Numerical simulation of the postformation evolution of a laminar vortex ring

Abstract

Direct numerical simulations are used to study the postformation evolution of a laminar vortex ring. The vortex structure is described by calculating the embedded boundaries of the vortex inner core, vortex core, and vortex bubble. The topology of the vortex ring is found to be self-similar during the entire postformation phase. We also show that extracting the vortex inner core provides an objective method in setting the upper value for the cutoff vorticity level separating the vortex from its tail. The computed power laws describing the decay of the translation velocity and integrals of motion (circulation, impulse, and energy) are shown to be consistent with both studies of Dabiri and Gharib [J. Fluid Mech. 511, 311 (2004)] and Maxworthy [J. Fluid Mech. 51, 15 (1972)]. We prove that the apparently different scaling laws reported in these two studies collapse if a virtual time origin is properly defined. Finally, the computationally generated vortex rings are matched to the classical Norbury–Fraenkel model and the recent model proposed by Kaplanski and Rudi [Phys. Fluids 17, 087101 (2005)]. The former model provides not only a good prediction of normalized energy and circulation but also a good estimation of individual integrals of motion. The latter model offers, in addition to a good prediction of integral quantities, a more accurate description of the vortex ring topology when comparing the contours of the inner cores or vortex signatures. Both models underestimate the volume of fluid carried inside the vortex bubble.