^{1,a)}and F. Kaplanski

^{2,b)}

### Abstract

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 *et al.* (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.

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.

I. INTRODUCTION

II. SOLUTION OF INITIAL-VALUE PROBLEM

III. KINETIC ENERGY AND TRANSLATION VELOCITY

IV. SAFFMAN’S FORMULA FOR MATURED STAGE

V. COMPARISON OF VORTICITY DISTRIBUTION WITH EXPERIMENTAL DATA

VI. FORMATION NUMBER

VII. SUMMARY

### Key Topics

- Vortex rings
- 66.0
- Vortex dynamics
- 46.0
- Reynolds stress modeling
- 21.0
- Viscosity
- 14.0
- Numerical solutions
- 11.0

## Figures

Dependence on the Reynolds number of temporal evolution of the traveling speed at small times. The upper thick solid line is Saffman’s formula (30). The solid line is the large-Reynolds-number asymptotics (46) and the lower thin solid line is the present result (45) for early-time behavior at low Reynolds numbers. The dashed lines are the values read off from the graph of numerical simulations conducted by Stanaway *et al.* (Ref. 19) from above.

Dependence on the Reynolds number of temporal evolution of the traveling speed at small times. The upper thick solid line is Saffman’s formula (30). The solid line is the large-Reynolds-number asymptotics (46) and the lower thin solid line is the present result (45) for early-time behavior at low Reynolds numbers. The dashed lines are the values read off from the graph of numerical simulations conducted by Stanaway *et al.* (Ref. 19) from above.

Temporal variation of the traveling speed (43) of the vortex ring. The left dashed line is the small-time asymptotics (45) and the right dashed line is the large-time asymptotics (48).

Temporal variation of the traveling speed (43) of the vortex ring. The left dashed line is the small-time asymptotics (45) and the right dashed line is the large-time asymptotics (48).

The distance traversed by vortex ring (50) as a function of time. The left dashed line is the small-time asymptotics (51) and the right dashed line is the large-time asymptotics (52). The upper horizontal line designates the upper bound on traveling distance (53).

The distance traversed by vortex ring (50) as a function of time. The left dashed line is the small-time asymptotics (51) and the right dashed line is the large-time asymptotics (52). The upper horizontal line designates the upper bound on traveling distance (53).

Comparison of the traveling speed with Saffman’s matured-stage formula. The solid line is calculated from Eq. (43). The dashed lines draw Eq. (56) with and given by Eq. (59) (above) and with and (below).

Comparison of the traveling speed with Saffman’s matured-stage formula. The solid line is calculated from Eq. (43). The dashed lines draw Eq. (56) with and given by Eq. (59) (above) and with and (below).

Comparison of vorticity distribution (24) (dashed lines) with Gaussian approximation (62) (solid lines) at and 0.222.

Comparison of vorticity distribution (24) (dashed lines) with Gaussian approximation (62) (solid lines) at and 0.222.

Comparison of radial vorticity profile (24) with the experimental result (Ref. 34), at , for a vortex ring created from a piston-nozzle system. Here, is the diameter of the orifice and is the velocity of discharged jet. The measurement was made at the cross section of distance from the orifice and at the estimated time .

Comparison of radial vorticity profile (24) with the experimental result (Ref. 34), at , for a vortex ring created from a piston-nozzle system. Here, is the diameter of the orifice and is the velocity of discharged jet. The measurement was made at the cross section of distance from the orifice and at the estimated time .

Comparison of axial vorticity profile (24) with the experimental result (Ref. 34) at .

Comparison of axial vorticity profile (24) with the experimental result (Ref. 34) at .

The normalized energy as a function of time. The solid line corresponds to definition (67) and the dashed line to , the right-hand side of Eq. (68).

The normalized energy as a function of time. The solid line corresponds to definition (67) and the dashed line to , the right-hand side of Eq. (68).

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