An extension of the hydrodynamic source-sink method for axisymmetric bodies

Abstract

Real flow patterns are produced by formally placing a pair of conjugate complex sources at conjugate complex points on the axis of symmetry. These complex singularities are shown to be equivalent to a non-uniform distribution of real doublets on a real disc. Reciprocal relationships are formulated between these new singularities and the well-known simple source ring and vortex ring. While the latter are simpler physically, the new type of singularity is easier to handle in mathematical analysis, involving only square roots instead of elliptic integrals. Sufficient conditions are determined under which an axisymmetric body may be generated by a real distribution of sources and sinks along the axis of symmetry, and the formula for the source intensity is given when these conditions are satisfied. An example deals with the flow about all oblate spheroid.