End Depth under Zero-Inertia Conditions

The terminal depth and velocity conditions for flow in an open channel ending in a free overfall are investigated under the assumption that inertial effects are everywhere negligible. When inertial terms are retained, as in the Saint-Venant or varied-flow equations, these terminal conditions are both at critical. When not only the inertial terms but the depth-gradient term too is dropped, depth and velocity at the brink of the overfall are at normal. Under the zero-inertia assumption, end depth is found to be zero and the velocity correspondingly infinite. This notion is tested in a series of steady-flow comparisons with results obtained from the varied-flow equation. The results of the zero-inertia assumption lie close to those of the varied-flow equation, when the flow conditions are characterized by low Froude numbers. In application to the problem of discharge from a lake into a channel ending in an overfall, the traditional trial-and-error solution is replaced by a new direct solution read off from a graph.