Effect of free stream vorticity on spatial stability of incompressible boundary layers with nonparallel effects

Abstract

The Falkner–Skan boundary layer is perturbed by an imposed mean free-stream vorticity. The stability of the perturbed flow to Tollmien–Schlichting waves is considered within the framework of the quasiparallel approximation (boundary layer growth terms are included but the x variation of disturbance characteristics is neglected). Falkner–Skan boundary-layer flows with Hartree pressure gradient parameter β=1.0, 0.8, 0.6, 0.1, 0.0, −0.1, and −0.14 are considered. It is found that positive free-stream vorticity is stabilizing and negative free-stream vorticity is destabilizing. For positive free-stream vorticity and starting at β=−0.14, the combined effects of the nonparallel flow effects and positive free-stream vorticity on the critical Reynolds number first decreases as β becomes more positive, reaches a minimum, and then increases again. Similarly, for negative free-stream vorticity, the effect on the critical Reynolds number first decreases as β becomes more positive, reaches a minimum value, and then increases to an asymptotic value. At a given β, the critical Reynolds number increases as the free-stream vorticity parameter becomes more positive. A limited amount of computations for flat plate shows that the transition Reynolds number, as determined from Smith’s e9 method, is strongly affected by free-stream vorticity.