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).
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 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 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).
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