Potential Flow over Weirs of Moderate Curvature

A review is given of some salient studies of critical flow over round-crested weirs, in particular Lenau's extension of Matthew's results. The basic equations of two-dimensional, incompressible, potential flow are recapitulated and applied to general free-surface, gravity flows. Approximate solutions in the form of series expansions about the free surface for ψ, ln v, and Θ in terms of x and y are derived. An assumed free surface profile of simple exponential form is investigated and is shown to represent a weir type of flow provided certain relationships exist between the free surface equation parameters. These are related to the ratio of total head above crest to radius of curvature of that crest (H/R) and the coefficient of discharge and the weir surface profile are deduced as series functions of that ratio. Compatibility with the Lenau theory is shown. This method brings out both the attractions and difficulties inherent in an inverse method remaining within a cartesian framework.