Linear and nonlinear instabilities of a Blasius boundary layer perturbed by streamwise vortices. Part 2. Intermittent instability induced by long-wavelength Klebanoff modes

This paper presents theoretical results on the instability of a Blasius boundary layer perturbed by Klebanoff modes (i.e. the low-frequency streaks known to be induced by free-stream turbulence). Herein, the Klebanoff distortions are modelled as the signature of a three-dimensional convected gust that may be either isolated or periodic along the spanwise direction. Relatively weak Klebanoff fluctuations can produce O(1) changes to the near-wall curvature of the base flow profile and, hence, fundamentally alter the nature of its instability characteristics. The perturbed flow is shown to support instabilities that are predominantly inviscid and have significantly larger growth rates and characteristic frequencies than the Tollmien-Schlichting (T-S) modes of an unperturbed Blasius flow. The spanwise mode shape of instabilities in the perturbed flow is determined by the Schrodinger equation, with a potential function that corresponds to the skin friction perturbation due to the Klebanoff distortion. The growth rates of these modes are determined by the near-wall torsion of the perturbed flow. The unsteadiness of the Klebanoff distortion is shown to be a crucial element in determining the overall instability characteristics. A localized Klebanoff distortion supports both sinuous and varicose modes of instability, but the sinuous modes are generally more unstable than the varicose modes. Overall, the instability is intermittent in time and localized in space, being confined to certain parts of the modulation cycle and within a specific window(s) along the streamwise direction. In particular, the dominant sinuous modes appear only during the phase in which a low-speed streak dominates the Klebanoff distortion. A periodic distortion supports spatially quasi-periodic modes through a parametric resonance mechanism. The theoretically predicted instability modes share some key features with the unstable disturbances measured in recent experiments, such as the relatively high frequencies, growth rates that depend on the level of free-stream turbulence, small rate of spreading in the lateral direction and, above all, their intermittency in space and time. Non-equilibrium critical-layer theory is used to track a localized sinuous mode through two distinct stages of nonlinear evolution, which eventually terminates in a singularity that indicates the onset of fully nonlinear yet primarily inviscid disturbance dynamics.