Sketch of the formation of a vortex ring generated by a piston/cylinder arrangement and geometric parameters of the ring during the postformation phase.
Simulation for . (a) Time evolution of the total (solid line) and vortex ring (dashed line) circulation normalized by the maximum value . Symbols are extracted from Fig. 7 of Zhao et al. (Ref. 16); (○) experimental data of Gharib et al. (Ref. 12); (◻) numerical result of Zhao et al. (Ref. 16). (b)–(d) Contour lines (, , ) of normalized vorticity before , close to , and after the pinch-off of the vortex ring from its tail. The gray patches are used to compute the circulation of the leading vortex ring.
Evolution of the normalized vorticity field during the postformation phase for different stroke ratios (contour lines: , , ).
Simulation for . Normalized vorticity field at . The dashed thick line represents the boundary of the inner core calculated as the locus of points at which the local tangential velocity is maximum. Velocity profiles through the center of the vortex: (cut line B) and (cut line A).
Simulation for . Contour of the inner core for different time instants. Local normalized coordinates (, ) in the frame of reference attached to the center of the vortex.
Simulation for . Variation in time of the axial and radial dimensions of the inner core and normalized vorticity corresponding to points , , , and (see Fig. 4 for definitions).
Simulation for . (a) Contours of the inner core , vortex core , and vortex bubble at . (b) Variation in time of the area ratios (dashed line) and (solid line).
Evolution of main integrals of motion and corresponding power-law fit. Simulations for and different stroke ratios (○), (△), and (◻).
Determination of the virtual origin from the time variation of . Simulations for different stroke ratios: (○), (△), and (◻).
Verification of the exponential decay equation (10) of the translation velocity with the downstream position of the vortex ring. Plot using semilogarithmic axis for (○), (△), and (◻).
Fit of the vortex numerically obtained for , to ideal models of Norbury and Fraenkel and of Kaplanski and Rudi. Isocontours of the normalized vorticity [(a)–(c)] and isocontours of the normalized stream function [(d)–(f)]. Boundary contours of the vortex core (g), vortex inner core (h), and vortex bubble (i) for the simulated vortex ring (solid line), Norbury–Fraenkel fit (dashed line) and Kaplanski–Rudi fit (dash-dotted line). The center of the simulated vortex is also represented.
Fit to ideal vortex models. Radial vorticity profiles through the center of the vortex ring. Identification of vortex centers .
Fit to ideal vortex models. Signature (Ref. 34) of the vortex ring normalized by its maximum value.
Norbury–Fraenkel ideal vortex model. Variation of the nondimensional quantities with the parameter . Numerical tabulated values from Norbury (Ref. 6) (solid line), asymptotic approximations of Fraenkel (Ref. 7) (dashed line) and Norbury (Ref. 6) (dash-dotted line). The points represent values corresponding to the simulated vortex ring for at .
Numerical parameters for the injection programs and corresponding integral quantities evaluated using slug-model equations (6) and (7).
Exponents for the power laws describing the main integrals of motion (see also Fig. 8).
Exponents for the power laws describing the main integrals of motion. The value of the virtual origin is calculated from Fig. 9.
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