A model equation for steady surface waves over a bump

Abstract

The objective of this paper is to study the solutions of a model equation for steady surface waves on an ideal fluid over a semicircular or semielliptical bump. For upstream Froude number F> 1, we show that the numerical solution of the equation has two branches and there is a cut-off value of F below which no solution exists. For F < 1, the problem is reformulated to overcome the so-called infinite-mass dilemma. A branch of solutions and a cut-off value of F, above which no solution exists, are found. Furthermore, we also obtain a branch of hydraulic-fall solutions which decrease monotonically from upstream to downstream.

Steady capillary-gravity waves on the interface of a two-layer fluid over an obstruction-forced modified K-dV equation

“…If we used the long wave approximation as discussed in [4] and [5], we would obtain the forced K-dV equation…”

“…Thus the forced K-dV equation studied in [4] and [5] fails when p is near h 2 . Since the case for p not near h 2 has been studied in [4] and [5], in the following we assume p = h 2 + Er 2 and c = u 0 + 2A which implies A, = 0 and is the critical speed to have the long wave approximation. Finally we obtain the so-called forced modified K-dV equation (FMK-dV);…”

“…An asymptotic theory for smallamplitude steady flow past an obstruction has been developed by Shen, Shen and Sun [4] and Shen [5] and the model equation governing the flow is the Forced Korteweg-de Vries equation (FK-dV)…”

“…To find a solution in xl -1, we need only consider (33) in -l1-x<-0 subject to (i7'(x)) 2 = -a,7q4 l6+a 2 7q 2 at x= -1 and q'(x)=0 at x=0. This problem can be solved numerically by a shooting method and the phase shift x is then determined by (36) for x = -1.…”

A Study on Inviscid Flow with a Free Surface over an Undulating Bottom

“…Forbes (1985) studied the free-surface flow by considering a submerged point vortex in a single-layer fluid of infinite depth. Shen et al (1989) obtained the numerical solution of the problem involving an inviscid fluid flow over a semi-circular as well as a semielliptical obstacles. Dias and Vanden-Broeck (2002) solved the steady free-surface flow problem numerically, and demonstrated that there exist supercritical flows with waves downstream only.…”