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Simplified equations for the interaction of nearly parallel vortex filaments

Published online by Cambridge University Press:  26 April 2006

Rupert Klein ,
Andrew J. Majda  and
Kumaran Damodaran
Rupert Klein
Affiliation:
Institut fur Technische Mechanik, RWTH, Templergraben 64, 52056, Aachen, Germany
Andrew J. Majda
Affiliation:
Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
Kumaran Damodaran
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

New simplified asymptotic equations for the interaction of nearly parallel vortex filaments are derived and analysed here. The simplified equations retain the important physical effects of linearized local self-induction and nonlinear potential vortex interaction among different vortices but neglect other non-local effects of self-stretching and mutual induction. These equations are derived systematically in a suitable distinguished asymptotic limit from the Navier–Stokes equations. The general Hamiltonian formalism and conserved quantities for the simplified equations are developed here. Properties of these asymptotic equations for the important special case involving nearly parallel pairs of interacting filaments are developed in detail. In particular, strong evidence is presented that for any filament pair with a negative circulation ratio, there is finite-time collapse in a self-similar fashion independent of the perturbation but with a structure depending on the circulation ratio. On the other hand, strong evidence is presented that no finite-time collapse is possible for perturbations of a filament pair with a positive circulation ratio. The present theory is also compared and contrasted with earlier linear and nonlinear stability analyses for pairs of filaments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 288 , 10 April 1995 , pp. 201 - 248
Copyright
© 1995 Cambridge University Press

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