Content uploaded by Kerry Hourigan
Author content
All content in this area was uploaded by Kerry Hourigan
Content may be subject to copyright.
J. Fluid Mech. (2005), vol. 534, pp. 23–38. c
2005 Cambridge University Press
doi:10.1017/S0022112005004313 Printed in the United Kingdom
23
The evolution of a subharmonic mode
in a vortex street
By G. J. S H E A R D1†, M. C. THOMPSON
1,
K. HOURIGAN
1AND T. LEWEKE
2
1Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical
Engineering, Monash University, Melbourne, Victoria 3800, Australia
2Institut de Recherche sur les Ph´
enom`
enes Hors ´
Equilibre (IRPH ´
E), UMR 6594 CNRS/Universit´
es
Aix-Marseille I & II, 12 avenue G´
en´
eral Leclerc, F-13003 Marseille, France
(Received 11 February 2004 and in revised form 9 November 2004)
The development of a subharmonic three-dimensional instability mode in a vortex
street is investigated both numerically and experimentally. The flow past a ring is
considered as a test case, as a previous stability analysis has predicted that for a range
of aspect ratios, the first-occurring instability of the vortex street is subharmonic. For
the flow past a circular cylinder, the development of three-dimensional flow in the
vortex street is known to lead to turbulent flow through the development of spatio-
temporal chaos, whereas subharmonic instabilities have been shown to cause a route to
chaos through the development of a period-doubling cascade. The three-dimensional
vortex street in the flow past a ring is analysed to determine if a subharmonic
instability can alter the route to turbulence for a vortex street.
A linear stability analysis and non-axisymmetric computations are employed to
compute the flow past a ring with an aspect ratio ar = 5, and comparisons with
experimental dye visualizations are included to verify the existence of a subharmonic
mode in the wake. Computations at higher Reynolds numbers confirm that the
subharmonic instability does not initiate a period-doubling cascade in the wake.
1. Introduction
Several studies have provided predictions of the mechanism for the route to chaos in
flows associated with the vortex street behind a bluff body. In Braun, Feudel & Guzdar
(1998), an externally driven row of two-dimensional vortices was studied numerically,
and it was determined that the flow developed chaos through a period-doubling
cascade. A period-doubling of the flow behind a cylinder with a mild variation in
diameter was observed in the experiments by Rockwell, Nuzzi & Magness (1991).
Early computations of the unsteady flow past a circular cylinder by Tomboulides,
Triantafyllou & Karniadakis (1992) suggested that the first-occurring three-
dimensional transition caused a period-doubling of the wake. The low-dimensional
computations of Noack & Eckelmann (1994a,b) predicted a similar bifurcation in
the wake. These computations predicted that the first-occurring three-dimensional
instability developed with a spanwise wavelength of approximately 1.7d–2d, where d
is the diameter of the cylinder. The results of these studies have since been proved
erroneous (Barkley & Henderson 1996), owing to either an inadequate spanwise
†Author to whom correspondence should be addressed: Greg.Sheard@eng.monash.edu.au
24 G. J. Sheard, M. C. Thompson, K. Hourigan and T. Leweke
computational domain, or an inadequate number of degrees of freedom. The earlier
studies do, however, provide tantalizing qualitative evidence of a third possible
perturbation mode in a vortex street.
The experimental flow visualizations of the wake behind a circular cylinder
provided by Williamson (1988, 1996a,b) showed that in fact the first-occurring three-
dimensional instability had a spanwise wavelength of approximately 3d–4d.That
observed wavelength was consistent with the wavelength predicted by the linear
stability analysis of the two-dimensional vortex street behind a circular cylinder by
Barkley & Henderson (1996). They also predicted that the Floquet multiplier of
the instability bifurcated through +1 on the real axis, which is consistent with a
regular instability rather than the subharmonic instability predicted by earlier studies.
These predictions were verified by the three-dimensional computations by Thompson,
Hourigan & Sheridan (1994, 1996), Henderson & Barkley (1996) and Henderson
(1997), which showed that the first-occurring three-dimensional instability had a
spanwise wavelength in the range 3d–4dat the onset of the transition, and that no
period-doubling was observed.
A linear stability analysis of the vortex street behind a square cross-section cylinder
by Robichaux, Balachandar & Vanka (1999) predicted the existence of a subharmonic
instability in the wake, which suggested that a period-doubling cascade may be
initiated in vortex streets behind alternative geometries. A detailed stability analysis
by Blackburn & Lopez (2003) later predicted that the instability in question occurred
with a complex-conjugate Floquet mode, rather than a subharmonic mode, but the
potential for instabilities additional to Modes A and B to exist in the wakes behind
cylinders of non-circular cross-section was nevertheless verified.
