Motion of an elliptical vortex under applied periodic strain

TLDR: The effect of subjecting a uniform elliptical vortex to a periodically varying plane straining field is considered in this article, where it is shown that at frequencies corresponding to the natural frequency of vibration of the vortex and their harmonics, a resonance phenomenon takes place which destabilizes the apparently stable stationary vortex in finite time, causing it to flip from state (a) into states (b) or (c), in which action of instabilities associated with a higher, nonelliptical, mode of deformation of the Vortex boundary by disturbances in the flow field would lead to disinteg

Abstract: The effect of subjecting a uniform elliptical vortex to a periodically varying plane straining field is considered. For plane steady straining fields, it is known that a Rankine‐type vortex core of uniform vorticity and elliptical shape can exist in a state in which it (a) is steady and stationary, (b) rotates about its axis or nutates about a fixed axis, or (c) elongates indefinitely, smearing the core into a thin layer. In state (a), for sufficiently weak straining fields, a vortex of small enough aspect ratio of the ellipse persists under such a plane strain, being robust to small two‐dimensional disturbances present in the flow field. It is shown that if, however, a small periodically varying component is added to the basic straining field, then at frequencies corresponding to the natural frequency of vibration of the vortex and their harmonics, a resonance phenomenon takes place which destabilizes the apparently stable stationary vortex in finite time, causing it to flip from state (a) into states (b) or (c), in which action of instabilities associated with a higher, nonelliptical, mode of deformation of the vortex boundary by disturbances in the flow field would lead to disintegration of the vortex structure. In fact it is found that the vortex also flips to states (b) and (c) for a range of non‐natural frequencies of oscillation of the straining field. The effect of the periodic plane straining on the three‐dimensional Crow instability is also discussed.