Global time evolution of an axisymmetric vortex ring at low Reynolds numbers

An initial-value problem of the Navier-Stokes equation is solved, at small Reynolds numbers, for evolution of an axisymmetric vortex ring. The traveling speed is written down in closed form over the whole time range, in terms of the generalized hypergeometric functions, for a vortex ring starting with infinitely thin core. We make a thorough asymptotic analysis of this solution. Three stages are identified, namely, initial, matured, and decaying stages. At the initial stage when the core is very thin, correction terms are found to Saffman's early-time formula [Stud. Appl. Math. 449, 371 (1970)]. The augmented formula establishes a lower bound on traveling speed of vortex rings starting from delta-function cores and exhibits an excellent agreement with the numerical simulation, at a small Reynolds number, conducted by Stanaway (NASA Technical Memorandum No. 101041, 1988). At the matured and decaying stages, the traveling speed is found to be closely fitted by Saffman's matured-stage formula, over a very wide time range, by an adjustment of disposable parameters in his formula. The traveling distance as a function of time is also deduced in closed form, and a simple relation of the maximum distance traversed during the whole life, being finite, is found with the viscosity, the initial circulation, and the initial ring radius. The formation number for an optimal vortex ring, estimated based on our solution, compares well with the experiments and numerical simulations.