A model for the pinch-off process of the leading vortex ring in a starting jet

Modifications have been made to an analytical model proposed by Shusser & Gharib (J. Fluid Mech., vol. 416,2000b, pp. 173-185) for the vortex ring formation and pinch-off process in a starting jet. Compared with previous models, the present investigation distinguishes the leading vortex ring from its trailing jet so as to consider the details of the pinch-off process in terms of the properties of the leading vortex ring, which are determined by considering the flux of kinetic energy, impulse and circulation from the trailing jet into the leading vortex ring by convective transportation. A two-stage process has been identified before the complete separation of the leading vortex ring from its trailing jet. The first stage involves the growth of the leading vortex ring by absorbing all the ejected fluid from the nozzle until certain optimum size is achieved. The second stage is characterized by the appreciable translational velocity of the leading vortex ring followed by a trailing jet. The leading vortex ring is approximated as a Norbury vortex ring with growing characteristic core radius E such that dimensionless energy a, as well as its translational velocity and penetration depth, can be estimated. By applying the Kelvin-Benjamin variational principle, the pinch-off process is signified by two time scales, i.e. the formation number, which indicates the onset of the pinch-off process, and the separation time, which corresponds to the time when the leading vortex ring becomes physically separated from the trailing jet and is therefore referred to as the end of the pinch-off process. The effect of nozzle geometry, i.e. a straight nozzle or a converging nozzle, has also been taken into account by using different descriptions of the growth of the trailing jet. The prediction of the formation number and the characteristics of the vortex ring are found to be in good agreement with previous experimental results on starting jets with straight nozzles and converging nozzles, respectively.