No abstract

The subcritical flow of a stream over a bottom obstruction or depression is considered with particular interest in obtaining solutions with no downstream waves. In the linearised problem this can always be achieved by superposition of multiple obstructions, but it is not clear whether this is possible in a full nonlinear problem. Solutions computed here indicate that there is an effective nonlinear superposition principle at work as no special shape modifications were required to obtain wave-cancelling solutions. Waveless solutions corresponding to one or more trapped waves are computed at a range of different Froude numbers and are shown to provide a rather elaborate mosaic of solution curves in parameter space when both negative and positive obstruction heights are included.

We examine a problem in which a line sink causes a disturbance to an otherwise uniform flowing stream of infinite depth. We consider the fully non-linear problem with the inclusion of surface tension and find the maximum sink strength at which steady solutions exist for a given stream flow, before examining non-unique solutions. The addition of surface tension allows for a more thorough investigation into the characteristics of the solutions. The breakdown of steady solutions with surface tension appears to be caused by a curvature singularity as the flow rate approaches the maximum. The non-uniqueness in solutions is shown to occur for a range of parameter values in all cases with non-zero surface tension.

This paper re-examines the problem of the flow of a fluid of finite depth over two Gaussian-shaped obstructions on the stream bed. A weakly nonlinear analysis in the form of the Korteweg–de Vries equation is used to compare with the results of the fully nonlinear problem. The main focus is to find waveless subcritical solutions, and contours showing the obstruction height and separation values that result in waveless solutions are found for different Froude numbers and different obstruction widths.