A number of experimental studies (Monson 1983; Bearman & Takamoto 1988;
Leweke & Provansal 1994, 1995) have shown that an axisymmetric vortex street is
observed in the flow past open rings (i.e. rings with a mean diameter greater than
the cross-section diameter). Numerical studies of the flow past open rings by Sheard,
Thompson & Hourigan (2001, 2003b) predicted that the axisymmetric annular vortex
street was unstable to a subharmonic instability in addition to regular instabilities
analogous to Modes A and B for a circular cylinder (Williamson 1988). This additional
subharmonic instability was referred to as Mode C, and was predicted to have an
azimuthal wavelength of approximately 1.8d. Mode C can be clearly distinguished
from Modes A and B in that streamwise vorticity in the wake is observed to switch
sign from one period to the next.
It is convenient to define an aspect ratio that describes the geometry of a ring
by ar =D/d,whereDis the mean ring diameter, and dis the diameter of the
circular ring cross-section. A Reynolds number for the flow past a ring is given
by Re =U∞d/ν,whereU∞is the uniform free-stream velocity of the flow and νis
the kinematic viscosity of the fluid. The numerical stability analysis presented in
Sheard et al. (2003b) predicted that for the flow past rings with aspect ratios in the
range 4 .ar .8, the Mode C instability was the first-occurring non-axisymmetric
instability in the vortex street. Non-axisymmetric computations of the flow past open
rings were reported in Sheard et al. (2003a, 2004). Those studies showed that for
the flow past a ring with ar = 5, the transition to non-axisymmetric flow occurred
at Re ≈161, and that the wake for Reynolds numbers slightly above this threshold
was periodic over two shedding cycles of the vortex street. This verified that the non-
axisymmetric wake developed from a subharmonic Mode C instability, and a period-
doubling had occurred. These previous studies were primarily aimed at characterizing
the three-dimensional instability modes and the nature of the associated transitions.
Evolution of a subharmonic mode in a vortex street 25
The size of the computational domain was always chosen to precisely isolate a
given mode, and interactions with other possible instability modes were therefore
suppressed.
The work reported here assesses, for the first time, the evolution and stability
of the non-axisymmetric Mode C wake structures in realistic configurations. This
is achieved through computations with an azimuthal domain size large enough to
capture also the Mode A instability, and through experiments involving an entire ring.
In this study, non-axisymmetric computations of the flow past a ring with ar ≈5
are compared with experimental dye visualizations using the same configuration, to
determine whether the computed subharmonic instability exists and can be observed
in a real flow. In a further step, computations are performed at higher Reynolds
numbers, in order to investigate if the route to chaos for the vortex street is altered
by the first-occurring subharmonic Mode C instability.
2. Methodology
This investigation comprised both numerical and experimental components. These
are described in the sections that follow.
2.1. Numerical methods
The numerical computations presented in this study were performed with a
formulation in cylindrical-polar coordinates of a spectral-element technique which
has been employed previously to compute the non-axisymmetric flow past a sphere
by Thompson, Leweke & Provansal (2001). The technique solved the unsteady non-
axisymmetric incompressible Navier–Stokes equations, which are written in terms of
non-dimensional variables as
∂u
∂t =−(u·∇)u−∇p+1
Re ∇2u, (2.1)
∇·u= 0. (2.2)
Equations (2.1) and (2.2) are determined from principles of conservation of momen-
tum and mass, respectively. By convention (Leweke & Provansal 1995; Sheard et al.
2003b, 2004), the spatial coordinates are non-dimensionalized by the diameter of the
ring, d. The subscript ∞represents free-stream conditions, and the velocity field u
is non-dimensionalized by U∞. The kinematic pressure pis given by P/ρ, where
Pis the pressure non-dimensionalized by P∞,andρis the (constant) density non-
dimensionalized by ρ∞.Re =U∞d/ν is the Reynolds number, and the time variable
tis non-dimensionalized by d/U∞. A Strouhal number is defined as St =fd/U
∞,
where fis the shedding frequency.
The numerical technique employed a weighted-residual method for spatial
discretization of the elements in the (r,z)-plane of the computational domain. A
Fourier expansion of the velocity and pressure fields in the azimuthal direction
was employed to compute the non-axisymmetric variation in the flow. In the (r,z)-
plane, the mesh elements consisted of Lagrangian tensor-product polynomials, with
node points which corresponded to the Gauss–Lobatto–Legendre collocation points.
Temporal integration was performed with a three-step splitting scheme as described
in Karniadakis, Israeli & Orszag (1991).
To model the flow past a ring, a mesh was constructed with 459 elements comprising
81 nodes per element. A mesh was created to model a ring with ar =4.94 for consist-
ency with the ring employed in the experimental study. The mesh in the (r,z)-plane
26 G. J. Sheard, M. C. Thompson, K. Hourigan and T. Leweke
(a)(b)
Figure 1. Plots of the mesh used to model the flow past a ring with ar =4.94. Sub-elements
are included in the plots, and in (a) the entire computational domain in the (r,z)-plane is
shown,andin(b) detail of the mesh in the vicinity of the cross-section of the ring is shown.
is shown in figure 1. In the azimuthal direction, 32 Fourier planes were used, which
provided spatial discretization of 16 Fourier modes.
For the mesh displayed in figure 1, flow is computed from left to right, and a zero
normal velocity condition is imposed at the axis located at the bottom of the mesh.
