^{1}and Charles H. K. Williamson

^{2}

### Abstract

We present results from an experimental study of the long-wavelength “Crow” instability of a counter-rotating vortex pair. Employing a vortex generator in a water tank, comprising rotating horizontal computer-controlled flaps, we follow the vortices, marked by laser-induced fluorescence, using dual simultaneous light-sheets to determine the growth rate of the Crow instability as a function of the perturbation wavelength. In order to make a meaningful comparison to theory [S. C. Crow, AIAA J.8, 2172 (1970); S. E. Widnall, Annu. Rev. Fluid Mech.4, 141 (1975)], one requires, as input to the theory, the distribution of circumferential velocity and thereby the “equivalent” core size of the vortices. These distributions are measured using particle image velocimetry. The resulting agreement of the growth rates, between theory and experiment, appears to be very good. Of relevance to this study, we compute a stability diagram using the exact expression for the self-induced rotation speed of perturbation waves on the vortices.Measurement of the nondimensional reconnection time, when the vortex pair evolves into a series of rings at later times, is compared to existing numerical simulations, and we find evidence to suggest it varies with the inverse curvature of the vortices where they approach each other. The major axes of the resulting elliptical vortex rings switch with their minor axes, as the rings descend in the fluid, leading to a surprising phenomenon where the rings reconnect for the second time. By considering the conservation of impulse, and a linear relationship between the major axis length and the vortex spacing, we find that the relative descent speed of the rings increases with Reynolds number. It is coincidentally only at our chosen value that the descent speed of the subsequent rings appears to be close to the initial speed of the vortex pairs. Finally, the paper presents clear visualizations of the Crow instability phenomenon.

The financial support from the Deutsche Forschungsgemeinschaft (Grant No. Le 972/1-1) (T.L.), the U.S. Office of Naval Research (Contract No. N00014-95-1-0332) (C.H.K.W.), and the NATO (Contract No. CRG 970259) is gratefully acknowledged.

I. INTRODUCTION

II. EXPERIMENTAL DETAILS

III. GROWTH OF THE LONG-WAVELENGTH INSTABILITY

A. General features of the instability

B. Relevant theoretical aspects

C. Comparison between theory and experiment

IV. VORTEX RECONNECTION

V. VORTEX DYNAMICS AT LATE TIMES

A. Vortex topology: A second reconnection

B. Descent speed of the vortex system

VI. CONCLUSIONS

### Key Topics

- Rotating flows
- 193.0
- Vortex dynamics
- 37.0
- Vortex rings
- 30.0
- Flow instabilities
- 16.0
- Reynolds stress modeling
- 12.0

## Figures

Schematic of the vortex pair generator. , .

Schematic of the vortex pair generator. , .

(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.

(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.

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) .

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) .

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

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

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.

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.

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 .

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 .

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.

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.

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 .

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 .

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

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

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

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

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.

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.

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).

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).

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.

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.

Sketch of the succession of different vortex configurations.

Sketch of the succession of different vortex configurations.

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 .

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 .

## Tables

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

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

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