Effects of long-wavelength Klebanoff modes on boundary-layer instability

It is known that low-frequency components of three-dimensional vortical disturbances in the free stream can be entrained into the boundary layer due to the nonparallel flow effect, producing significant distortion in the form of alternate thickening and thinning of the layer in the spanwise direction. This observation goes back to Dryden (1936) and Taylor (1939) who, in fact, suggested that the entrained vortex motion, rather than the Tollmien-Schlichting (T-S) instability, was the cause of transition to turbulence. The dispute continued until the experiments of Schubauer and Skramstad (1948), which were conducted by minimizing the free-stream perturbations, fully validated the instability theory of Tollmien (1929) and Schlichting (1933). Since then, most research efforts have focused on transition at low levels of free-stream turbulence. There has also been a significant amount of research on transition at moderate to high free-stream turbulence levels, primarily because of its relevance to turbomachinery flows. This has led to renewed interest in the findings of Dryden (1936) and Taylor (1939). Recent experimental studies (see e.g. Kendall 1985, Westin et al. 1994, Matsubara & Alfredsson 2001, and the references therein) show that the boundary layer filters out the high-frequency components of free-stream turbulence, while amplifying the lowfrequency parts of the signature. The distortion within the boundary layer is dominated by streamwise velocity fluctuations, which are manifested in the form of longitudinal vortices or streaks. These streaks are now referred to as Klebanoff modes, in recognition of the contribution of Klebanoff (1971). In this paper, we shall refer to them as Klebanoff distortions or fluctuations, so as to avoid possible confusion when genuine instability modes are being discussed. The boundary-layer response to small-amplitude unsteady vortical disturbances was calculated by Gulyaev et al. (1989) and Choudhari (1996) using linearized unsteady boundary-layer equations. Leib et al. (1999) pointed out that this approach is restricted to the region relatively close to the leading edge where the spanwise length scale of the perturbation is much larger than the local boundary-layer thickness. The continued growth of the perturbation amplitude and boundary-layer thickness implies that nonlinearity and cross-flow ellipticity will become significant sufficiently farther downstream, at which point the flow must be governed by boundary-region equations. For further work in the context of boundary-region equations, the reader is referred to the papers by Wundrow & Goldstein (2001), Goldstein & Wundrow (1998) and other references therein. Direct laboratory investigations of the transition process in the presence of Klebanoff fluctuations have been made by a number of investigators. At moderate levels of freestream turbulence, Arnal & Juillen (1978) and Kendall (1990) have observed intermittent appearance of wave packets inside the boundary layer. While their exact origin remains unclear, a series of experiments conducted by Kendall (1991, 1998) has helped reveal