A numerical solution of an integral equation satisfied by the velocity distribution around a body of revolution in axial flow

Abstract

Knowledge of the pressure distribution on a body moving in a fluid is required in a variety of applications, such as the prediction of cavitation inception conditions and the study of boundary-layer development. For many purposes the effects of viscosity can be neglected and it is sufficient to consider inviscid incompressible flow. In general this is essentially the problem of solving Laplace's equation with the condition of no flow through the surface of the body, but in the special case of a smooth, simply connected body of revolution in uniform axial flow it can be reduced (Ref. 1) to finding the solution of a linear integral equation. This equation is satisfied by the velocity-distribution function on the body surface from which the pressure distribution can be found by using Bernoulli's equation. This report describes a numerical solution of the integral equation which is suitable for an automatic digital computer and which has been programmed for the Ferranti 'Pegasus'. Illustrative examples for the particularly difficult case of bodies of revolution with flat heads are given in the Appendix. The integral equation will not be derived here but it should be remarked that the mathematical model is a system of co-axial vortex rings (replacing the body) at rest in an axial, inviscid, incompressible, uniform flow of unit speed.