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 solve the receptivity problem by integrating the compressible boundary-region equations complemented by appropriate initial and boundary conditions which synthesize the influence of the free-stream vortical flow. In the limit of high frequencies, the triple-deck equations are also solved and their results compared with the solution of the boundary-region equations.
The boundary-layer perturbations, in the proximity of the leading edge, 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 and 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.
The theoretical analysis also shows that the vortices move towards the wall as the Mach number increases when the Gortler number is large.
These results are confirmed by the receptivity analysis, which additionally clarifies that the temperature perturbations respond to this reversed behavior 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 lower than the free-stream streamwise wavelength.
During the initial development of the vortices, only the receptivity calculations are accurate. Downstream where the free-stream disturbances have fully decayed, the growth rate and wavelength are computed accurately by the eigenvalue analysis, although the correct amplitude 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.
The receptivity analysis is also used to compute the neutral curves generated by free-stream disturbances, i.e., curves that identify the region of growth and decay of the boundary-layer perturbations, for different Gortler numbers, Mach numbers, wavelengths, and low frequencies of the free-stream disturbance. The growth rate of the perturbation is used to identify if the boundary-layer instability is in the form of Klebanoff modes or Gortler vortices. A critical Gortler number can be identified below which Klebanoff modes are the only source perturbations, even when curvature is present. From the receptivity and eigenvalue formulation we define a streamwise-dependent receptivity coefficient and discuss the N-factor approach for transition prediction.
Finally, the equations the triple-deck analysis reveals that the curvature effects do not play a role in the limit of high frequencies, which is also confirmed by the boundary-region results.
