Free-surface flow over an obstruction in a channel

Two-dimensional steady potential flow over a semicircular obstacle at the bottom of a channel is considered. The problem is solved numerically by using an integrodifferential equation formulation derived by Forbes and Schwartz [J. Fluid Mech. 114, 299 (1982)]. This equation is reduced to a set of algebraic equations by a difference method and solved by Newton’s method together with parameter variation. The numerical results for subcritical flows agree with those of Forbes and Schwartz. However, it is found that supercritical solutions exist only for values of the Froude number greater than some particular value. Furthermore, for some values of the Froude number, there are two supercritical solutions. One is a perturbation of a uniform stream whereas the other is a perturbation of a solitary wave.