An iteration method for integral equations arising from gravity flows with free surface

Gravity potential flows with free surface still present considerable difficulties in non-linear mathematical problems. Previous researchers using analytic function theory could consider only simple geometrical boundaries. Analyzing curvilinear solid boundaries by means of analytic theory is a difficult problem that has not been solved. In this paper, using Muskhelishvili's singular integral equation theory, we turn the gravity flow problem into the Riemann-Hilbert problem. Taking the length of the streamline of the boundary as the independent variable and the velocity potential of the boundary as the function to be determined, we avoid the difficulty that the angle of the curved fixed part is unknown. Following the difference method and the finite element method, we develop a new numerical method that is suitable for complex solid boundaries and overcome the difficulties encountered in applying analytic function theory. Under known discharge, the convergence and stability of the method have been proved and an estimation of error has been obtained. The method has been successfully applied to the calculation of the flow past spillway buckets. The calculated values agree well with the measured results.