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.