Solution breakdown in a family of self-similar nearly inviscid axisymmetric vortices

Many axisymmetric vortex cores are found to have an external azimuthal velocity v, which diverges with a negative power of the distance v to their axis of symmetry. This singularity can be regularized through a near-axis boundary layer approximation to the Navier-Stokes equations, as first done by Long for the case of a vortex with potential swirl, v similar to r(-1). The present work considers the more general situation of a family of self-similar inviscid vortices for which v similar to r(m-2), where m is in the range 0 < m < 2. This includes Long's vortex for the case m = 1. The corresponding solutions also exhibit self-similar structure, and have the interesting property of losing existence when the ratio of the inviscid near-axis swirl to axial velocity (the swirl parameter) is either larger (when 1 < m < 2) or smaller (when 0 < m < 1) than an m-dependent critical value. This behaviour shows that viscosity plays a key role in the existence or lack of existence of these particular nearly inviscid vortices, and supports the theory proposed by Hall and others on vortex breakdown. Comparison of both the critical swirl parameter and the viscous core structure for the present family of vortices with several experimental results under conditions near the onset of vortex breakdown show a good agreement for values of m slightly larger than 1. These results differ strongly from those in the highly degenerate case m = 1.