Fluid Dynamics Research 3 (198X) 1333139

North-Holland 133

Bifurcation of an elliptic vortex ring

Yuko OSHIMA *, Naoki IZUTSU * *. Koichi OSHIMA ** and A.K.M.F. HUSSAIN * * *

* Ochanomrru Unroers~ty, Otsuka. Bunkyo-ku. Tokyo

* * The Institute of Spcrce trnd Astromutrcal Science, T&o

* * * Uniuersl~v of Hourton. Texus, USA

Abstract. Time-dependent vorticity fields of elliptic vortex rings of aspect ratios 2, 3 and 4 were measured by

means of hot-wire anemometry. The time evolution of their vorticity fields was analyzed and the processes of

vortex ring formation, advection, interaction and decay, and the mechanism of vortex bifurcation are studied.

The following crosslinking model is proposed: A thick vertical region composed of many equivalent vortex

filaments with distributed cores is initially formed at the orifice and they behave as inviscid filaments. The

elliptic ring deforms and the end parts of its major axis get closer. Then, the vortex filaments interact at the

touching point and the ring partially bifurcates. Almost simultaneously, turbulent spot appears at this point, and

propagates around the ring cross section. thus preventing further bifurcation. And it becomes a turbulent blob.

This model is also supported by numerical simulation by a high-order vortex method and the Navier-Stokes

solution.

1. Introduction

In order to investigate interaction phenomena of vortex elements, such as mutual induction, entanglement and crosslinking, elliptic rings with various aspect ratios (AR) are studied experimentally as well as numerically using a high-order discrete vortex method (Oshima, 1986) and numerical simulation of the Navier-Stokes equations (Chen, 1987). Here, only the experimental aspect of this study will be discussed. This problem was previously studied by Dhanak (1981) using a single discrete vortex approximation, by Ting (1986) using asymptotic expansion, by Takaki (1984) based on analytical consideration of the vortex interaction mechanism and by Ashurst (1987) using numerical simulation of the Navier-Stokes equations. From these works, it has been shown that bifurcation occurs when the aspect ratio of the elliptic vortex rings is larger than about 5, although the detailed mechanism was not clear.

2. Experimental setup

Schematic views of an elliptic vortex ring are shown in fig. 1, together with the coordinate system. All the physical quantities are non-dimensionalized using the radius equivalent to a circular orifice with the same area and the speed of the ejected fluid at the center of orifice. The Reynolds number based on these parameters is about 6000.

A vortex ring is released from an elliptic orifice attached to a speaker box which is impulsively driven by electric pulses with variable waveforms. Time-dependent measurements

of flow fields of vortex rings generated through three orifices of AR = 2, 3 and 4 were carried out using a computer-controlled vorticity measuring system (Izutsu, 1986). An X-wire probe is traversed in the x-, y- and z-directions through the flow fields, and time sequential data at each measuring point are conditionally sampled by triggering on the speaker driving pulse. Two kinds of measurements are carried out for each orifice. One is fine mesh measurements, in which two orthogonal planes through the center axis are traversed and the u- and W-, or u- and

0169-5983/88/$2.75 0 1988, The Japan Society of Fluid Mechanics

134 Y. Oshima et al. / Bifurcrrtion of an elhptrc uortex rrng

Fig. 1. Schematic view of an elliptic vortex

ring and the coordinate system.

Table 1

Conditions of measurement

Case Aspect T At Ax A) A; x Y i ratio (ms) (ms) (mm) (mm) (mm) (mm) (mm) (mm)

