By using a recently developed numerical method, we explore in detail the possible inviscid equilibrium flows for a Kármán street comprising uniform, large-area vortices. In order to determine stability, we make use of an energy-based stability argument (originally proposed by Lord Kelvin), whose previous implementation had been unsuccessful in determining stability for the Kármán street [P. G. Saffman and J. C. Schatzman, “

Stability of a vortex street of finite vortices
,” J. Fluid Mech.117, 171
186
(1982)]. We discuss in detail the issues affecting this interpretation of Kelvin's ideas, and show that this energy-based argument cannot detect subharmonic instabilities. To find superharmonic instabilities, we employ a recently introduced approach, which constitutes a reliable implementation of Kelvin's stability ideas [P. Luzzatto-Fegiz and C. H. K. Williamson, “
Stability of conservative flows and new steady fluid solutions from bifurcation diagrams exploiting a variational argument
,” Phys. Rev. Lett.104, 044504 (2010)]
. For periodic flows, this leads us to organize solutions into families with fixed impulse I, and to construct diagrams involving the flow energy E and horizontal spacing (i.e., wavelength) L. Families of large-I vortex streets exhibit a turning point in L, and terminate with “cat's eyes” vortices (as also suggested by previous investigators). However, for low-I streets, the solution families display a multitude of turning points (leading to multiple possible streets, for given L), and terminate with teardrop-shaped vortices. This is radically different from previous suggestions in the literature. These two qualitatively different limiting states are connected by a special street, whereby vortices from opposite rows touch, such that each vortex boundary exhibits three corners. Furthermore, by following the family of I = 0 streets to small L, we gain access to a large, hitherto unexplored flow regime, involving streets with L significantly smaller than previously believed possible. To elucidate in detail the possible solution regimes, we introduce a map of spacing L, versus impulse I, which we construct by numerically computing a large number of steady vortex configurations. For each constant-impulse family of steady vortices, our stability approach also reveals a single superharmonic bifurcation, leading to new families of vortex streets, which exhibit lower symmetry.

1.
D. G.
Bohl
and
M. M.
Koochesfahani
, “
MTV measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency
,”
J. Fluid Mech.
620
,
63
88
(
2009
).
2.
M. S. T. G. S.
Triantafyllou
and
D. K. P.
Yue
, “
Hydrodynamics of fishlike swimming
,”
Annu. Rev. Fluid Mech.
32
,
33
53
(
2000
).
3.
C. H. K.
Williamson
, “
Vortex dynamics in the cylinder wake
,”
Annu. Rev. Fluid Mech.
28
,
477
539
(
1996
).
4.
M.
Horowitz
and
C. H. K.
Williamson
, “
Vortex-induced vibration of a rising and falling cylinder
,”
J. Fluid Mech.
662
,
352
383
(
2010
).
5.
A.
Andersen
,
U.
Pesavento
, and
Z. J.
Wang
, “
Unsteady aerodynamics of fluttering and tumbling plates
,”
J. Fluid Mech.
541
,
65
90
(
2005
).
6.
C. H. K.
Williamson
and
R.
Govardhan
, “
Vortex-induced vibrations
,”
Annu. Rev. Fluid Mech.
36
,
413
455
(
2004
).
7.
T. L.
Morse
and
C. H. K.
Williamson
, “
Prediction of vortex-induced vibration response by employing controlled motion
,”
J. Fluid Mech.
634
,
5
39
(
2009
).
8.
C. H. K.
Williamson
and
A.
Roshko
, “
Vortex formation in the wake of an oscillating cylinder
,”
J. Fluids Struct.
2
,
355
381
(
1988
).
9.
T.
Schnipper
,
A.
Andersen
, and
T.
Bohr
, “
Vortex wakes of a flapping foil
,”
J. Fluid Mech.
633
,
411
423
(
2009
).
10.
T.
von Kármán
, “
über den Mechanismus des Widerstands, den ein bewegter Korper in einer Flüssigkeit erfährt. 1. Teil
,”
Gött. Nachr. Math. Phys. Kl.