On the other boundaries, the following conditions were imposed: a uniform free-
stream velocity at the upstream and transverse boundaries, a zero tangent velocity at
the downstream boundary, and a zero velocity on the ring surface. From the centre
of the ring cross-section, the distance to the upstream, transverse and downstream
boundaries of the computational domain are 15d,30dand 25d, respectively. The
grid-resolution study performed for the computational study by Sheard et al. (2004)
verified that the computational domain sizes and distribution of elements as employed
in the present study provide solutions accurate to within 1 % for the Reynolds number
range considered in the present study (Re .320).
Non-axisymmetric computations were typically initiated from periodic axisymmetric
solutions, with a small random perturbation (of order 10−3) added to the velocity
field. With a perturbation of this magnitude the evolution of the instabilities was
observed to be initially linear, and 50 to 100 shedding cycles were required for the
three-dimensional wake to fully evolve.
For visualization of the computed wakes, simulated particles were included in the
computations. The particles were integrated forward in time using a second-order
Adams–Bashforth technique, and the particle velocities were obtained by polynomial
interpolation (of the spectral-element order) within the elements. For integration of
the particle positions, between 100–120 time steps were employed per shedding cycle
to minimize time-stepping errors in the computations. The injection points of the
simulated particles were located in the vicinity of the flow separation points around
the ring cross-section to mimic the entrainment of dye into the wake in experiments.
To hasten the dispersal of particles in the flow, the particles were injected 0.04dfrom
the surface of the ring.
2.2. Experimental method
Two techniques for experimental flow visualization have been employed in previous
studies for the wakes behind rings. In one study, Monson (1983) observed rings
falling through a tank of water. The rings were coated with a dye that entrained into
Evolution of a subharmonic mode in a vortex street 27
60 cm
Ring
Pulleys
50 cm (× 50 cm)
Glass
water
tank
Stepper
motor
Figur e 2. A schematic representation of the experimental apparatus, which shows the water
tank, the buoyant ring, the submerged pulley system and the stepper motor. Note that the
diagram is not to scale.
the wake, highlighting the flow structures present. This method has the advantage
of not imparting any spurious perturbation on the wake owing to the presence of
tethers attached to the ring, but has a major drawback in the inability to maintain a
perpendicular orientation of the ring to the vertical direction of motion. This effect
is especially noticeable for asymmetric wake visualization, where the non-uniform
distribution of drag around the ring incites a wobble in the orientation of the ring.
Leweke & Provansal (1995) overcame this difficulty by fixing their rings within a
wind tunnel with fine tethers. The attachment of the tethers provided no noticeable
effect on the wake, owing to the large difference in scale between the tether diameter
and their placement around the circumference of the ring. The azimuthal distribution
of these wire anchors had no observable effect on the wake structures in the flow-
visualization images presented by Leweke & Provansal (1995).
The experimental method employed in the present study differed somewhat from
those of these previous studies. Because of the desire to monitor the wakes behind
rings with aspect ratios in the range 4 .ar .8, a technique was developed whereby
a buoyant ring coated in Fluorescein dye was dragged vertically downward in a water
tank. The technique of wake visualization by coating a body in a fluorescent dye was
pioneered in the studies of the wake behind a circular cylinder by Williamson (1985,
1988, 1989, 1992).
The experimental rig consisted of a vertical tank 600 mm high and 500 mm square
at the base. A schematic representation of the set-up of the experimental apparatus
is shown in figure 2. The horizontal inclination of the ring was maintained by the
placement of three equi-spaced tethers which were attached to the upstream surface
of the ring. The three tethers were affixed to a tow line approximately 10dupstream
of the ring. The tow line traversed a near-frictionless pulley system at the base of
the tank, and returned to the surface near to the tank wall. The tow line was wound
onto a spool which was machined to a uniform diameter. The spool was driven
at a constant velocity by a computer-controlled stepper motor which employed
28 G. J. Sheard, M. C. Thompson, K. Hourigan and T. Leweke
40 000 steps per revolution. This kept the ring velocity constant to a high degree
of accuracy, which maintained a constant Reynolds number for the flow. Regular
temperature measurements were made to ensure that accurate estimations of the
kinematic viscosity of the fluid and the Reynolds number were made.
Flow visualization was performed by illuminating the dye with a laser light source
(wavelength 540 nm). Images were captured with a Canon EOS 300 35 mm SLR
camera fitted with a 22–55 mm f/4.0–f/5.6 USM zoom lens. 400 ASA colour film
stock was employed, and the photographs were digitized with a Nikon Coolscan II
2700 dpi film scanner.
The computer-controlled motion of the ring was initiated from a static position
approximately 1cm below the surface of the tank, and buoyancy forces maintained
a vertical alignment of the ring. Initially, a fast acceleration was applied to the ring,
which resulted in the evolution of a uniform axisymmetric vortex street in the wake.
This condition was a fundamental requirement for the evolution of linear asymmetric
instability modes of the wake.
