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Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes

Published online by Cambridge University Press:  26 April 2006

M. J. Shelley ,
D. I. Meiron  and
S. A. Orszag
M. J. Shelley
Affiliation:
Computational and Applied Mathematics Program, Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
D. I. Meiron
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, USA
S. A. Orszag
Affiliation:
Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
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Abstract

The phenomenon of vortex reconnection is analysed numerically and the results are compared qualitatively with the predictions of a model of reconnection recently proposed by Saffman. Using spectral methods over both uniform and strained meshes, numerical simulations are performed of two nearly parallel, counter-rotating vortex tubes, over the range of Reynolds numbers Re = 1000–3500. The calculations utilizing a uniform mesh are performed for Re ≤ 1500 with a resolution of 128 points in each direction. The calculations utilizing a stretched mesh are performed for 1500 < Re ≤ 3500 with a resolution of up to 160 points in each direction and with a fourfold stretching about the region of reconnection. We present results for the variation of the maximum of vorticity, the time to reconnection, and other diagnostics of this flow as functions of the Reynolds number. From numerical simulation of the model equations, we infer and demonstrate the existence of exact solutions to the model to which its solutions arising from more general initial conditions are attracted at late times. In the limit of infinite Reynolds number, the model predicts eventual saturation of the axial strain, a feature observed in the recent work of Pumir & Siggia and also observed in our full numerical simulations. In this respect the model captures the observed local dynamics of vortex stretching. However, because the global effects of external flows are not included in the model, the model predicts that the axial strain eventually decays and the maximum vorticity grows linearly at late times. In contrast, from the full simulations, we see the possible emergence of the behaviour of the axial strain at infinite Reynolds number. As our simulations are affected by non-local effects, we do observe saturation of the strain but no subsequent decay. It is also shown analytically that the model predicts a reconnection time which varies logarithmically with increasing Reynolds number. Comparison with the full numerical simulations shows a much slower variation of the reconnection time with increasing Reynolds number than predicted by the model. Other points of agreement and disagreement between the Saffman model and the simulations are discussed, Reconnection is also discussed from the point of view of its relation to the possible onset of nearly singular behaviour of the Euler equation. In agreement with the recent numerical results of Pumir & Siggia, our results suggest that no singularity in the vorticity will form in a finite time for this initial condition.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 246 , January 1993 , pp. 613 - 652
Copyright
© 1993 Cambridge University Press

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