On the relation between Long’s vortex and general self-similar vortex cores

Abstract

We have found that the existence/nonexistence behavior of self-similar axisymmetric vortices studied by Fernandez-Feria, Fernandez de la Mora, and Barrero [J. Fluid Mech. 305, 77 (1995)], is not particularly different from that for Long’s vortex. In fact, in both cases the existence of a solution appears to depend exclusively on the value of the flow force in a disk of radius r at the inflow divided by the square of the circulation of the inflow at the same radius. However, for Long’s vortex the value of this quotient does not affect the ratio of the azimuthal and axial velocities at the edge of the layer, which it does for all other self-similar axisymmetric vortices. This result provides a simple explanation of the existence properties for the general self-similar axisymmetric vortices, quite consistent with the corresponding properties for Long’s vortex.