J. Fluid Mech. (2003), vol. 492, pp. 147–180.
c
2003 Cambridge University Press
DOI: 10.1017/S0022112003005512 Printed in the United Kingdom
147
From spheres to circular cylinders: the stability
and flow structures of bluff ring wakes
By G. J. S H E A R D,M.C.THOMPSONAND K. HOURIGAN
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical
Engineering, Monash University, Melbourne, Victoria 3800, Australia
(Received 18 July 2002 and in revised form 23 April 2003)
The low-Reynolds-number wake dynamics and stability of the flow past toroids placed
normal to the flow direction are studied numerically. This bluff body has the attrac-
tive feature of behaving like the sphere at small aspect ratios, and locally like the
straight circular cylinder at large aspect ratios. Importantly, the geometry of the
ring is described by a single parameter, the aspect ratio (Ar), defined as a ratio of
the torus diameter to the cross-sectional diameter of the ring. A rich diversity of
wake topologies and flow transitions can therefore be investigated by varying the
aspect ratio. Studying this geometry allows our understanding to be developed as to
why the wake transitions leading to turbulence for the sphere and circular cylinder
differ so greatly. Strouhal–Reynolds-number profiles are determined for a range of
ring aspect ratios, as are critical Reynolds numbers for the onset of flow separation,
unsteady flow and asymmetry. Results are compared with experimental findings
from the literature. Calculated Strouhal–Reynolds-number profiles show that ring
wakes shed at frequencies progressively closer to that of the straight circular cylinder
wake as aspect ratio is increased from Ar =3. For Ar > 8, the initial asymmetric
transition is structurally analogous to the mode A transition for the circular cylinder,
with a discontinuity present in the Strouhal–Reynolds-number profile. The present
numerical study reveals a shedding-frequency decrease with decreasing aspect ratio for
ring wakes, and an increase in the critical Reynolds numbers for flow separation and
the unsteady flow transition. A Floquet stability analysis has revealed the existence
of three modes of asymmetric vortex shedding in the wake of larger rings. Two
of these modes are analogous to mode A and mode B of the circular cylinder wake,
and the third mode, mode C, is analogous to the intermediate wavelength mode found
in the wake of square section cylinders and circular cylinder wakes perturbed by a
tripwire. Furthermore, three distinct asymmetric transition modes have been identified
in the wake of small aspect ratio bluff rings. Fully developed asymmetric simulations
have verified the unsteady transition for rings that exhibit a steady asymmetric wake.
1. Introduction
Wake flows of two-dimensional bluff-body geometries, and the inherent transitions
with increasing Reynolds number from steady two-dimensional wake flow, through
unsteady and three-dimensional flows, to fully turbulent wakes have been of interest
to researchers for many decades. A recent comprehensive review of the work on
the circular cylinder wake has been provided by Williamson (1996). The wake
transitions for another widely studied bluff body, the sphere, are markedly different
(e.g. Johnson & Patel 1999; Ormi
`
eres & Provansal 1999; Tomboulides & Orszag 2000;
148 G. J. Sheard, M. C. Thompson and K. Hourigan
Thompson, Leweke & Provansal 2001a). There are also relevant related studies into
the wakes from other body geometries such as the square cross-sectioned cylinder (e.g.
Robichaux, Balachandar & Vanka 1999), and long rectangular plates (e.g. Hourigan,
Thompson & Tan 2001; Mills, Sheridan & Hourigan 2002, 2003). Although these
studies indicate some similarities in the bifurcations and wake dynamics from different
bodies, there are also significant differences that warrant further investigation. In this
paper, we are especially interested in the differences between the sphere and circular
cylinder wakes which show major differences in the wake bifurcations as a function
of Reynolds number. The characteristics of these geometries are now presented. The
characteristics of both the mode C instability for the square cylinder and the wakes
of bluff rings are also presented.
1.1. The sphere wake
Amajor difference in the wake transition behaviour of the sphere and circular cylinder
wakes is that the sphere wake becomes asymmetrical prior to a transition to unsteady
flow (Magarvey & Bishop 1961a, b), whereas the cylinder wake becomes unsteady
before asymmetric structures become present in the wake (Williamson 1988a, b).
Forthe sphere wake, the transition from attached to separated flow at the rear of
the sphere has been interpolated from direct numerical simulations to be Re
S1
=20
(Tomboulides, Orszag & Karniadakis 1993; Johnson & Patel 1999; Tomboulides &
Orszag 2000). On increasing the Reynolds number, the wake remains steady and
axisymmetric up to Re
S2
= 211 (Johnson & Patel 1999). Magarvey & Bishop (1961b)
provided early flow visualizations of a liquid sphere falling through a liquid phase.
