The stability of boundary layers on curved surfaces and surfaces involving abrupt changes

Abstract

This thesis is concerned with the effect that boundary-layer instabilities have on laminar- turbulent transition over a commercial aircraft wing. We consider the effect that changing the structure of a wing’s surface may have on these instabilities. This thesis is separated into two parts, each concerning a different instability. Firstly our focus is on Tollmien-Schlichting waves; we investigate how abrupt changes may affect boundary-layer transition. The abrupt changes considered are junctions between rigid and porous surfaces. A local scattering problem is formulated; the abrupt changes cause waves to scatter in a subsonic boundary layer. The mechanism is described mathematically by using a triple-deck formalism, while the analysis across the junctions is based in a Wiener- Hopf factorisation. The impact of the wall junctions is characterised by a transmission coefficient, defined as the ratio of the amplitudes of the transmitted and incident waves. From our analysis we determine the effectiveness of porous strips in delaying transition. In the second part of this thesis we concentrate on a curved wing. Over curved sur- faces Go ̈rtler vortices may be generated; our focus is on long-wavelength Go ̈rtler vortices and the effect of changing curvature. The flow is described using a three-tiered system that balances the displacement and centrifugal forces. Two different problems concerning Go ̈rtler vortices are investigated, firstly we consider the effect of slowly varying curvature. Using a WKB approximation we derive multi-scale systems of equations, allowing us to find leading-order analytic solutions. The second problem concerning curved surfaces considers the effect of long-wavelength Go ̈rtler vortex-wave interaction. We use vortex-wave interaction theory to describe the evo- lution of this nonlinear interaction over a concave surface, where the curvature is modified in the streamwise direction. Analytical solutions are found for the vortex-induced shear stress and the wave pressure amplitude, using these solutions we solve for the remaining variables numerically.