Progressive Deformation of a Perturbed Line Vortex Filament

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Abstract
When a straight two‐dimensional vortex filament, which is laid on the x axis, is perturbed by a three‐dimensional distortion, it deforms progressively by its own induction. The progressive deformation is numerically obtained in this paper for a localized distortion y = a exp(−x2) and for a periodic distortion y = 2a cos[¼(πx)], a being the amplitude relative to the lateral extent of the distortion. When a is small, the Gaussian distortion causes a helical deformation which first moves in and then moves away toward far ends along the vortex filament, whereas the central portion where the disturbance was originally located, subsides and straightens. The plane of the sinusoidal distortion simply rotates in the direction opposite to that of the translation of fluid in the undisturbed vortex. The retrograde rotation is the same as that formulated by Kelvin. For these cases of small amplitude, a linearlized theory is also put forward. When a2 is large compared with unity, on the contrary, a nonlinear effect comes in, causing higher‐order deformations to take place in both cases near the tip of the distorted pattern. This substantiates in part the author's experimental observation of the progressive deformation of a vortex loop in the final stage of boundary‐layer transition. A possible mechanism of three‐dimensional amplification of initially small perturbation in a shear flow is also discussed.

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