Dr Mark Blyth (University of East Anglia): Critical free surface flow over localised topography
Two-dimensional free-surface flow over a localised bottom topography at critical Froude number (F=1) is examined with an emphasis on calculating steady, forced solitary-wave solutions. In particular we focus on the case of a Gaussian dip topography. Most of the focus is on the weakly-nonlinear limit where a forced KdV equation is applicable. Boundary-layer theory is used to construct asymptotic solutions in appropriate limits. For KdV essentially the problem boils down to solving a second order nonlinear ordinary differential equation with a forcing term and a single parameter quantifying the amplitude of the topography. This equation has a rich solution space with a large (probably infinite) number of solution branches. The asymptotics for small amplitude topography reveal some interesting features, for example an internal boundary layer which mediates a change from exponential to algebraic decay of the free-surface tail in the far-field. Traditional boundary-layer theory fails beyond the first two solution branches, where the surface profiles feature multiple waves trapped over the dip. Solitary waves on the first few solution branches are also found for the fully nonlinear potential flow problem using a conformal mapping method. The stability of the steady solutions will also be discussed.