Theory of optimum shapes in free-surface flows. Part 2. Minimum drag profiles in infinite cavity flow

The problem considered here is to determine the shape of a symmetric twodimensional plate so that the drag of this plate in infinite cavity flow is a minimum. With the flow assumed steady and irrotational, and the effects due to gravity ignored, the drag of the plate is minimized under the constraints that the frontal width and wetted arc-length of the plate are fixed. The extremization process yields, by analogy with the classical Euler differential equation, a pair of coupled nonlinear singular integral equations. Although analytical and numerical attempts to solve these equations prove to be unsuccessful, it is shown that the optimal plate shapes must have blunt noses. This problem is next formulated by a method using finite Fourier series expansions, and optimal shapes are obtained for various ratios of plate arc-length to plate width.