Improved calculations of waterfalls and weir flows

Abstract

Many works have considered two-dimensional free-surface flow over the edge of a plate, forming a waterfall, and with uniform horizontal flow far upstream. The flow is assumed to be steady and irrotational, whilst the fluid is assumed to be inviscid and incompressible. Gravity is also taken into account. In particular, amongst these works, numerical solutions for supercritical flows have been computed, utilising conformal mappings as well as a series truncation and collocation method. We present an extension to this work where a more appropriate expression is taken for the assumed form of the complex velocity. The justification of this lies in the behaviour of the waterfall flow far downstream and the wish to better encapsulate the parabolic nature of such a free-falling jet. New numerical results will be presented, demonstrating the improved shape of the new free-surface profiles. These numerical solutions will also be validated through comparisons with asymptotic solutions, in particular for flows with larger Froude numbers. For flows with Froude numbers closer to 1, we demonstrate that the revised complex velocity ansatz should be employed in place of the asymptotic solution. We present further adjustments to the method that lead to enhanced coefficient decay. The aforementioned adjustments are also applied to supercritical weir flows and similar improvements to the jet shape can be observed.