Waveless subcritical flow past symmetric bottom topography

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.

A note on waveless subcritical flow past symmetric bottom topography

“…This paper follows the work of Forbes [8] and Hocking et al [11], who found solutions to the nonlinear problem of subcritical flow over a semi-elliptical obstruction on the stream bed with no downstream waves, and Holmes et al [12], who found waveless nonlinear solutions for subcritical flow over two Gaussian obstructions. Forbes [8] calculated the ellipse height and length values that would produce waveless solutions at a Froude number of F = 0.5, with these results presented as contours in parameter space.…”

“…Cases involving obstructions of both positive and negative heights will be examined. In an earlier paper [12], the authors considered the fully nonlinear problem, and contours in parameter space representing waveless solutions were found. Here, the problem is revisited using a weakly nonlinear analysis in the form of the Korteweg-de Vries (KdV) equation, and comparisons are made with the results of the fully nonlinear problem.…”

“…Holmes et al [12] found the Gaussian obstruction height and separation values that resulted in waveless solutions for a range of Froude numbers from F = 0.5 to 0.7. Contours in parameter space representing waveless solutions were found for obstructions of positive and negative heights.…”

“…Waveless subcritical solutions are of particular interest and will be the main focus of this paper. As an extension to the work presented by Holmes et al [12], the suitability of the KdV equation for this problem is investigated by computing waveless solutions for flow over two Gaussian obstructions and comparing with the fully nonlinear solutions. Comparisons are made at different values of the Froude number starting with Froude numbers close to F = 1, before examining the effect of obstruction width.…”

On steady non-breaking downstream waves and the wave resistance

“…The converse statements of this corollary are not true because the waveless solutions with δ = 0, Fr = Fr d exist both for Fr < 1 (see Forbes 1982;Maklakov 1995;Holmes et al 2013) and Fr > 1 (see Forbes & Schwartz 1982;Vanden-Broek 1987). But the solutions with δ = 0, Fr < 1 exist only if the parameters of the disturbance are chosen by a special way.…”