Free surface flow over bottom topography
Date
2018
Authors
Keeler, Jack Samuel
Editors
Advisors
Binder, Benjamin
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Theses
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Abstract
This thesis explores flow in a channel over a bottom topography. In particular, the
problem of finding the shape of the unknown free-surface if the bottom topography is prescribed
forms the main problem of this thesis. In chapter 2 the forced Korteweg De- Vries equation is
derived from first principles as a model partial differential equation that determines the shape of
the unknown free-surface profile in terms of the topogra- phy. A discussion of the steady solution
space when the forcing is highly localised is also presented. In chapter 3 flow over bottom
topography at critical Froude number (when F = 1) is examined. For large amplitude negative
Gaussian forcing, asymptotic solutions are constructed using boundary layer theory; one point of
interest here is an internal layer away from the origin which mediates a change from exponential
decay away from the central dip to algebraic decay in the far-field. Intriguingly, solutions with
different numbers of waves trapped around the central dip are also found for large amplitude
topography but these cannot be captured by the boundary-layer analysis. In fact a seemingly
infinite sequence of solution branches is uncovered using numerical methods and a nonlinear
multiple-scales technique, and in general the solution for any given topography amplitude is
non-unique. In addition to these results the stability of the steady solutions is examined using
numerical simulations, linear stability analysis and formal stability analysis. In chapter 4 the
issue of existence of steady solutions is analysed for an algebraically decaying topography at
critical flow speed. For this topography the analysis is subtle and numerical solutions have to be
treated with care. In chapter 5 the solution space is studied for varying Froude number for flow
over a corrugated topography where a rich solution space is discovered. Finally, preliminary work
on the three-dimensional analogue of the fKdV equation, namely the fKP equa- tion is presented,
including a novel result regarding three-dimensional solitary waves
that decay in all spatial directions.
School/Discipline
School of Mathematical Sciences
Dissertation Note
Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2018
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