Vortex formation in a free boundary layer according to stability theory

Abstract

An attempt is made to explain the formation of vortices in free boundary layers by means of stability theory using a hyperbolic-tangent velocity profile. The vorticity distribution of the disturbed flow, as obtained by the inviscid linearized stability theory, is discussed. The path lines of particles which are initially placed along straight lines parallel to the x-axis are calculated. Lines connecting the positions of these particles give an impression of the instant shape of the disturbed flow. With increasing time the boundary layer becomes thinner in certain regions and thicker in others. A special line—originally positioned at the critical layer—shows in the thicker region a tendency to roll up. Also extrema of the vorticity are located there. Finally, these results are compared with those which can be expected from the non-linear Helmholtz equation. Disagreement is found in the neighbourhood of the critical layer. Using the non-linear stability theory of Stuart up to the third-order terms, the vorticity distribution shows the tendency expected from the non-linear equation.