The ring was machined from wood, which had a density approximately half that
of water. The surface was sanded with a fine grit sandpaper to a finish that was
smooth to the touch. This was deemed sufficient owing to both the low Reynolds
number of the experiments, which negated surface roughness effects, and the fact that
the ring surface was to be painted with dye. The mean inner and outer diameters of
the ring were measured by digital callipers to be 31.9mm and 48.1 mm, respectively.
From these measurements the mean ring diameter, D=40.0 mm, and the mean cross-
section diameter, d=8.1 mm, were determined. As an independent verification of
these measurements, the ring cross-section was measured directly at several locations
around the ring, and was determined to be d=8.1±0.1 mm. These measurements
verify that the ring used in the experiments had an aspect ratio of ar ≈4.94. The
circumference along the ring centreline non-dimensionalized with respect to the ring
cross-section (d)isπD=15.5d.
The measured mean ring cross-section diameter was d=8.1×10−3m, and the
assumed kinematic viscosity was ν=1.0×10−6m2s−1based on a water temperature
of 23.5◦C. Therefore, the velocity (U) range at which the ring was required to be
towed to achieve the Reynolds number range (above the critical Reynolds number for
Mode C and below Mode A) was 2.0cms
−1<U<2.42 cm s−1. These velocities were
extremely low, and left the experiment susceptible to slightly non-uniform conditions,
such as convection within the tank, imperfections on the body of the ring, and the
wake disruption imposed by the towing line upstream of the ring. These problems
were minimized through careful machining of the geometry, the use of extremely fine
thread (diameter 0.19 mm) to attach the ring to the tow line, and by allowing sufficient
time before a run for the ring to reach a vertical equilibrium point in the tank, and the
fluid to become motionless to the limit of observation. It should be noted that there
was no observable correlation between the location of the tether attachment points
and the azimuthal phase of the non-axisymmetric wake structures, suggesting that the
effect of the tethers on the evolution of the instabilities was qualitatively negligible.
The relatively short run over which the ring is towed is limited in a practical sense
by the size of the water tank, but also by the development of asymmetric modes as the
ring nears the end of its run. This asymmetry was observed on many occasions, and is
thought to be associated with the development of spatiotemporal chaos in the vortex
street. The instability tended to manifest itself through irregularities in the otherwise
axisymmetric vortex rings in the wake, leading to the inception of a transverse motion
of the ring. This nonlinear instability evolved from background noise in the tank,
Evolution of a subharmonic mode in a vortex street 29
and appears to be the asymptotic state of the wake. Thus the linear modes can only
be observed clearly shortly after they initially evolve. This is similar to observations
of the wake of a circular cylinder (Williamson 1996a), where Mode A is rapidly
disrupted by irregularities in the vortex street. The transverse motion of the ring bore
some resemblance to the wakes behind free-falling rings (Monson 1983), which in
some cases showed similar helical and transverse shedding patterns. The visualizations
of flow behind a fixed ring by Leweke & Provansal (1995) also showed shedding of
single, double and triple-helix modes for all aspect ratios investigated (10 &ar &30),
but the employment of a fixed ring in their experiments allowed these modes to be
produced in a stable state, whereas in the present set-up, the motion of the ring and
the resulting transverse shedding modes tended to dominate over long times.
3. Results
As a preface to the study of the development of three-dimensional modes in the
wake, images are provided in figure 5 to show that axisymmetric wakes could be
obtained reliably with the experimental rig. In the figure, the experimental flow at
Re = 100 is compared with a computed flow that remained axisymmetric, as the
Reynolds number was below the transition Reynolds number for non-axisymmetric
flow. The computational domain in this instance modelled the entire ring. A laser light
sheet was employed in the experiment to highlight the cross-section of the shedding
wake, and in the computation, particles were injected to mimic this visualization
technique. Numerical studies (Sheard et al. 2003b, 2004) predict that the wake behind
a ring with ar = 5 will become unstable to a linear Mode C instability at Re = 161,
followed by a Mode A instability at Re = 194. The respective azimuthal mode
numbers of these instability modes are m=9 and m= 4, respectively, corresponding
to azimuthal wavelengths of λd=1.75dand λd=3.93d. For further details of the
symmetry and critical Reynolds numbers of these modes, see figure 11 in Sheard
et al. (2003b) and table 1 in Sheard et al. (2004). The non-axisymmetric computations
in this study capture Mode C with m=8, owing to the size of the computational
domain, but it should be stressed that this mode number still lies within the bandwidth
of the Mode C instability.
A precise measurement of the transition Reynolds number for the development of
non-axisymmetric flow was not intended to be made using the experimental apparatus
employed. It should be noted that observations of a non-axisymmetric instability of
the vortex street was made for Reynolds numbers in the range 150 .Re .170.
The nature of the non-axisymmetric instabilities that develop from the axisymmetric
vortex street was analysed numerically, and experimental dye visualization was
used to corroborate the numerical computations. The results of these studies are
presented in the sections that follow. First, computations are reported which show
the period-doubling caused by the evolution of the Mode C instability in the
vortex street behind a ring with ar =5 at Re = 190. Secondly, simulated-particle
computations and experimental dye visualization methods are employed to verify
that the Mode C instability can be reproduced experimentally. Finally, computations
at higher Reynolds numbers are reported to examine whether a period-doubling
cascade is in fact initiated in the vortex street behind a ring.
