Hoyle, Jonathan Michael; (1992) Extensions to the theory of finite-time breakdown of unsteady interactive boundary layers. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis aims to expand upon the existing works of Brotherton-Ratcliffe and Smith (1987), and Smith (1988), which consider the structure of a collapse in the solution of various unsteady, interactive boundary-layer flows at high Reynolds numbers. Applications include transition and dynamic stall in boundary layers in aerodynamics, liquid-layer flows and flows over roughness elements. Initially we discuss the extension of the theory into three spatial dimensions and this proceeds successfully along the lines of the first named paper above. The resultant three-dimensional pressure equation describes the nature of the breakdown form and is a second-order, partial differential equation of mixed type. Asymptotic solutions for the far-field are found which are then incorporated into a numerical approach to the problem. Various numerical schemes and solutions are presented. On considering the more general pressure-displacement relation addressed in Smith (1988) the three-dimensional case remains consistent/ although the analysis of the critical layer produced becomes very involved, so much so that the leading-order jump condition for the streamwise and normal velocities is difficult to describe analytically. We then return to the two-dimensional case to consider the modification of the breakdown structure closer to the critical time. The new physical influence that enters the problem in this case is the emergence of normal pressure gradient effects influencing the lower-order terms. This problem is addressed and we find that a triple derivative term now enters the two-dimensional pressure relation to give an extended Korteweg-de-Vries equation. This modifies the assumed monotonic pressure behaviour until a zero pressure gradient point is approached, and here the working again needs alteration, rather than going on to produce the well-known soliton solutions. An analysis of the pressure behaviour about this point indicates the beginnings of a vortex wind-up effect. The latter is examined for the case of a small velocity jump being produced in the critical layer, although the general situation is more difficult to tackle.

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