Three-dimensional distortions of a vortex filament with axial velocity

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Abstract
Three-dimensional motion of a thin vortex filament with axial velocity, embedded in an inviscid incompressible fluid, is investigated. The deflections of the core centreline are not restricted to be small compared with the core radius. We first derive the equation of the vortex motion, correct to the second order in the ratio of the core radius to that of curvature, by a matching procedure, which recovers the results obtained by Moore & Saffman (1972). An asymptotic formula for the linear dispersion relation is obtained up to the second order. Under the assumption of localized induction, the equation governing the self-induced motion of the vortex is reduced to a nonlinear evolution equation generalizing the localized induction equation. This new equation is equivalent to the Hirota equation which is integrable, including both the nonlinear Schrödinger equation and the modified KdV equation in certain limits. Therefore the new equation is also integrable and the soliton surface approach gives the N-soliton solution, which is identical to that of the localized induction equation if the pertinent dispersion relation is used. Among other exact solutions are a circular helix and a plane curve of Euler's elastica. This local model predicts that, owing to the existence of the axial flow, a certain class of helicoidal vortices become neutrally stable to any small perturbations. The non-local influence of the entire perturbed filament on the linear stability of a helicoidal vortex is explored with the help of the cutoff method valid to the second order, which extends the first-order scheme developed by Widnall (1972). The axial velocity is found to discriminate between right- and left-handed helices and the long-wave instability mode is found to disappear in a certain parameter range when the successive turns of the helix are not too close together. Comparison of the cutoff model with the local model reveals that the non-local induction and the core structure are crucial in making quantitative predictions.

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