TY - JOUR

T1 - Global time evolution of an axisymmetric vortex ring at low Reynolds numbers

AU - Fukumoto, Yasuhide

AU - Kaplanski, Feliks

N1 - Funding Information:
The authors are grateful to Professor Tsutomu Kambe for fruitful discussions and to an anonymous referee for invaluable comments. Assistance by Mr. Ryu Sasaki is also acknowledged. Y.F. was supported in part by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science. F.K. was supported by the Estonian Science Foundation (Project No. ETF6832). F.K. would also like to express his gratitude to the Japan Society for the Promotion of Science for the invitation and for the financial support for his one and a half month stay at Kyushu University.

PY - 2008/5

Y1 - 2008/5

N2 - 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.

AB - 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.

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U2 - 10.1063/1.2925682

DO - 10.1063/1.2925682

M3 - Article

AN - SCOPUS:44649193145

SN - 1070-6631

VL - 20

JO - Physics of Fluids

JF - Physics of Fluids

IS - 5

M1 - 053103

ER -