13
,
509
517
(
1911
);
reprinted in:
Collected works of Theodore von Kármán
(
Butterworth
,
London
,
1956
), Vol. 1, pp.
324
330
.
11.
V. V.
Meleshko
and
H.
Aref
, “
A bibliography of vortex dynamics 1858-1956
,”
Adv. Appl. Mech
41
,
197
292
(
2007
).
12.
T.
von Kármán
, “
über den Mechanismus des Widerstands, den ein bewegter Korper in einer Flüssigkeit erfährt. 2. Teil
,”
Gött. Nachr. Math. Phys. Kl.
13
,
547
556
(
1912
);
reprinted in:
Collected works of Theodore von Kármán
(
Butterworth
,
London
,
1956
), Vol. 1, pp.
331
338
.
13.
P. G.
Saffman
and
J. C.
Schatzman
, “
Stability of a vortex street of finite vortices
,”
J. Fluid Mech.
117
,
171
186
(
1982
).
14.
D. I.
Meiron
,
P. G.
Saffman
, and
J. C.
Schatzman
, “
The linear two-dimensional stability of inviscid vortex streets of finite-cored vortices
,”
J. Fluid Mech.
147
,
187
212
(
1984
).
15.
S.
Kida
, “
Stabilizing effects of finite core on Kármán vortex street
,”
J. Fluid Mech.
122
,
487
504
(
1982
).
16.
J.
Jiménez
, “
On the linear stability of the inviscid Kármán vortex street
,”
J. Fluid Mech.
178
,
177
194
(
1987
).
17.
J. R.
Kamm
, “
Shape and stability of two-dimensional uniform vorticity regions
,” Ph.D. dissertation (
California Institute of Technology
, Pasadena,
1987
).
18.
P. G.
Saffman
and
J. C.
Schatzman
, “
Properties of vortex street of finite vortices
,”
SIAM J. Sci. Stat. Comput.
2
,
285
295
(
1981
).
19.
H. M.
Blackburn
,
F.
Marques
, and
J. M.
Lopez
, “
Symmetry breaking of two-dimensional time-periodic wakes
,”
J. Fluid Mech.
522
,
395
411
(
2005
).
20.
J. C.
Schatzman
, “
A model for the von Kármán vortex street
,” Ph.D. dissertation (
California Institute of Technology
, Pasadena,
1981
).
21.
P. G.
Saffman
and
R.
Szeto
, “
Structure of a linear array of uniform vortices
,”
Stud. Appl. Math.
65
,
223
248
(
1981
).
22.
P.
Luzzatto-Fegiz
and
C. H. K.
Williamson
, “
An accurate and efficient method for computing uniform vortices
,”
J. Comp. Phys.
230
,
6495
6511
(
2011
).
23.
P.
Luzzatto-Fegiz
and
C. H. K.
Williamson
, “
Stability of conservative flows and new steady fluid solutions from bifurcation diagrams exploiting a variational argument
,”
Phys. Rev. Lett.
104
,
044504
(
2010
).
24.
P.
Luzzatto-Fegiz
and
C. H. K.
Williamson
, “
Determining the stability of steady two-dimensional flows through imperfect velocity-impulse diagrams
,” J. Fluid Mech. (in press).
25.
P.
Luzzatto-Fegiz
and
C. H. K.
Williamson
, “
Stability of elliptical vortices from “Imperfect-Velocity-Impulse” diagrams
,”
Theor. Comput. Fluid Dyn.
24
,
181
188
(
2010
).
26.
W.
Thomson
, “
Vortex statics
,”
Proc. R. Soc. Edinb.
9
,
59
73
(
1876
);
W.
Thomson
,
Philos. Mag.
10
,
97
109
(
1880
).
27.
P. G.
Saffman
,
Vortex Dynamics
(
Cambridge University Press
,
Cambridge
,
1992
).
28.
