense velocity and temperature perturbations. The high-Gortler-number asymptotic analysis reveals the condition for which the Gortler vortices start to grow. The Mach number is destabilizing when the spanwise diffusion is negligible and stabilizing when the boundary-layer thickness is comparable with the spanwise wavelength of the vortices. When the Gortler number is large, the theoretical analysis also shows that the vortices move towards the wall as the Mach number increases. These results are confirmed by the receptivity analysis, which additionally clarifies that the temperature perturbations respond to this reversed behaviour further downstream than the velocity perturbations. A matched-asymptotic composite profile, found by combining the inviscid core solution and the near-wall viscous solution, agrees well with the receptivity profile sufficiently downstream and at high Gortler number. The Gortler vortices tend to move towards the boundary-layer core when the flow is more stable, i.e. as the frequency or the Mach number increase, or when the curvature decreases. As a consequence, a region of unperturbed flow is generated near the wall. We also find that the streamwise length scale of the boundary-layer perturbations is always smaller than the free-stream streamwise wavelength. During the initial development of the vortices, only the receptivity calculations are accurate. At streamwise locations where the free-stream disturbances have fully decayed, the growth rate and wavelength are computed with sufficient accuracy by the eigenvalue analysis, although the correct amplitude and evolution of the Gortler vortices can only be determined by the receptivity calculations. It is further proved that the eigenvalue predictions of the growth rate and wavenumber worsen as the Mach number increases as these quantities show a dependence on the wall-normal direction. We conclude by qualitatively comparing our results with the direct numerical simulations available in the literature.