1 2:l 51 0.2 1.5 1.5 - 3-54 1.5-144

51 0.2 1.5

2 3:l 51 0.2

51 0.2 1.5 3 4:l 51 0.2 _

51 0.2 1.5

4 2:l 39.6 0.6 2.6

5 3:l 39.6 0.6 2.8

6 4:l 39.6 0.6 3.0

_ 1.5 3-54 _ 1.5-144

1.5 1.5 _ - 3-54 1.5-144

_ 1.5 3-54 1.5-144

1.5 1.5 - 3-54 1.5-144

_ 1.5 3-54 1.5-144

2.6 2.6 o-39 o-39 1.5-123.7

2.8 2.8 O-42 O-42 1.5-119.1

3.0 3.0 o-45 o-45 1.5L118.5

w-components are recorded. In the coarse mesh measurements, all 3-dimensional grid points are scanned and all the three components of the velocity are measured. The grid spacings, flow field scanned, time step, and measuring time span are listed in the table 1. Time sequential, color-coded vorticity distributions are displayed on CRT, and have been converted into an animation movie. Also, the integrated physical quantities such as the translational velocity, geometry, circulation, impulse, energy and enstrophy over the whole flow field are computed from these data.

Besides the velocity measurements, the flow fields are visualized by the Schlieren method using carbon-dioxide gas or by a smoke tracer method. Note that the visualized patterns represent the contours of the material which was originally ejected, emphasizing their edge, and correspond neither to the vorticity contour nor to the vortex filaments.

3. Results and discussions

Perspective views of an elliptic vortex ring of AR = 3 during its time evolution are schematically shown in fig. 2. A series of Schlieren photographs, which correspond to the time

Y. Oshrma et al. / Bifurcafion qfan elliptrc oorfex rrng 135

Fig. 2. Time evolution of an elliptic ring

of AR = 3.

instants illustrated in fig. 2, is presented in fig. 3; the upper (a) and the lower (b) pictures correspond to the x- and y-planes, respectively, at the same instant. The local advection of a vortex filament is in the direction of its binormal, and its velocity is inversely proportional to its radius of curvature. As a consequence, both end parts of the major axis (the +x and -x ends) move ahead of the rest of the ring (see t = 8 ms). Then, the vortex deforms like the seam of a tennis ball (see t = 16 ms).

Elliptic 3 : 1

t = 8ms 16 24 32 40 Fig. 3. Schlieren photographs of the x- and y-planes at the same time instant.

136 Y. Oshrma et al. / Bifurcutmn of an elliptic uortex rrng

Elliptic 2~1 ?x (cm)

Z

12-

6-

44

I

2

1 ”

Fig. 4. Trajectories of the vorticity centroid.

1 Elliptic 3:l 1 Elliptic 4:l (cm)

1 ...... X-plane

13 - Y-plane

F

6 1

2

1 0 1234

y yw:

(cm)

i

.: yplane

13 -Y-plane

At t = 24 ms, the ring mutually touches and bifurcates into two rings; however, bifurcation is only partial. When the two parts of the interacting ring touch, adjacent opposite signed vorticities cancel, producing crosslinking of only the outer part of the adjacent vortex filament. This reconnected part moves rapidly normal to the plane of the two initial filaments due to curvature-dependent self-induction. This leaves two parts of the original filaments intact (i.e. they are not crosslinked) while an inner part of them has crosslinked. We call this partial bifurcation. For the case of AR = 2, such partial bifurcation does not occur.

At almost the same time instant, an abrupt breakdown into turbulence occurs at the point of contact of the crosslinking. This turbulence then propagates around the vortex cross section at the interaction region (see t = 32 ms). That is, transition prevents complete bifurcation. Turbulence propagates around the entire vortex forming a turbulent blob (see t = 40 ms).

The trajectories of the vorticity centroid in the x- and y-planes are shown in fig. 4, in which a pair of the open and solid circles connected by a straight line denotes the positions at the same instant. Note that the translational velocity of the larger axis is higher than that of the smaller axis and that the deformation of the ring is smaller for AR = 2 than for higher AR, and the crosslinking does not occur.

The vorticity core radius of initially formed vortex ring is relatively large, about one half of the equivalent ring radius. Both the core radius and the vorticity distribution change consider- ably during the evolution of the ring. Figs. 5 and 6 are time variations of the perspective and front views of the vorticity contours above a specified threshold. The bifurcation of the ring is not seen at this high threshold value, but is observable at lower threshold values. With respect to the same threshold value, the vorticity core for AR = 4 is thinner then that for AR = 2. This is not unexpected because the partial bifurcation cancels some of circulation and because at higher AR, self-induction is stronger and turbulence breakdown is faster and more intense, thus turbulence diffusion being stronger.

