TY - JOUR

T1 - Asymptotic equations for the stretching of vortex filaments in a background flow field

AU - Klein, Rupert

AU - Majda, Andrew J.

AU - McLaughlin, Richard M.

PY - 1992

Y1 - 1992

N2 - Recently, two of the authors have derived [Physica D 49, 323 (1991)] and analyzed [Physica D 53, 267 (1991)] a new asymptotic equation for the evolution of small-amplitude short-wavelength perturbations of slender vortex filaments in high Reynolds number flows. This asymptotic equation differs significantly from the familiar local self-induction equation in that it includes some of the nonlocal effects of self-stretching of the filament in a simple fashion. Here, through systematic asymptotic expansions, the authors derive a modification of this asymptotic equation that incorporates the important additional effects of strain and rotation from a general background flow field. The main requirement on the background flow is that it does not displace the unperturbed background filament. The new asymptotic equations exhibit in a simple fashion the direct competition in filament dynamics between internal effects such as self-induction and self-stretching and external effects of background flows involving strain and rotation. Solutions of these asymptotic equations revealing various aspects of this competition are analyzed in detail through both theory and numerical simulation. An application is also presented for the nonlinear stability of a columnar vortex to suitable perturbations in a straining, rotating, background environment.

AB - Recently, two of the authors have derived [Physica D 49, 323 (1991)] and analyzed [Physica D 53, 267 (1991)] a new asymptotic equation for the evolution of small-amplitude short-wavelength perturbations of slender vortex filaments in high Reynolds number flows. This asymptotic equation differs significantly from the familiar local self-induction equation in that it includes some of the nonlocal effects of self-stretching of the filament in a simple fashion. Here, through systematic asymptotic expansions, the authors derive a modification of this asymptotic equation that incorporates the important additional effects of strain and rotation from a general background flow field. The main requirement on the background flow is that it does not displace the unperturbed background filament. The new asymptotic equations exhibit in a simple fashion the direct competition in filament dynamics between internal effects such as self-induction and self-stretching and external effects of background flows involving strain and rotation. Solutions of these asymptotic equations revealing various aspects of this competition are analyzed in detail through both theory and numerical simulation. An application is also presented for the nonlinear stability of a columnar vortex to suitable perturbations in a straining, rotating, background environment.

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U2 - 10.1063/1.858467

DO - 10.1063/1.858467

M3 - Article

AN - SCOPUS:33746712680

VL - 4

SP - 2271

EP - 2281

JO - Physics of fluids. A, Fluid dynamics

JF - Physics of fluids. A, Fluid dynamics

SN - 0899-8213

IS - 10

ER -