Response of a compressible laminar boundary layer to free-stream vortical disturbances

As a first step towards understanding the role of free-stream turbulence in laminar-turbulent transition, we calculate the fluctuations induced by free-stream vortical disturbances in a compressible laminar boundary layer. As with the incompressible case investigated by Leib et al. (J. Fluid Mech. vol. 380, 1999, p. 169), attention is focused on components with long streamwise wavelength. The boundary-layer response is governed by the linearized unsteady boundary-region equations in the typical streamwise region where the local boundary-layer thickness delta* is comparable with the spanwise length scale Lambda of the disturbances. The compressible boundary-region equations are solved numerically for a single Fourier component to obtain the boundary-layer signature. The root-mean-square of the velocity and mass-flux fluctuations induced by a continuous spectrum of free-stream disturbances are computed by an appropriate superposition of the individual Fourier components. Low-frequency vortical disturbances penetrate into the boundary layer to form slowly modulating streamwise-elongated velocity streaks. In the compressible regime, vortical disturbances are found to induce substantial temperature fluctuations so that 'thermal streaks' also form. They may have a significant effect on the secondary instability. The calculations indicate that for a vortical disturbance with a relatively large Lambda, the induced boundary-layer fluctuation ultimately evolves into an amplifying wave. This is due to a receptivity mechanism, in which a vortical disturbance first excites a decaying quasi-three-dimensional Lam-Rott eigensolution. The latter then undergoes wavelength shortening to generate a spanwise pressure gradient, which eventually converts the Lam-Rott mode into an exponentially growing mode. The latter is recognized to be a highly oblique Tollmien-Schlichting wave. A parametric study suggests that this receptivity mechanism could be significant when the free-stream Mach number is larger than 0.8.