A direct iteration method for the calculation of the velocity distribution of bodies of revolution and symmetrical profiles

Abstract

An integral equation is derived for the velocity on the surface of a given body of revolution or a given symmetrical profile in longitudinal flow, using the generation of a body by a vortex layer on its surface. The equation can be solved by the usual iteration method for linear integral equations of the second kind. The method of generating a body by a vortex layer leads to formulae expressing the components of the velocity outside the body by means of its pressure distribution. In general, the numerical work is greater for this method than for the indirect methods which use essentially a generation of the body by a distribution of sources and sinks on its axis. On the other hand, the method is not restricted to the case in which the analytic continuetion of the flow into the interior of the body does not meet singularities outside the axis. The only requirement is that the shape of the body must have a continuous tangent, whereas its curvature may have isolated discontinuities. Numerical examples are given for the two-dimensional case of a semi-infinite plate of constant thickness with a semi-circular leading edge, and for the three-dimensional case of a semi-infinite cylinder with three different heads (hemispherical, 2 caliber ogival and ¼ caliber rounded). The calculation is in good agreement with experimental results. Some numerical tables are given in order to facilitate the calculation of the kernel of the integral equation which forms the major part of the numerical work.