A method is presented whereby the translational velocity of a vortex ring can be approximated from the total circulation, impulse, and kinetic energy of the vortex system. Assuming a uniform vorticity density, these bulk quantities define a unique stable vortex ring configuration, and the translational velocity can be inferred from this configuration and the system scaling. Here, the accuracy of this approximation is presented for vortex rings formed from starting jets, and the translational velocity is also characterized as it relates to the driving parameters. The translational velocity is well approximated for a wide range of experimentally generated vortex rings. It is observed that starting jets with a converging radial velocity create vortex rings with a significantly higher translational velocity. The converging radial velocity was observed to increase translational velocity by as much as 30% over parallel jet flows with identical volume flux and nozzle diameter, but the exact increase is specific to the nozzle arrangement and driving conditions.

References

References
1.
Shariff
,
K.
, and
Leonard
,
A.
,
1992
, “
Vortex Rings
,”
Ann. Rev. Fluid Mech.
,
34
, pp.
235
279
.10.1146/annurev.fl.24.010192.001315
2.
Fraenkel
,
L.
,
1970
, “
On Steady Vortex Rings of Small Cross-Section in an Ideal Fluid
,”
Proc. R. Soc. Lond.
,
316
, pp.
29
62
.10.1098/rspa.1970.0065
3.
Hicks
,
W. M.
,
1884
, “
On the Steady Motion and Small Vibrations of a Hollow Vortex
,”
Phil. Trans. A
,
175
, pp.
161
195
.10.1098/rstl.1884.0008
4.
Saffman
,
P. G.
,
1970
, “
The Velocity of Viscous Vortex Rings
,”
Stud. Appl. Math.
,
49
(
4
), pp.
371
379
.
5.
Lamb
,
H.
,
1945
,
Hydrodynamics
,
Dover
,
New York
.
6.
Moffatt
,
H.
, and
Fukumoto
,
Y.
,
2000
, “
Motion and Expansion of a Viscous Vortex Ring. Part 1. A Higher-Order Asymptotic Formula for the Velocity
,”
J. Fluid Mech.
,
417
, pp.
1
45
.10.1017/S0022112000008995
7.
Hill
,
M. J. M.
,
1894
, “
On a Spherical Vortex
,”
Phil. Trans. Roy. Soc. A
,
185
, pp.
213
245
.10.1098/rsta.1894.0006
8.
Maxworthy
,
T.
,
1972
, “
The Structure and Stability of Vortex Rings
,”
J. Fluid Mech.
,
51
(
1
), pp.
15
32
.10.1017/S0022112072001041
9.
Maxworthy
,
T.
,
1977
, “
Some Experimental Studies of Vortex Rings
,”
J. Fluid Mech.
,
80
, pp.
465
495
.10.1017/S0022112077002171
10.
Stewart
,
K. C.
,
Niebel
,
C. L.
,
Jung
,
S.
, and
Vlachos
,
P. P.
,
2012
, “
The Decay of Confined Vortex Rings
,”
Exp. Fluids
,
53
, pp.
163
171
.10.1007/s00348-012-1277-5
11.
Krueger
,
P. S.
,
2009
, “
Vortex Ring Velocity and Minimum Separation in an Infinite Train of Vortex Rings Generated by a Fully Pulsed Jet
,”
Theor. Comput. Fluid Dyn.
,
24
, pp.
291
297
.10.1007/s00162-009-0130-9
12.
Troolin
,
D. R.
, and
Longmire
,
E. K.
,
2010
, “
Volumetric Velocity Measurements of Vortex Rings From Inclined Exits
,”
Exp. Fluids
,
48
, pp.
409
420
.10.1007/s00348-009-0745-z
13.
Mohseni
,
K.
,
2001
, “
Statistical Equilibrium Theory of Axisymmetric Flows: Kelvin's Variational Principle and an Explanation for the Vortex Ring Pinch-Off Process
,”
Phys. Fluids
,
13
(
7
), pp.
1924
1931
.10.1063/1.1368850
14.
Mohseni
,
K.
, and
Gharib
,
M.
,
1998
, “
A Model for Universal Time Scale of Vortex Ring Formation
,”
Phys. Fluids
,
10
(
10
), pp.
2436
2438
.10.1063/1.869785
15.
Mohseni
,
K.
,
2006
, “
A Formulation for Calculating the Translational Velocity of a Vortex Ring or Pair
”.
