Upstream influence

Abstract

The phenomenon under investigation occurs in various flow systems characterized by being dispersive to small-amplitude waves, but such that phase and group velocities approach the same finite limit at extreme wavelengths: for example, water in an open channel, density-stratified fluid flowing between horizontal boundaries, and rotating fluid contained in a tube. It is well known in each of these examples that, when a solid body is moved steadily at a subcritical velocity (i.e. less than the long-wave limit) relative to the undisturbed fluid, the body experiences resistance accountable to the continual development of a pattern of waves on the leeward side. But in these circumstances there is a second effect, upstream influence, consisting of a disturbance in the form of a uniform long wave that extends to a steadily increasing distance ahead of the body; and this has for some time been the subject of a controversy, particularly with regard to the question whether or not it is always present. The aim of the paper is to establish the principle that this second effect is inevitably coexistent with the first, being an essential component of the mechanism of wave resistance.

On the effects of non-linearity in free-surface flow about a submerged point vortex

“…We suspect that the mean free-surface elevation for this wave cycle is slightly below that of the Stokes waves which must be formed infinitely far downstream, and that the values of ~ in Table 1 are therefore a little low. However, the Table clearly indicates that 7/ is a uniformly increasing function of c. Note that the opposite trend is often the case in water of finite depth; for a discussion of this case the reader is referred to Benjamin [14], Salvesen and yon Kerczek [6] and Forbes [15]. We turn now to perhaps the more interesting case of negative circulation.…”

Laboratory measurements of deep-water breaking waves

“…The mean momentum density around the packet, after the transients have moved away from the packet, is theoretically zero in deep water and slightly negative in shallower water, according to equation (A1.54) (Whitham, 1977;Benjamin, 1970).…”

Nonlinear and Wave Theory Contributions of T. Brooke Benjamin (1929–1995)

“…Perhaps because this effect is often too small to observe, it used to be neglected in relevant flow calculations. But as Benjamin (1970) pointed out, these upstream perturbations have a controlling influence on many geophysical and engineering flows. He analyzed disturbances created in slowly moving subcritical currents whose speed U is less than that of the speed c of long waves, for the particular cases of open-channel water flow, horizontal stably stratified flows, and rotating flows (with the current running parallel to the axis of rotation).…”