Walton, Andrew Gerard; (1991) Theory and computation of three-dimensional nonlinear effects in pipe flow transition. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis consists of eight chapters. Following an introduction to the subject in Chapter One, Chapter Two describes a nonlinear instability of the fully-developed Hagen- Poiseuille flow, at large Reynolds number. It is seen that the disturbance waves, which have a long wavelength, are connected with a similar instability found by Smith and Bodonyi. In this chapter it is suggested how such an instability may be found by numerical integration of the governing equations. Chapter Three concentrates on the developing flow nearer the pipe entrance and investigates the effects of the interaction of a Tollmien-Schlichting (TS) wave of small amplitude with its induced streamwise vortex flow in such a situation. A linear analysis of the resulting interaction equations is subsequently performed and it is shown that spatial disturbances can experience exponential growth. Chapter Four is devoted to the flow near a symmetry line in the pipe, and the simplification to the interaction equations of Chapter Three that results in this case. These equations are used in Chapter Five, in which two possibilities for the ultimate downstream form of the flow produced by the vortex-TS interaction are considered. The first possibility is that the interaction results in a three-dimensional separation singularity at a finite distance downstream. A number of examples are outlined, illustrating the likely forms of the associated velocity profiles. The second option discussed in this chapter concerns the possibility that a strong attachment may take place in the flow, also at a finite distance downstream. Another possibility for the ultimate form of the interaction is analysed in Chapter Six. In this case the flow acquires a similarity form and numerical results for both the symmetry line and asymmetric versions of the interaction are presented. In Chapter Seven we return to the situation where the flow in the pipe is of a more developed nature, and we construct the vortex-wave interaction equations for this case. The resulting time-dependent equations are integrated numerically for the cases of axisymmetric and three-dimensional disturbances. Finally, Chapter Eight summarizes the findings of the thesis and suggests further avenues for related research.

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