Theory of High-Aspect-Ratio Planing Surfaces

Abstract

Abstract : A high-aspect-ratio planing surface gliding on a stream of an infinitely deep, incompressible, inviscid and gravity-free fluid is treated. This complicated problem is decomposed into two relatively simpler boundary- value problems: (1) The near-field boundary-value problem is valid only in the neighborhood of the planing surface. The problem is solved by the classical hodograph method. The second-order inner problem is also shown to be a plane, irrotational flow and the solution is obtained by following the same procedure as in the first-order inner solution. (2) The far-field boundary-value problem is valid only far away from the planing surface. The first-order outer solution is shown to be a trivial uniform flow. The outer velocity potential is defined in the whole space by harmonic continuation. The second-order solution is then shown to be similar to a lifting-line solution. The unknown strength of singularities is obtained by matching of the velocity potential. Then a matching of the free-surface deflection provides a height reference for the planing surface. The location of the planing surface with respect to the undisturbed free surface is uniquely defined. In order to obtain a unique second-order solution, it is necessary to solve the third-order outer solution. The detail of this solution is presented. A numerical solution for a planing plate of arbitrary angle of attack is presented. A downwash correction is also included. It is shown mathematically that the present theory can be applied to V-shape or general-shape planing surfaces with curvature in the spanwise direction.