Linear and nonlinear instabilities of a Blasius boundary layer perturbed by streamwise vortices. Part 1. Steady streaks

Abstract

This paper investigates the stability properties of a Blasius boundary layer perturbed by a small-amplitude steady three-dimensional distortion, which may be isolated or periodic along the spanwise direction. It is shown that once the strength of the distortion exceeds a threshold, the perturbed flow becomes inviscidly unstable. An isolated distortion which features a dominant low-speed streak induces a localized mode, while a periodic distortion supports spatially quasi-periodicmodes through a parametric resonance. The frequencies and growth rates of both types of mode are much higher than those of the viscous Tollmien–Schlichting (T–S) waves. For moderate distortions, the instability modes can be viewed as kinds of modified T–S waves, which amplify at rates in excess of the viscous instability. For a localized distortion, these modes do not reduce to the usual T–S wavesin the zero-distortion limit. The nonlinear development of the inviscid modes is also studied, and is foundto be governed by a slightly modified version of the evolution equation derived by Wu (1993). A numerical study suggests that nonlinearity has a strong destabilizing effect, and ultimately leads to an explosive growth in the form of a finite-distance singularity. The present theoretical model is found to capture qualitatively some key experimental observations.