Exact actuator disk solutions for non-uniform heavy loading and slipstream contraction

Abstract

A semi-analytical method has been developed to solve for the inviscid incompressible flow induced by a heavily loaded actuator disk with non-uniform loading. The solution takes the contraction of the slipstream fully into account. The method is an extension of the analytical theory of Conway (1995) for the linearized actuator disk and is exact for an incompressible perfect fluid. The solutions for the velocities and stream function are given as one-dimensional integrals of expressions containing complete elliptic integrals. Any load distribution with bounded radial gradient can be treated. Results are presented here for both contra-rotating and normal propellers. For the special case of a contra-rotating propeller with a parabolic velocity profile in the ultimate wake, the vorticity in the slipstream is shown to be the same as in the analytically tractable spherical vortex of Hill (1894) and the related family of steady vortices explored by Fraenkel (1970, 1972) and Norbury (1973).