Marshall, Thomas James; (2004) A study of three-dimensional effects in end-stage boundary layer transition. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

A theoretical study into the three-dimensional effects at the end-stages of transition of laminar flow to fully turbulent flow in a boundary layer is presented. Various theoretical approaches have been broadly successful in capturing and explaining the early stages of the entry of three-dimensionality but a description of the late stages eludes satisfactory theoretical description. The particular aspect of this that we concentrate on here is the inclusion of three-dimensional effects which are vital for the generation of so-called "Omega" (ω) vortices. These herald the last stages of transition and the subsequent interactions between them and the fluid closer to the wall causes rapid transition to turbulence. Asymptotic methods are used to derive a system of nonlinear integro-differential equations, which extends previous work to three-dimensions. In particular they describe the capture of fluid in spanwise vortices of finite spanwise extent centred about a critical layer. The behaviour of these derived equations is studied. An analytical study of the critical layer equations gives a condition for the existence of a non-zero critical layer jump term. An expression for the spanwise critical layer jump term is found but it involves a triple repeated integral and a computational study is required to make further progress. The feedback of the developing vortices onto the flow is found to be particularly strong, emphasizing the importance of the entrance of three-dimensionality earlier in the flow development. This leads to the breakdown of the proposed theory and new scalings are needed. An explanation for this strong feedback is presented. However further analytic progress is made through a consideration of a relatively weak interaction between the critical layer and the bulk of the flow. This is analogous to delaying the feedback mechanism and we study the behaviour of recirculating flow within a vortex by considering two active timescales.

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