The Axisymmetric Prandtl-Batchelor Eddy Behind a Circular Disc in a Uniform Stream

Abstract

Analytical support is given to Fornberg's numerical evidence that the steady axially symmetric flow of a uniform stream past a bluff body has a wake eddy which tends towards a large Hill's spherical vortex as the Reynolds number tends to infinity. The viscous boundary layer around the eddy resembles that around a liquid drop rising in a liquid, especially if the body is a circular disc, so that the boundary layer on it does not separate. This makes it possible to show that if the first-order perturbation of the eddy shape from a sphere is small then the eddy diameter is of order R1/5 times the disc diameter, where R is the Reynolds number based on the disc diameter. Previous authors had suggested R1/3 and ln R, but they appear to have made unjustified assumptions.