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Experiments on long-wavelength instability and reconnection of a vortex pair
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Image of FIG. 1.
FIG. 1.

Schematic of the vortex pair generator. , .

Image of FIG. 2.
FIG. 2.

(a) Profile of the vertical (azimuthal) velocity component on a line going through both vortex centers. A superposition of two Oseen vortices [dotted lines, Eq. (1)] is a good representation of the measured data. (b) Sketch of the different vorticity distributions considered. Thick line: realistic (Gaussian) distribution. Dashed line: Rankine vortex equivalent to the Gaussian vortex concerning the long-wave dynamics. Both profiles have the same total circulation. is the peak vorticity of the Gaussian profile.

Image of FIG. 3.
FIG. 3.

Visualization of the vortex pair evolution due to the long-wavelength instability for and . The axial wavelength is forced to be . The pair is seen in front view, i.e., the vortices move toward the observer. (a) , (b) , (c) .

Image of FIG. 4.
FIG. 4.

Evolution of the instability seen in side view; the vortices move down. Images were taken simultaneously with the ones in Fig. 3.

Image of FIG. 5.
FIG. 5.

Stability diagram for symmetric bending-wave perturbations of a vortex pair. (a) Result obtained with the long-wave approximation of the self-induced rotation rate (Ref. 3) valid only for (i.e., above the diagonal in this diagram). (b) Result obtained with the exact rotation rate. Regions with have no physical relevance, since the cores are overlapping there.

Image of FIG. 6.
FIG. 6.

Measurement of the geometry and motion of the vortex system. (a) Schematic of the visualization set-up. (b) Example of a simultaneous visualization of the perturbed pair in two planes at . (c) Schematic of the vortex geometry. (d) Growth of the perturbation amplitude for . The cores touch close to .

Image of FIG. 7.
FIG. 7.

Growth rate of the long-wavelength instability as a function of the normalized axial wavelength. Results from the present experiments: symbols represent measurements made in the range . The instability could not be forced to take on the wavelengths marked by a square symbol. The line gives the theoretical prediction for a core size , which is representative for all experiments.

Image of FIG. 8.
FIG. 8.

Evolution of the (a) mean and (b) minimum vortex spacing at , , and . Four stages can be identified leading to the formation of vortex rings: ① vortex generation (moving plates), ② linear and ③ nonlinear growth of the Crow instability, ④ overlapping cores and reconnection. The dashed line reproduces a result from Klein et al. (Ref. 9) for inviscid ultrathin vortex pairs perturbed at .

Image of FIG. 9.
FIG. 9.

Visualization of the flow in the reconnection plane [plane ② in Fig. 6(a)]. (a) , (b) .

Image of FIG. 10.
FIG. 10.

Contours of axial vorticity in the reconnection plane at . (a) , (b) . Contours are separated by .

Image of FIG. 11.
FIG. 11.

Evolution of the circulation in the reconnection plane for and definition of the reconnection time . Measurements for the two vortices of the pair are represented by different symbols.

Image of FIG. 12.
FIG. 12.

Reconnection time as a function of , which is roughly proportional to the inverse curvature of the vortices. ◼: present study, ▲: Melander and Hussain (Ref. 51), ●: Marshall et al. (Ref. 53), ▼: Garten et al. (Ref. 44).

Image of FIG. 13.
FIG. 13.

Visualization (front view) of the long-time evolution of the flow. (a) , (b) . The major axis of the oval vortex rings has changed its orientation by 90° between the two images.

Image of FIG. 14.
FIG. 14.

Sketch of the succession of different vortex configurations.

Image of FIG. 15.
FIG. 15.

Descent of the vortex system. (a) Average vertical position of the large-scale vortex structures as a function of time for and . ○: , ●: . For , the flow consists of a series of oscillating vortex rings. (b) Average descent speed of the vortex rings resulting from the Crow instability as a function of Reynolds number. Symbols show the measurements from (a), and the thick solid line shows the prediction according to Eq. (15) with . The shaded area represents the results for the conditions accessible in our experiments: , , and .


Generic image for table
Table I.

Compilation of data from four vortex reconnection studies. : core size at the onset of reconnection.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Experiments on long-wavelength instability and reconnection of a vortex pair