Vortex formation out of two-dimensional orifices

The understanding of the vortex formation process is currently driving a novel attempt to evaluate the performance of fluid dynamics in biological systems. The concept of formation time, developed for axially symmetric orifices, is here studied in two-dimensional flows for the generation of vortex pairs. The early stage of the formation process is studied with the single vortex model in the inviscid limit. Within this framework, the equation can be written in a universal form in terms of the formation time. The single vortex model properly represents the initial circular spiralling vortex sheet and its acceleration for self-induced motion. Then, an analysis is performed by numerical simulation of the two-dimensional Navier-Stokes equations to cope with the spatially extended vortex structure. The results do not show the pinch-off phenomenon previously reported for vortex rings. The two-dimensional vortex pair tends to a stably growing structure such that, while it translates and extends longitudinally, it remains connected to the sharp edge by a shear layer whose velocity is always about twice that of the leading vortex. At larger values of the Reynolds number the instability of the shear layer develops small-scale vortices capable of destabilizing the coherent vortex growth. The absence of a critical formation number for two-dimensional vortex pairs suggests further considerations for the development of concepts of optimal vortex formation from orifices with variable curvature or of a tapered shape.