Davis, Dominic Andrew Robert; (1992) On linear and nonlinear instability in boundary layers with crossflow. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

The instability and transition of an incompressible boundary layer along a flat surface are considered theoretically and computationally in various linear and nonlinear regimes, with crossflow being the major new concern. The characteristic Reynolds number is taken as a large parameter throughout. The majority of the work describes so-called vortex/wave interactions which involve short-scale/long-scale interaction phenomena and divide roughly into two categories, namely for weakly nonlinear and strongly nonlinear theory. In Chapters Two to Five, the former theory is applied to two low-amplitude three-dimensional (3D) Tollmien-Schlichting waves present in a 3D boundary layer. The nonlinear interaction, near to the lower branch of the neutral curve, is controlled by a partial-differential system for the vortex flow together with an ordinary-differential equation for each wave pressure. Three distinct possibilities emerge for the nonlinear behaviour of the flow solution downstream - an algebraic finite-distance singularity, far-downstream exponential wave-decay or irregular oscillations - depending on the input amplitudes upstream, the wave angles, the size of the perturbed basic-flow wall-shear and the size of the crossflow. In Chapter Six, the interactive 3D boundary-layer equations from lower-branch theory are examined in the double limit of small streamwise distances and high frequencies, whereupon the equations are converted into a quasi-two-dimensional form. A nonlinear periodic solution for the displacement decrement is subsequently obtained. Chapters Seven and Eight describe strongly nonlinear Rayleigh-wave/vortex interactions for a slightly 3D boundary layer. Here, as the crossflow is increased, the more important part of the nonlinear interaction is concentrated in a relatively thin buffer layer in the midst of the boundary layer. The problem reduces to solving the generalised Rayleigh-wave pressure equation subject to a third-derivative wave-pressure discontinuity across the buffer layer, and a solution is obtained near the streamwise input station.

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