Bioinspir. Biomimet.
,
1
, pp.
S57
S64
.10.1088/1748-3182/1/4/S08
16.
Krieg
,
M.
, and
Mohseni
,
K.
,
2013
, “
Modelling Circulation, Impulse and Kinetic Energy of Starting Jets With Non-Zero Radial Velocity
,”
J. Fluid Mech.
,
719
, pp.
488
526
.10.1017/jfm.2013.9
17.
Norbury
,
J.
,
1973
, “
A Family of Steady Vortex Rings
,”
J. Fluid Mech.
,
57
(
3
), pp.
417
431
.10.1017/S0022112073001266
18.
Norbury
,
J.
,
1972
, “
A Steady Vortex Ring Close to Hill's Spherical Vortex
,”
Proc. Camb. Phil. Soc.
,
72
, pp.
253
284
.10.1017/S0305004100047083
19.
Fraenkel
,
L.
,
1972
, “
Examples of Steady Vortex Rings of Small Cross-Section in an Ideal Fluid
,”
J. Fluid Mech.
,
51
(
1
), pp.
119
135
.10.1017/S0022112072001107
20.
Friedman
,
A.
, and
Turkington
,
B.
,
1981
, “
Vortex Rings: Existence and Asymptotic Estimates
,”
Trans. Am. Math. Soc.
,
268
, pp.
1
37
.10.1090/S0002-9947-1981-0628444-6
21.
Gharib
,
M.
,
Rambod
,
E.
, and
Shariff
,
K.
,
1998
, “
A Universal Time Scale for Vortex Ring Formation
,”
J. Fluid Mech.
,
360
, pp.
121
140
.10.1017/S0022112097008410
22.
Krieg
,
M.
, and
Mohseni
,
K.
,
2012
, “
Modelling Circulation, Impulse, and Kinetic Energy of Starting Jets With Non-Zero Radial Velocity
,”
J. Fluid Mech.
,
719
, pp.
488
526
.10.1017/jfm.2013.9
23.
Glezer
,
A.
, and
Coles
,
D.
,
1990
, “
An Experimental Study of Turbulent Vortex Rings
,”
J. Fluid Mech.
,
211
, pp.
243
283
.10.1017/S0022112090001562
24.
Weigand
,
A.
, and
Gharib
,
M.
,
1997
, “
On the Evolution of Laminar Vortex Rings
,”
Exp. Fluids
,
22
, pp.
447
457
.10.1007/s003480050071
25.
Cater
,
J. E.
,
Soria
,
J.
, and
Lim
,
T. T.
,
2004
, “
The Interaction of the Piston Vortex With a Piston-Generated Vortex Ring
,”
J. Fluid Mech.
,
499
, pp.
327
343
.10.1017/S0022112003006980
26.
Fukumoto
,
Y.
, and
Kaplanski
,
F.
,
2008
, “
Global Time Evolution of an Axisymmetric Vortex Ring at Low Reynolds Numbers
,”
Phys. Fluids
,
20
(
5
), p.
053103
.10.1063/1.2925682
27.
Willert
,
C.
, and
Gharib
,
M.
,
1991
, “
Digital Particle Image Velocimetry
,”
Exp. Fluids
,
10
, pp.
181
193
.10.1007/BF00190388
28.
Raffel
,
M.
,
Willert
,
C. E.
, and
Kompenhans
,
J.
,
1998
,
Particle Image Velocimetry
,
Springer
,
Heidelberg/New York
.
29.
Saffman
,
P.
,
1992
,
Vortex Dynamics
,
Cambridge University Press
,
Cambridge, UK
.
30.
Lim
,
T.
, and
Nickels
,
T.
,
1995
, “
Vortex Rings
,”
Fluid Vortices
,
S. I.
Green
, ed.
Kluwer Academic Publishers
,
Dordrecht, The Netherlands
.
31.
Marshall
,
J. S.
,
2001
,
Inviscid Incompressible Flow
,
Wiley
,
New York
, pp.
260
292
.
32.
Kelvin
,
L.
,
1867
, “
The Translatory Velocity of a Circular Vortex Ring
,”
Phil. Mag.
,
4
, pp.
511
512
.
33.
Kelvin
,
L.
,
1910
,
Mathematical and Physical Papers
, Vol.
IV
,
Cambridge University Press
,
Cambridge, UK
.
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