On the asymptotic matching procedure predicting the formation number

A detailed review of the asymptotic matching procedures predicting the formation number of vortex rings is presented. The original studies of Mohseni and Gharib [“A model for universal timescale of vortex ring formation,” Phys. Fluids 10(10), 2436–2438 (1998).], Shusser and Gharib [“A new model for inviscid vortex rings,” in 30th AIAA Fluid Dynamical Conference (1999).], and Linden and Turner [“The formation of ‘optimal’ vortex rings, and the efficiency of propulsion devices,” J. Mech. Fluids, 427, 61–72 (2001).] are applied to the extended slug-flow model for orifice starting jets and the Kaplanski model of isolated vortex rings. A predicted formation number of 3.5 in the modified non-dimensional time frame is found when the closure assumption in terms of the translational ring speed is chosen, which is consistent with experimental evidence. In addition, particle image velocimetry was performed to assess the validity of the closure assumptions of Mohseni and Gharib and Shusser and Gharib. First, it was further demonstrated that the modified slug-flow model provides an appropriate scaling for the kinematics of orifice-generated vortex rings. Second, the measurements provide experimental support to the method of Shusser and Gharib rather than the method of Mohseni and Gharib. This is further demonstrated by data extracted from the literature. To summarize, in order to predict the formation number, it is recommended to use the extended slug-flow model and the Kaplanski model of isolated vortex rings along with the closure assumption of Shusser and Gharib.