A solution for three-dimensional vortex flows with strong circulation

Abstract

The Navier-Stokes equations for a viscous, incompressible fluid are considered for a steady, axisymmetric flow composed of a strong rotation combined with radial sink flow which exhausts axially inside a finite radius. The equations are reduced to two coupled partial differential equations in terms of the stream function and circulation. The equations contain three dimensionless parameters: the radial Reynolds number, a characteristic ratio of mass flow per unit lenght to circulation, and a characteristic ratio of an axial dimension to a radial dimension. The product of these last two dimensionless parameters is used as a new expansion parameter for generating an asymptotic series solution. To zeroth order in this parameter, the solution for the stream function is a linear distribution between two axial boundary values. First-order correction terms are calculated for a specific example.In discussing these equations the limitations of the exact solutions due to Donaldson & Sullivan (1960) and Long (1961) are noted. These exact solutions are contrasted with the approximate treatment of this type of vortex originated by Einstein & Li (1951) and generalized by Deissler & Perlmutter (1958).