The pre-transitional Klebanoff modes and other boundary-layer disturbances induced by small-wavelength free-stream vorticity

Abstract

The response of the Blasius boundary layer to free-stream vortical disturbances of the convected gust type is studied. The vorticity signature of the boundary layer is computed through the boundary-region equations, which are the rigorous asymptotic limit of the Navier–Stokes equations for low-frequency disturbances. The method of matched asymptotic expansion is employed to obtain the initial and outer boundary conditions. For the case of forcing by a two-dimensional gust, the effect of a wall-normal wavelength comparable with the boundary-layer thickness is taken into account. The gust viscous dissipation and upward displacement due to the mean boundary layer produce significant changes on the fluctuations within the viscous region. The same analysis also proves useful for computing to second-order accuracy the boundary-layer response induced by a three-dimensional gust with spanwise wavelength comparable with the boundary-layer thickness. It also follows that the boundary-layer fluctuations of the streamwise velocity match the corresponding free-stream velocity component. The velocity profiles are compared with experimental data, and good agreement is attained.The generation of Tollmien–Schlichting waves by the nonlinear mixing between the two-dimensional unsteady vorticity fluctuations and the mean flow distortion induced by localized wall roughness and suction is also investigated. Gusts with small wall-normal wavelengths generate significantly different amplitudes of the instability waves for a selected range of forcing frequencies. This is primarily due to the disparity between the streamwise velocity fluctuations in the free stream and within the boundary layer.