3.1. The primary non-axisymmetric instability
To show that the evolution of the Mode C instability in the wake causes a period-
doubling bifurcation, the shedding frequency of the flow was monitored from
30 G. J. Sheard, M. C. Thompson, K. Hourigan and T. Leweke
0.190
(b)(a)
0.185
0.180
0.175 350 450 550 650 750 850
St
t
Figure 3. The non-axisymmetric flow past a ring computed with ar = 5 and an azimuthal
domain of 3.9dat Re = 190. (a) An isosurface plot of the saturated wake. Flow is upwards
and the ring is located at the bottom of the frame. Streamwise vorticity isosurfaces (at levels
±0.4) are indicated by light and dark isosurfaces, and a pressure isosurface is also shown
(level −0.1) on the right-hand side of the wake only. Two repetitions of the computational
domain in the azimuthal direction are shown. (b) The peak-to-peak frequency variation in
point velocity (+) and drag force (solid line).
peak-to-peak computations of the drag force and the velocity at a point in the
wake approximately 4ddirectly downstream of the ring cross-section. Note that the
drag force is a global quantity of the flow whereas the point velocity is a local
quantity.
The flow past a ring with ar = 5 was computed at Re = 190. The computational
domain had an azimuthal span of 3.9d, which was large enough to capture each of the
Mode A, B and C instabilities. In order to adequately resolve all the non-axisymmetric
modes in the flow that were not damped by viscous diffusion (e.g. see Henderson
1997), 32 Fourier planes were employed in the computation. Figure 3 shows the
non-axisymmetric wake structure and the peak-to-peak frequency variation as the
non-axisymmetric flow evolved to saturation.
The isosurface plot in figure 3(a) shows that the saturated wake at Re = 190
comprised two azimuthal repetitions of Mode C wake structures over the comput-
ational domain, each with an azimuthal wavelength λd≈2d. Inspection of the stream-
wise vorticity isosurfaces shows that the sign of the vorticity alternates from one
shedding cycle to the next. The plot in figure 3(b) shows that the evolution of non-
axisymmetric flow caused a period-doubling in the wake. The plot also shows that
the period-doubling bifurcation does not alter the periodicity of the computed drag
force. Hence the non-axisymmetry of each successive shedding cycle is 180◦out of
phase with that of the previous cycle.
Evolution of a subharmonic mode in a vortex street 31
u
0.5 0.7 0.9 1.1 1.3
(b)(a)
t
v
800 820 840
–0.4
–0.2
0
0.2
0.4
0.6
0.8
–0.4
–0.2
0
0.2
0.4
0.6
0.8
Figur e 4. Phase plots of the saturated non-axisymmetric Mode C wake at Re = 190 for the
flow past a ring with ar =5. The u-andv-velocities were recorded in the wake at a location
approximately 4ddirectly behind the ring cross-section. The v-velocity plotted (a) against time,
and (b) against the u-velocity.
The period-doubling that is observed through the evolution of the Mode C
instability in the wake of a ring with ar = 5 is verified by the phase plots presented
in figure 4. Figure 4(a) shows the time history of the v-velocity at a point in the
wake approximately 4ddownstream of the ring, and figure 4(b) plots the v-velocity
against the u-velocity. The u-andv-velocity components refer to the axial and
radial directions, respectively. In both plots, only data at saturation are shown. In
figure 4(a), the period-doubling of the point-velocity signal in the wake is shown by
the alternation between two peak velocities, which renders the period of the wake to
be approximately T=11.0. In figure 4(b), the period-doubling of the saturated wake
is shown by the two distinct loops traced by the point velocity in u–vspace over two
shedding cycles.
3.2. Comparison between experimental and computational flows
The aim of the experiments performed for this study was to obtain visual evidence
of a non-axisymmetric vortex street with an azimuthal wavelength in the range
1.6d.λd.2.0d, corresponding to the predicted Mode C instability. A number of
experiments were performed at Reynolds numbers beyond the critical Reynolds
number (Re ≈161) for the onset of non-axisymmetric flow in the wake of a ring with
ar ≈5. It was difficult to obtain a uniform axisymmetric vortex street in the wake
of the ring. The low velocities that were required to obtain the necessary Reynolds
numbers meant that asymmetrical wakes were sometimes produced. In the cases
where an axisymmetric vortex street formed after the impulsive start of the ring,
non-axisymmetric flows did evolve with an azimuthal wavelength that was consistent
with the Mode C instability.
32 G. J. Sheard, M. C. Thompson, K. Hourigan and T. Leweke
(a)(b)
Figur e 5. For caption see facing page.
(a) (i)(a) (ii)
(b) (i)(b) (ii)
(c) (i)(c) (ii)
Figur e 6. For caption see facing page.
