Approximate solution of gravity-affected flow from planar sluice gate at high Froude number

Abstract

An approximate method is developed for solution of the full nonlinear equations governing irrotational flow with a free surface under a sluice gate, of small slope, at high Froude number. Schwarz-Christoffel transformation is used to map the region, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique, for small inclination angle, are used as a basis for the approximate solution of the problem. General equations, in integral form, for any order of approximation are obtained. A complete solution, up to first-order approximation, for the downstream free surface profile for different values of inclination angle, depth ratio and Froude number is discussed and illustrated. The contraction coefficient for different values of inclination angle has been found. The curvature of the downstream free surface has been obtained in general form and plotted as a function of the spatial variable for different values of the inclination angle. The results obtained are sufficiently accurate for practical purposes in case of high Froude number and small slope of the gate.