Receptivity of Unsteady Compressible Görtler Flows to Free-Stream Vortical Disturbances

The perturbations triggered by free-stream vortical disturbances in compressible boundary layers developing over concave surfaces are investigated by asymptotic methods and numerically. We utilize an asymptotic framework based on the limits of high Reynolds and Gortler numbers and solve the receptivity problem by integrating the initial-boundary value problem coupled with appropriate initial and boundary conditions, synthesizing the effect of the free-stream vortical disturbances. The conditions for which the Gortler vortices start to grow are revealed and we find that 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. 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 an eigenvalue method, although the correct amplitude and shape of the Gortler vortices can only be determined by the receptivity calculations. We find that Klebanoff modes always evolve from the leading edge, the Gortler vortices dominate when the influence of the curvature becomes significant, and highly-oblique Tollmien-Schlichting waves may precede the Gortler vortices for moderate Gortler numbers. We also obtain the neutral stability curves, i.e. curves that distinguish spatially growing from spatially decaying perturbations at different streamwise locations. For relatively high frequencies the triple-deck formalism allows us to confirm the numerical results of the negligible influence of the curvature on the Tollmien-Schlichting waves when the Gortler number is an order-one quantity. Experimental data for compressible Gortler flows are mapped onto our neutral-curve graphs and earlier theoretical results are compared with our predictions.