Year

2017

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

The flow of a fluid over topography in the long wavelength, weakly nonlinear limit is considered, for both isolated obstacles and steps or jumps. The upstream flow velocity is assumed to be close to a linear long wave velocity of the unforced flow, so that the flow is near resonant. Higher order nonlinear, dispersive and nonlinear-dispersive terms beyond the Korteweg-de Vries approximation are included, so that the flow is governed by a forced extended Korteweg-de Vries equation.

For the isolated obstacle, modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found and used to study the influence of the higher-order terms on the resonant flow, which increases for steeper waves. These modulation theory solutions are compared with numerical solutions of the forced extended Korteweg-de Vries equation for the case of surface water waves. Good comparison is obtained between theoretical and numerical solutions, for properties such as the upstream and downstream solitary wave amplitudes and the widths of the bores. They are also compared with numerical solutions of the forced extended Benjamin-Bona-Mahony equation, which is asymptotically equivalent to the forced extended Korteweg-de Vries equation, but is numerically stable for higher amplitude waves.

The usefulness of uniform soliton theory is also considered, for waves generated by an obstacle. It is based on the conservation laws of the extended Korteweg-de Vries equations for mass and energy and assumes that the upstream wavetrains is composed of solitary waves. We compare the solutions with theoretical and numerical solutions of the forced extended Korteweg-de Vries equation and the forced extended Benjamin-Bona- Mahony equation, to fully assess this approximation method for upstream solitary wave amplitude and wave speed.

The flow of a fluid over a step or jump is also examined, and is a variation on the problem of flow over an isolated obstacle. Higher-order modulation theory solutions, based on the extended Korteweg-de Vries equation, for the undular bores generated upstream and downstream of the forcing are found. It is shown that an upstream propagating undular bore is generated by a positive step and formed by an elevation upstream of the step, and a downstream propagating undular bore is generated by a negative step and formed by a depression downstream of the step. An excellent comparison is obtained between the analytical and numerical solutions.

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