T. B.
Benjamin
, “
The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics
,” in
Applications of Methods of Functional Analysis to Problems in Mechanics
(
Springer
,
New York
,
1976
), pp.
8
29
.
29.
D. G.
Dritschel
, “
The stability and energetics of corotating uniform vortices
,”
J. Fluid Mech.
157
,
95
134
(
1985
).
30.
P. G.
Saffman
and
R.
Szeto
, “
Equilibrium shapes of a pair of equal uniform vortices
,”
Phys. Fluids
23
,
2339
2342
(
1980
).
31.
P.
Luzzatto-Fegiz
and
C. H. K.
Williamson
, “
Resonant instability in two-dimensional vortex arrays
,”
Proc. Roy. Soc. A
467
,
1164
1185
(
2011
).
32.
J.
Lamb
and
J. A.G.
Roberts
, “
Time-reversal symmetry in dynamical systems: a survey
,”
Physica D
112
,
1
39
(
1998
).
33.
D. I.
Pullin
, “
Contour dynamics methods
,”
Annu. Rev. Fluid Mech.
24
,
89
115
(
1992
).
34.
R. T.
Pierrehumbert
and
S. E.
Widnall
, “
The structure of organized vortices in a free shear layer
,”
J. Fluid Mech.
102
,
301
313
(
1981
).
35.
D. I.
Pullin
, “
The nonlinear behaviour of a constant vorticity layer at a wall
,”
Annu. Rev. Fluid Mech.
108
,
401
421
(
1981
).
36.
D. G.
Dritschel
, “
The nonlinear evolution of rotating configurations of uniform vorticity
,”
J. Fluid Mech.
171
,
157
182
(
1986
).
37.
E. A.
Overman
, “
Steady-state solutions of the euler equations in two dimensions II. Local analysis of limiting V-states
,”
SIAM J. Appl. Math.
46
,
765
800
(
1986
).
38.
W.
Thomson
, “
Vibrations of a columnar vortex
,”
Proc. R. Soc. Edinburg
10
,
443
450
(
1880
);
W.
Thomson
,
Philos. Mag.
10
,
155
168
(
1880
).
39.
D. G.
Dritschel
, “
Nonlinear stability bounds for inviscid, two-dimensional, parallel or circular flows with monotonic vorticity, and the analogous three-dimensional quasi-geostrophic flows
,”
J. Fluid Mech.
191
,
575
581
(
1988
).
40.
R. T.
Pierrehumbert
, “
A family of steady, translating vortex pairs with distributed vorticity
,”
J. Fluid Mech.
99
,
129
144
(
1980
).
41.
H.
Aref
,
M. A.
Stremler
, and
F.
Ponta
, “
Exotic vortex wakes–point vortex solutions
,”
J. Fluid Struct.
22
,
929
940
(
2006
).
42.
V. I.
Arnol'd
,
Catastrophe Theory
(
Springer-Verlag
,
Berlin Heidelberg
,
1984
).
43.
R.
Govardhan
and
C. H. K.
Williamson
, “
Modes of vortex formation and frequency response of a freely vibrating cylinder
,”
J. Fluid Mech.
420
,
85
130
(
2000
).
44.
D. G.
Crowdy
and
C. C.
Green
, “
Analytical solutions for von Karman streets of hollow vortices
,”
Phys. Fluids
23
,
126602
(
2011
).
45.
Y.
Fukumoto
, “
The three-dimensional instability of a strained vortex tube revisited
,”
J. Fluid Mech.
493
,
287
318
(
2003
).
46.
P. G.
Saffman
and
S.
Tanveer
, “
The touching pair of equal and opposite uniform vortices
,”
Phys. Fluids
25
,
1929
1930
(
1982
).
47.
H. M.
Wu
,
E. A.
Overman
, and
N. J.
Zabusky
, “
Steady-state solutions of the Euler equations in two dimensions: Rotating and translating V-states with limiting cases. I. Numerical algorithms and results
,”
J. Comp. Phys.
53
,
42
71
(
1984
).
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