Time variations of the circulation measured in the x- and y-planes are shown in figs. 7(a-c), where T+ and r_ mean the circulations with clockwise and counter-clockwise rotations, respectively. The appearance of r_ is due to the vortex core bifurcation, and its value increases

Y. Oshima et al. / Bifurcatron of an elliptic vortex ring 137

Fig. 5. Perspective view of vorticity contours of threshold value of 1.5. Non-dimensionalized time is from bottom to up

(a) 1.5, 3.5,5.4, 8.4, 10.9, (b) 1.6,4.2,7.1,10.6, (c) 1.5,4.1,6.8,10.7, respectively.

Fig. 6. Front views of vorticity contours

corresponding to the lower 3 of fig. 5.

6 6 10 12 ,L , 0 2 4 6 2 L 6 8 10 12 ,i, I

Fig. 7. Time variation of circulation in the orthogonal planes.

138 Y. Oshrmu et al. / Bifurcation of an elliptic vortex rrng

IO-

0-

6-

2 : 1

---- 3 :I -.-. 4 : 1

I ’

.2 4 6 0

Fig. 8. Time variation of the fluid impulse.

10 12 t

as the aspect ratio increases due to the stronger partial bifurcation. Total circulation (F, + r_) contained in the measured domain remains almost constant until the crosslinking occurs, but after then measured value decreases gradually for higher aspect ratio due to the vorticity cancellation by partial bifurcation and dissipative effect by turbulent mixing. Eventually, the elliptic vortex ring becomes a turbulent blob.

Time variations of the nondimensionaliz~d fluid impulse P = gj(jw x r d3r and the nondi- mensionalized enstrophy E = +jjjw" d3r are shown in figs. 8 and 9, respectively. After the

E

14 i

,,_.--. ‘.

12. ‘_... .... . . . . _-/- . . . . .._.____ LY....,’ ‘.., ,__.~~~ ‘1.

_______L_--- __-.____’

),.. _____/---+W7~.-.--__ ,_. _. _’ \ . 1;

1 / I 1' ..~.'.~.“-~____""-- -- __--- -'+, . . . . '. %\

10. if' --___ \ '\

:! x. '\ '\

'I 8. i AR

. . . . . . 2 : 1 6- / -_-_ 3..1

i -.-. 4 : 1

/ /’

72 2 4 6 8 10 12 t

Fig. 9. Time variation of the enstrophy.

Y. Oshima et al. / Bifurcation of an elliptrc vortex ring 139

initial formation process, the conservation of these physical quantities are confirmed regardless the aspect ratio, even though considerable variations of the vorticity distributions occur.

References

Ashurst, Wm.T. and Meiron, D.I. (1987) Numerical Study of Vortex Reconnection, Phys. Reu. Lett. 58 1632-1635.

Chen, H.L. (1987) Numerical Simulation of the Interaction of Vortex Rings in Viscous Fluid, ISAS Rep. Sp5 pp.

15-22.

Dhanak, M.R. and De Bemardinis, B. (1981) The Evolution of an Elliptic Vortex Ring, J. Fluid Mech. 109, 189-216.

Izutsu, N. Oshima, K. and Oshima, Y. (1986) Experimental Study of Interacting Vortex Rings, ISAS Rep. SP5 pp.

3-13. Oshima, Y. Noguchi, T. and Oshima, K. (1986) Numerical Study of Interaction of Two Vortex Rings, Fluid Dyn. Rex

I, 215-227.

Takaki, R. and Hussain, A.K.M.F. (1984) Dynamics of Entangled Vortex Filaments, Phys. Fluids 27, 761-763.

Ting, L. and Liu, G.C. (1986) Merging of Vortices with Decaying Cores and Numerical Solutions of Navier-Stokes

Equations, Lecture Notes in Physics 264, 612-616.