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Abstract

The hydrodynamic stability of flows in channels of slowly varying width is presented in this thesis. Two classes of channel are considered, a straight-wall divergent channel and a channel with curved walls. A particular curved-wall channel is taken, whose walls have equations

y = +[1 + tanh (ex) , (not formatted correctly; read the abstract on the pdf for the correct version).

where y is the cross-stream variable, x the downstream variable, and e is constant. These are the same physical situations as were considered by Eagles & Weissman (1975) (straight-wall channel) and Eagles & Smith (1980) (curved-wall channel). However the results presented here are obtained by a completely different method than was used in either paper.

The straight-wall channel is taken first. Having introduced the new co-ordinates and set up the governing equations for two-dimensional flow, the steady-state solution for the velocity profile is obtained numerically. The linearized equation for the disturbance stream function is obtained, and this solved by first obtaining the finite difference analogue equations and solving these by Gauss elimination. Growth rates based on the disturbance stream function, the disturbance kinetic energy density and the relative kinetic energy density are defined, and then the results presented. Comparisons are made with results from Eagles & Weissman with favourable agreement, and a graph of the critical Reynolds number based on the relative kinetic energy density versus the semi-divergence angle of the channel is presented.

For the curved-wall channel, the same order is followed as for the straight-wall channel. The steady state stream function equation is solved by two methods, one numerical and the other an expansion method. The results for the disturbance stream function and critical Reynolds number show reasonable agreement with those of Eagles & Smith. The critical Reynolds number drawn against E is shown.

Finally, the effect of a localized initial perturbation is studied, which necessitates the solution of an initial-boundary-value problem, the linearized equation for the disturbance stream function. This was first replaced by two coupled-equations, one of which was solved by an explicit finite difference scheme, the other (Poisson's equation) by a direct method based on Fourier analysis. A graph showing the evolution of the disturbance as it travels downstream is presented.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Department of Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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