Evolution of a subharmonic mode in a vortex street 33
Simulated-particle computations were performed to corroborate with the experi-
mental observations of the possible Mode C wakes. Some experiments at both
Re = 200 and Re = 210 produced a consistent parallel vortex-shedding pattern with
an azimuthal wavelength of 1.4d.λd.2d, with non-axisymmetric structures that
were periodic in the azimuthal direction. These Reynolds numbers were above the
threshold for the development of the Mode A instability, but the observed wavelength
of the non-axisymmetry in the wakes is consistent with the wavelength of the Mode
C instability. Since these Reynolds numbers are only slightly above the threshold for
the Mode A instability, it is clear that Mode C has a significantly larger growth rate
than Mode A in this situation.
In figure 6, the experimental dye visualizations of these non-axisymmetric wakes
are compared with the numerical particle trace computations of a saturated Mode C
wake. Figure 6(a)(i) is annotated to show the location of a vortex pair in the wake,
and the location of the ring. The computed wakes shown in figure 6 employed a
domain with an azimuthal size of approximately 1.9d. This isolated the pure Mode C
instability, and suppressed the evolution of longer-wavelength modes, such as is seen
in the far-wake region in figure 6(c)(i). Thus, Mode A will not be observed in the
simulated-particle computation images.
In figures 6(a)(i)and(b)(i), the experimental dye-visualization images were captured
as the ring was returning to the surface, and have been reproduced here upside-down
for consistency with the other figures. The important observation to make from these
visualizations is that in the near-wake region, the azimuthal span of the repeating
asymmetric structures and the orientation of the wavy deformations of the vortex
suggests that a Mode C instability was captured in each case. In figure 6(c)(i), the
experimental dye-visualization is less conclusive than those in parts (a)(i)and(b)(i).
A less-definite waviness of the vortices was observed than at Re = 200. The near-wake
region shows evidence of a poorly developed non-axisymmetric perturbation with a
wavelength of approximately 1.4d. This is too long, and at too low a Reynolds number
to be attributed to the Mode B instability, but is too short to be a Mode A instability.
Again, it must be concluded that this perturbation is associated with the subharmonic
Mode C instability. Further downstream, the vortices exhibit a deformation consistent
with the wavelength of a Mode A instability (approximately 4d). Although it is likely
that insufficient time has transpired for the non-axisymmetric instabilities to develop
fully in these images. they nevertheless show strong evidence confirming that the wake
behind the ring with ar =4.94 is indeed unstable to a non-axisymmetric instability
with a wavelength consistent with the Mode C instability.
It remains to be shown whether the difficulty in capturing well-defined non-
axisymmetric structures experimentally at Re = 210 was a by-product of the
limitations imposed by the experimental method, or a result of a change in the
respective stability of the Mode A and C instabilities in the wake. This problem is
addressed in the next section, where computations with an azimuthal domain size of
Figure 5. The axisymmetric wake behind a ring with ar =4.94 at Re = 100.
(a) Experimental and (b) numerical flows are shown at a similar stage in the shedding cycle.
Figur e 6. Experimental (i) and numerical (ii ) visualization of a ring with ar =4.94. (a, b)
Flow at Re = 200. (c) Flow at Re = 210. Flow is upwards in each frame. The images are
annotated to show the azimuthal wavelength of the instability.
34 G. J. Sheard, M. C. Thompson, K. Hourigan and T. Leweke
approximately 4dat a higher Reynolds number are presented. These computations
suggest a nonlinear transition to a subsequent non-axisymmetric flow regime
dominated by the Mode A instability.
The non-dimensional times at which the experimental dye-visualization images
in figure 6 were obtained varied between approximately 45 and 50 time units from
the impulsive initiation of the ring motion. Figure 3 shows that at Re = 190, the
computed non-axisymmetric wake instabilities took significantly longer to evolve,
as they developed from a very small random perturbation to the axisymmetric flow
field. Why, then, is the time scale for the evolution of the experimentally observed
wakes so short? Two factors somewhat mitigate this problem: first, the experiments
were conducted at Reynolds numbers greater than Re = 190, thus reducing the time
required for the mode to develop owing to the higher growth rate in the linear regime
of the instability; and secondly, conditions in the tank provided a three-dimensional
perturbation of significant magnitude from which the non-axisymmetric mode could
develop more rapidly. Despite these points, however, it must be stressed that the dye-
visualization images presented in figure 6 do not show the fully developed periodic
wakes at those Reynolds numbers. Instead, the images are considered to be indicative
of the non-axisymmetric features that will dominate the wake at the given Reynolds
numbers.
3.3. The secondary non-axisymmetric instability
The computations presented in the previous sections verified that the evolution of
the subharmonic Mode C instability in a vortex street caused a period-doubling of
the wake. For a period-doubling cascade, an increase in the control parameter of the
system leads to additional period-doubling bifurcations. The control parameter for
the flow past a ring is the Reynolds number, and in this section, computations at
a higher Reynolds number are presented to determine whether the period-doubling
bifurcation leads to a period-doubling cascade in the wake. The flow past a ring
with ar = 5 was computed at Re = 220, again with an azimuthal span of 3.9d.In
figure 7(a), an isosurface plot of the saturated non-axisymmetric wake at Re = 220 is
presented. The plot shows that the Mode C wake observed at Re = 190 is replaced by
aModeAwakeatRe = 220. The azimuthal span of the non-axisymmetric wake was
3.9d, and the distribution of streamwise vorticity suggests that the non-axisymmetric
wake is periodic with the vortex street.
