New analysis and numerical values for the classical dam problem

Abstract

The classical, two dimensional, dam problem was solved analytically and independently by Davison in 1932 and Hamel in 1934 and involved double integrals. In 1938 Polubarinova-Kochina produced solutions involving single integrals and in 1940 showed that the Davison-Hamel solutions could be reduced to ones obtained by her method. Missing from the conversion was a linkage between scale constants which is given in this paper. Initial calculations by these authors were difficult because of evaluations of elliptic integrals and numerical quadratures. As a free boundary problem it is now a test case for a multitude of numerical methods. Surveying analytical and numerical approaches, it is found that both are limited in accuracies and are overwhelmingly concerned with the location of the free boundary and its seepage point. Analytical expressions are here given for the gradients near the three critical points of free boundary entry and exit and the intersection of seepage face and tailwater. New numerical results to six decimal places are given for tables and contours on and within the dam boundary for potential and stream functions and orthogonal potential gradients.