SOLUTION BREAKDOWN IN A FAMILY OF SELF-SIMILAR NEARLY INVISCID AXISYMMETRICAL 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 situa tion 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 vo rtex for the case m = 1. The corresponding solutions also exhibit self -similar structure, and have the interesting property of losing existe nce when the ratio of the inviscid near-axis swirl to axial velocity ( the swirl parameter) is either larger (when 1 < m < 2) or smaller (whe n 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 th e critical swirl parameter and the viscous core structure for the pres ent family of vortices with several experimental results under conditi ons near the onset of vortex breakdown show a good agreement for value s of m slightly larger than 1. These results differ strongly from thos e in the highly degenerate case m = 1.