Figure 7(b) shows a plot of the peak-to-peak frequency variation as the wake
saturates at Re = 220. The saturated flow computed at Re = 190 was used as an initial
condition for the computation at Re =220, and hence a behaviour consistent with the
flow after a period-doubling bifurcation is observed at the left-hand side of the plot.
The plot shows that the subharmonic non-axisymmetric wake was initially unstable,
and quickly reverted to a periodic state in which no evidence of a period-doubling
bifurcation was detected.
In figure 8, phase plots are presented for the saturated non-axisymmetric flow past
a ring with ar =5 at Re = 220. Figure 8(a) shows that the saturated wake is periodic
with the vortex street at Re = 220, and figure 8(b) shows that no period-doubling
can be observed in (u, v)-space. The dye-visualization images from experiments at
Re = 200 which were presented in figures 6(a)(i)and6(b)(i) showed that for Reynolds
numbers up to and including Re ≈200, non-axisymmetric structures consistent with
a Mode C instability were observed. In the dye visualization at Re = 210 presented
in figure 6(c)(i) little evidence of Mode C wake structures can be seen, and beyond
Evolution of a subharmonic mode in a vortex street 35
0.20
(b)(a)
0.19
0.18
0.17 0 100 200 300
St
t
Figur e 7. The non-axisymmetric flow past a ring computed with ar =5andanazimuthal
domain of 3.9dat Re = 220. (a) An isosurface plot of the saturated wake. (b) The peak-to-peak
frequency variation. Isosurface shading and plot symbols are as per figure 3.
u
0.6 0.8 1.0 1.2
(b)(a)
t
v
700 720 740
–0.4
–0.2
0
0.2
0.4
0.6
–0.4
–0.2
0
0.2
0.4
0.6
Figur e 8. Phase plots of the saturated non-axisymmetric Mode A wake at Re = 220 for the
flow past a ring with ar = 5. Velocities were recorded as in figure 4. (a)Thev-velocity is
plotted against time. (b)Thev-velocity is plotted against the u-velocity.
36 G. J. Sheard, M. C. Thompson, K. Hourigan and T. Leweke
approximately 5ddownstream, a waviness in the vortices is observed with a wavelength
consistent with the Mode A instability. The experimental observations support these
computations with an azimuthal domain size of approximately 4dat Re = 220,
showing that through some nonlinear process, the Mode A instability becomes the
dominant feature of the wake somewhere in the range 200 <Re <220, and the
period-doubling characteristics of the wake are lost.
4. Conclusions
The vortex street in the wake of a circular cylinder is unstable to regular three-
dimensional instabilities known as Modes A and B. The evolution of these instabilities
in the wake leads to the development of turbulent flow through the emergence of
fine scales in the wake, and the uniformity of the vortex street is lost through the
development of spatio-temporal chaos (Henderson 1997).
A recent analysis (Sheard et al. 2003b) of the flow around a ring has predicted, in
addition to the regular Mode A and B instabilities, an instability with respect to a
subharmonic non-axisymmetric perturbation known as Mode C. It was shown that
for a ring of aspect ratio ar = 5, this subharmonic instability is the first-occurring non-
axisymmetric instability as the Reynolds number is increased, and that its development
leads to a period-doubling of the vortex street behind the ring. In the present paper,
simulated-particle computations and experimental dye-visualization confirmed the
existence of the predicted non-axisymmetric Mode C wake in the flow past a ring
with ar ≈4.94 up to Re ≈210.
Computations with an azimuthal span sufficiently large to include each of the
Mode A, B and C instabilities in the flow past a ring with ar = 5 showed that the
period-doubling bifurcation associated with the evolution of the subharmonic Mode C
instability did not initiate a period-doubling cascade in the vortex street. With an
increase in Reynolds number, the Mode C wake structures in the vortex street were
replaced by Mode A wake structures. Therefore, it is proposed that the development
of turbulence in the vortex street behind a ring is independent of the order in which
the non-axisymmetric instabilities occur. Furthermore, from previous studies of the
vortex street behind a circular cylinder, and the present study of the vortex street
behind rings, it is speculated that the route to turbulence for a vortex street behind an
arbitrary body is independent of the periodicity of the three-dimensional instabilities
of the flow.
For future work, it would be useful to compute the vortex streets behind rings
with an azimuthal domain size of several multiples of the wavelength of the Mode
A instability. This would enable long-wavelength effects such as spatio-temporal
chaos to be computed. The route to chaos for the vortex streets behind rings
could then be computed for comparisons to the vortex street behind a circular
cylinder.
The computations presented in this paper were performed on the facilities provided
by the Victorian Partnership for Advanced Computing (VPAC) consortium. This
paper was supported by an ARC Linkage International Grant. G. J. S. received
financial assistance from a Postgraduate Publication Award while this paper was
in preparation. For his stay at Monash, T. L. received funding from the Australian
Research Council and CNRS in France.
