Free surface flow over bottom topography

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.