Instability of Blasius boundary layer in the presence of steady streaks

It is well known that the instability of boundary layers is sensitive to the mean velocity profile, so that a small distortion to the basic flow may have a detrimental effect on its stability. The main interest of the present paper is in investigating the mechanisms by which a relatively weak distortion can significantly affect the instability of an otherwiseuniform Blasius flow. Specifically, we shall address two issues: (a) how the TollmienSchlichting instability, which operates in the absence of any distortion, is modified by a weak distortion, and (b) whether or not a weak distortion is able to cause inviscid instability. Many factors can cause three-dimensional steady or unsteady distortions in the form of streamwise or longitudinal vortices. These include small steady or unsteady perturbations superimposed on the oncoming flow, imperfections at the leading edge, crossflow instability, and Görtler vortices induced by surface curvature, as well as certain excitation devices. Distortion of this kind also arises due to the nonlinear interaction between pairs of Tollmien-Schlichting waves. The resulting perturbed flows are spanwise-dependent but essentially unidirectional, i.e. the transverse velocity components are much smaller than the streamwise component. The instability of such transversely sheared flows has attracted a great deal of interest because it appears to be related to various aspects of the transition process, such as secondary instabilities and by-pass transition. A distortion of particular interest occurs when the boundary layer is subject to relatively high free-stream turbulence level. As was first observed by Dryden (1936) and Taylor (1939), small low-frequency three-dimensional perturbations in the free stream produce significant distortion within the boundary layer, leading to alternating thickening and thinning of the layer in the spanwise direction. Steady disturbances also cause a similar type of variation (Bradshaw 1965). Recent experimental studies show that the distortions are in the form of elongated streaks (see e.g. Kendall 1985, Westin et al. 1994, Alfredsson & Matsubara 2001, and references therein), now commonly referred to as the Klebanoff mode, as a tribute to the contribution of Klebanoff (1971). These experiments have provided fairly complete quantitative data about the characteristics of Klebanoff modes themselves. However, the instability of the streaks and its role in the transition process remain poorly understood. The main obstacle of course is the random nature (in both time and space) of free-stream disturbances and the associated Klebanoff motion, which make a quantitative study extremely hard. Numerous researchers instead investigated steady distortions, induced in a controlled manner. Hamilton & Abernathy (1994) used surface roughness elements to create single or multiple streamwise vortices. These vortices cause the distorted flow profile to have an