A note on a Galerkin technique for integral equations in potential flows

Abstract

The properties are studied of a Galerkin numerical solution of integral equations for an assumed singularity distribution or a velocity potential arising in potential flows around rigid bodies in incompressible aerodynamics, acoustics and surface waves. The body boundary is approximated by a collection of panels and the integral equation is averaged over each panel instead of being enforced at a ‘collocation’ point. For the resulting Galerkin synthesis the matrix equation obtained for the source distribution is the exact transpose of the corresponding equation obtained for the velocity potential on the body boundary, a property known to hold for the continuous operators. Moreover, the integrated hydrodynamic forces experienced by the body are shown to be identically predicted by the source-distribution method or by directly solving for the velocity potential.