Evolution of a subharmonic mode in a vortex street 37
REFERENCES
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake
of a circular cylinder. J. Fluid Mech. 322, 215–241.
Bearman,P.W.&Takamoto,M.1988 Vortex shedding behind rings and discs. Fluid Dyn. Res. 3,
214–218.
Blackburn,H.M.&Lopez,J.M.2003 On three-dimensional quasi-periodic Floquet instabilities
of two-dimensional bluff body wakes. Phys. Fluids 15, L57–L60.
Braun, R., Feudel, F. & Guzdar, P. 1998 Route to chaos for a two-dimensional externally driven
flow. Phys. Rev. E58, 1927–1932.
Henderson, R. D. 1997 Nonlinear dynamics and pattern formation in turbulent wake transition.
J. Fluid Mech. 352, 65–112.
Henderson, R. D. & Barkley, D. 1996 Secondary instability in the wake of a circular cylinder.
Phys. Fluids 8, 1683–1685.
Karniadakis,G.E.,Israeli,M.&Orszag,S.A.1991 High-order splitting methods for the
incompressible Navier–Stokes equations. J. Comput. Phys. 97, 414–443.
Leweke, T. & Provansal, M. 1994 Model for the transition in bluff body wakes. Phys. Rev. Lett.
72, 3174–3177.
Leweke, T. & Provansal, M. 1995 The flow behind rings: bluff body wakes without end effects.
J. Fluid Mech. 288, 265–310.
Monson, D. R. 1983 The effect of transverse curvature on the drag and vortex shedding of
elongated bluff bodies at low Reynolds number. Trans. ASME I: J. Fluids Engng 105, 308–
317.
Noack, B. R. & Ecke lmann , H. 1994aA global stability analysis of the steady and periodic cylinder
wake. J. Fluid Mech. 270, 297–330.
Noack, B. R. & Ecke lmann , H. 1994bA low-dimensional Galerkin method for the three-
dimensional flow around a circular cylinder. Phys. Fluids 6, 124–143.
Robichaux, J., Balachandar, S. & Vanka, S. P. 1999 Three-dimensional Floquet instability of the
wake of a square cylinder. Phys. Fluids 11, 560–578.
Rockwell, D., Nuzzi, F. & Magness, C. 1991 Period doubling in the wake of a three-dimensional
cylinder. Phys. Fluids A3, 1477–1478.
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2001 A numerical study of bluff ring
wake stability. In Proc. of the Fourteenth Australasian Fluid Mech. Conf. (ed. B. B. Dally),
pp. 401–404. Department of Mechanical Engineering, Adelaide University, SA 5005, Australia.
Adelaide University.
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2003aCriticality and structure of the asymmetric
vortex shedding modes of bluff ring wakes. In The 5th Euromech Fluid Mech. Conf.: Book of
Abstracts, p. 391. Centre des Congr `
es Pierre Baudis, Toulouse, France. Institut de M´
echanique
des Fluides de Toulouse.
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2003bFrom spheres to circular cylinders: the
stability and flow structures of bluff ring wakes. J. Fluid Mech. 492, 147–180.
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2004 From spheres to circular cylinders:
non-axisymmetric transitions in the flow past rings. J. Fluid Mech. 506, 45–78.
Thompson, M. C., Hourigan, K. & Sheridan, J. 1994 Three-dimensional instabilities in the wake of
a circular cylinder. In Proc. of the Intl Colloquium on Jets, Wakes and Shear Layers . CSIRO,
Melbourne, VIC, Australia.
Thompson, M. C., Hourigan, K. & Sheridan, J. 1996 Three-dimensional instabilities in the wake
of a circular cylinder. Expl Therm. Fluid Sci. 12, 190–196.
Thompson, M. C., Leweke, T. & Provansal, M. 2001 Kinematics and dynamics of sphere wake
transition. J. Fluids Struct. 15, 575–585.
Tomboulides, A. G., Triantafyllou, G. S. & Karniadakis, G. E. 1992 A new mechanism of period
doubling in free shear flows. Phys. Fluids A4, 1329–1332.
Williamson, C. H. K. 1985 Evolution of a single wake behind a pair of bluff bodies. J. Fluid Mech.
159, 1–18.
Williamson, C. H. K. 1988 The existence of two stages in the transition to three-dimensionality of
a cylinder wake. Phys. Fluids 31, 3165–3168.
38 G. J. Sheard, M. C. Thompson, K. Hourigan and T. Leweke
Williamson, C. H. K. 1989 Oblique and parallel mode of vortex shedding in the wake of a circular
cylinder at low Reynolds numbers. J. Fluid Mech. 206, 579–627.
Williamson, C. H. K. 1992 The natural and forced formation of spot-like ‘vortex dislocations’ in
the transition of a wake. J. Fluid Mech. 243, 393–441.
Williamson, C. H. K. 1996aMode A secondary instability in wake transition. Phys. Fluids 8,
1680–1682.
Williamson, C. H. K. 1996bThree-dimensional wake transition. J. Fluid Mech. 328, 345–407.
