Simplified asymptotic equations describing the nonlinear dynamics of perturbed pairs of parallel vortex filaments are derived and analyzed here. The derivations are general enough to allow for vortices of unequal strength, but emphasis here is on the antiparallel vortex pair. The simplified asymptotic equations account for both the internal effects of self‐induction and self‐stretching for each filament and also the external effects of mutual induction that lead to a nontrivial coupling of the perturbations of the two filaments. When these nonlinear equations are linearized at the unperturbed filament pair, the linearized stability theory of Crow [AIAA J. 8, 2172 (1970)] is recovered in a systematic fashion. The asymptotic equations are derived in a novel singular limit at high Reynolds numbers through assumptions similar to the authors’ recent theories [Physica D 49, 323 (1991); ibid. 53, 267 (1991); Phys. Fluids A 4, 2271 (1992)] for the dynamics of a single perturbed vortex filament. Through the Hasimoto transform [J. Fluid Mech. 51, 477 (1972)], these equations become two coupled perturbed nonlinear Schrödinger equations for a pair of filament functions. A series of numerical solutions of the asymptotic equations exhibits several new phenomena in the nonlinear instability of pairs of antiparallel vortex filaments.
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February 1993
This content was originally published in
Physics of Fluids A: Fluid Dynamics
Research Article|
February 01 1993
An asymptotic theory for the nonlinear instability of antiparallel pairs of vortex filaments
Rupert Klein;
Rupert Klein
Institut für Technische Mechanik, RWTH Aachen, Templergraben 55, W‐5100 Aachen, Germany
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Andrew J. Majda
Andrew J. Majda
Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544
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Phys. Fluids 5, 369–379 (1993)
Article history
Received:
May 26 1992
Accepted:
October 03 1992
Citation
Rupert Klein, Andrew J. Majda; An asymptotic theory for the nonlinear instability of antiparallel pairs of vortex filaments. Phys. Fluids 1 February 1993; 5 (2): 369–379. https://doi.org/10.1063/1.858860
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