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

Published online by Cambridge University Press:  26 April 2006

R. Fernandez-Feria ,
J. Fernandez de la mora  and
A. Barrero
R. Fernandez-Feria
Affiliation:
Universidad de Málaga, ETS Ingenieros Industriales, 29013 Málaga, Spain
J. Fernandez de la mora
Affiliation:
Yale University, Mechanical Engineering Dept., New Haven, CT 06520-2159, USA
A. Barrero
Affiliation:
Universidad de Sevilla, ETS Ingenieros Industriales, 41012 Sevilla, Spain
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Abstract

Many axisymmetric vortex cores are found to have an external azimuthal velocity v, which diverges with a negative power of the distance r 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∼r−1. The present work considers the more general situation of a family of self-similar inviscid vortices for which v∼rm−2, where m is in the range 0 n< m < 2. This includes Longs 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 1m < 2) or smaller (when 0m < 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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 305 , 25 December 1995 , pp. 77 - 91
Copyright
© 1995 Cambridge University Press

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