A criterion for identifying vortex ring pinch-off based on the Lagrangian coherent structures ͑LCSs͒ in the flow is proposed and demonstrated for a piston-cylinder arrangement with a piston stroke to diameter ͑L / D͒ ratio of Ϸ12. It is found that the appearance of a new disconnected LCS and the termination of the original LCS are indicative of the initiation of vortex pinch-off. The subsequent growth of new LCSs, which tend to roll into spirals, indicates the formation of new vortex cores in the trailing shear layer. Using this criterion, the formation number is found to be 4.1Ϯ 0.1, which is consistent with the predicted formation number of Ϸ4 of Gharib et al. ͓J. Fluid Mech. 360, 121 ͑1998͔͒. The results obtained using the proposed LCS criterion are compared with those obtained using the circulation criterion of Gharib et al. and are found to be in excellent agreement. The LCS approach is also compared against other metrics, both Lagrangian and Eulerian, and is found to yield insight into the pinch-off process that these do not. Furthermore, the LCS analysis reveals a consistent pattern of coalescing or "pairing" of adjacent vortices in the trailing shear layer, a process which has been extensively documented in circular jets. Given that LCSs are objective and insensitive to local errors in the velocity field, the proposed criterion has the potential to be a robust tool for pinch-off identification. In particular, it may prove useful in the study of unsteady and low Reynolds number flows, where conventional methods based on vorticity prove difficult to use. © 2010 American Institute of Physics. ͓doi:10.1063/1.3275499͔The formation of axisymmetric vortex rings is a widely occurring phenomenon both in nature and in industry. It is known that these vortex rings cannot grow indefinitely, but rather there is a physical limit to their size.1 Beyond this limit, vortex rings do not grow any further but "pinch off," and a trailing jet forms behind them. This limit implies the existence of an optimum vortex size: for optimum momentum transfer, rings must be made as large as possible while avoiding pinch-off.2 This optimum has important implications for natural and engineering flows, and hence vortex ring pinch-off has been extensively studied, principally by means of the vorticity field. However, in the more complex naturally occurring flows, the vorticity field tends to break down and diffuse, and existing criteria prove insufficient for robustly identifying pinch-off. In this paper we propose a criterion for pinchoff identification, based on Lagrangian coherent structures (LCSs), which could provide further insight into the structure of these complex natural flows. The criterion is demonstrated for a laboratory-generated vortex ring, and it is found to be in good agreement with the established criterion based on circulation.

The formation and pinch-off of non-axisymmetric vortex rings is considered experimentally. Vortex rings are generated using a non-circular piston-cylinder arrangement, and the resulting velocity fields are measured using digital particle image velocimetry. Three different nozzle geometries are considered: an elliptical nozzle with an aspect ratio of two, an elliptical nozzle with an aspect ratio of four and an oval nozzle constructed from tangent circular arcs. The formation of vortices from the three nozzles is analysed by means of the vorticity and circulation, as well as by investigation of the Lagrangian coherent structures in the flow. The results indicate that, in all three nozzles, the maximum circulation the vortex can attain is determined by the equivalent diameter of the nozzle: the diameter of a circular nozzle of identical cross-sectional area. In addition, the time at which the vortex rings pinch off is found to be constant along the nozzle contours, and independent of relative variations in the local curvature. A formation number for this class of vortex rings is defined based on the equivalent diameter of the nozzle, and the formation number for vortex rings of the three geometries considered is found to lie in the range of 3-4. The implications of the relative shape and local curvature independence of the formation number on the study and modelling of naturally occurring vortex rings such as those that appear in biological flows is discussed.

The nonlinear perturbation response of two families of vortices, the Norbury family of axisymmetric vortex rings and the Pierrehumbert family of two-dimensional vortex pairs, is considered. Members of both families are subjected to prolate shape perturbations similar to those previously introduced to Hill's spherical vortex, and their response is computed using contour dynamics algorithms. The response of the entire Norbury family to this class of perturbations is considered, in order to bridge the gap between past observations of the behaviour of thin-cored members of the family and that of Hill's spherical vortex. The behaviour of the Norbury family is contrasted with the response of the analogous two-dimensional family of Pierrehumbert vortex pairs. It is found that the Norbury family exhibits a change in perturbation response as members of the family with progressively thicker cores are considered. Thin-cored vortices are found to undergo quasi-periodic deformations of the core shape, but detrain no circulation into their wake. In contrast, thicker-cored Norbury vortices are found to detrain excess rotational fluid into a trailing vortex tail. This behaviour is found to be in agreement with previous results for Hill's spherical vortex, as well as with observations of pinch-off of experimentally generated vortex rings at long formation times. In contrast, the detrainment of circulation that is characteristic of pinch-off is not observed for Pierrehumbert vortex pairs of any core size. These observations are in agreement with recent studies that contrast the formation of vortices in two and three dimensions. We hypothesize that transitions in vortex formation, such as those occurring between wake shedding modes and in vortex pinch-off more generally, might be understood and possibly predicted based on the observed perturbation responses of forming vortex rings or dipoles.

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