The experimental layout enabled striking images of the trailing wakes to be captured,
as they were motionless in the reference frame of the camera. The transition to
asymmetry is through a regular bifurcation (i.e. steady to steady flow) of the m =1
azimuthal mode (Tomboulides et al. 1993; Tomboulides & Orszag 2000). Their studies
located the transition at Re
S2
= 212. In good agreement, the numerical stability
analysis of Natarajan & Acrivos (1993) also found the m =1 azimuthal mode to
undergo a regular bifurcation at Re
S2
= 210. Experiments and numerical simulations
(Johnson & Patel 1999) found the resulting wake to undergo a regular bifurcation
through a shift of the steady recirculating bubble behind the sphere from the axis.
Two threads of vorticity trail downstream from the recirculation bubble. This wake
structure has become known as the double-threaded wake, and has also been predicted
numerically by Tomboulides & Orszag (2000). The beautiful early dye visualizations of
Magarvey & Bishop (1961b)foundthatthe double-threaded wake exists in the range
200 < Re < 350. Since then, more accurate experiments and numerical simulations
have refined this range considerably, as described below.
The steady asymmetric wake undergoes a further transition to unsteady flow at
ahigher Reynolds number. Through stability analysis, Natarajan & Acrivos (1993)
found a time-dependent instability of the m =1 azimuthal mode at Re
S3
= 277.5, and
the visualizations from numerical simulations (Tomboulides et al. 1993; Johnson &
Patel 1999; Tomboulides & Orszag 2000) support this bifurcation scenario, with
unsteady wakes being observed for Re > 280. In all instances, the unsteady wake
consisted of vortex loops or hairpins shedding downstream from the sphere, in the
same plane as that of the initial steady asymmetric structures. Magarvey & Bishop
(1961b)observed this periodic wake pattern at Re = 350. An analysis of the transition
to the periodic wake was also performed by Magarvey & MacLatchy (1965). They
observed the equilibrium in transport of vorticity into and out of the near field
of the double-threaded wake in the approximate range 200 < Re < 300. For higher
Spheres to circular cylinders 149
Reynolds numbers, the formation of the periodic hairpin-shedding wake was required
to transport the vorticity generated behind the sphere downstream. The periodic wake
of the sphere remains planar-symmetric up to a Reynolds number of approximately
Re = 375, as observed numerically by (Mittal 1999a, b). Johnson & Patel (1999) esti-
mated the unsteady transition to occur in the range 270 < Re
S3
< 280.
Stability of the sphere wake has been studied using the complex wave amplitude
Landau equation (Ghidersa & Du
ˇ
sek 2000; Thompson et al. 2001a). The coefficients
of the linear and cubic terms of the Landau model were estimated from asymmetric
numerical simulations close to the transition Reynolds numbers. The initial asym-
metric transition was found to be a regular type transition, occurring at Re
S2
= 212,
and the subsequent transition was identified as being a Hopf transition at Re
S3
= 272.
The critical Reynolds numbers of the transitions are in agreement with previous
studies. The analysis demonstrates that both transitions were shown to be supercritical
(non-hysteretic).
Tomboulides et al. (1993) observed fine-scale flow structures in large-eddy numerical
simulations in the Reynolds-number range 500 < Re < 1000. Magarvey & Bishop
(1961b)observed a breakdown in periodicity of the hairpin shedding for Re > 600
also. These results are considered to mark the onset of turbulence, and hence are
beyond the scope of the present study.
1.2. The circular cylinder wake
The initial transition for the cylinder wake occurs with the separation of flow from
the rear of the cylinder, resulting in a steady recirculation bubble. This transition was
predicted by numerical stability analysis to occur at Re
C1
=5 (Noack & Eckelmann
1994b). The recirculation zone remains steady two-dimensional and symmetrical about
the centreline of the flow until a subsequent transition to periodic flow occurs. This
transition was predicted to occur at Re
C2
=54(Noack & Eckelmann 1994a); however,
the Galerkin method used appeared to have too few modes to capture the instability
accurately. The experimentally derived results of Williamson (1988a, 1989) at Re =49
are widely regarded as more accurate. Sheard, Thompson & Hourigan (2001) validated
this finding through application of a spectral-element method, achieving Re
C2
= 47,
in good agreement with Du
ˇ
sek, Frauni
´
e&Le Gal (1994), who obtainined a value
of Re
C2
=47.1throughnumerical simulation and the application of the theoretical
Landau model. Du
ˇ
sek et al. (1994) identified the transition as a Hopf bifurcation.
Two three-dimensional wake states are observed in the wake behind the circular
cylinder: oblique shedding and instabilities of the parallel vortex street. At Reynolds
numbers Re > 64, oblique shedding is observed (Williamson 1988a, 1996), where
the vortex rollers are shed at an angle to the cylinder resulting in a reduction of
Strouhal number. Oblique shedding is a phenomenon associated with the interaction
of nonlinear long-wavelength azimuthal modes disrupting the parallel vortex street,
and as such is beyond the scope of the present work.
Experiments have found the parallel periodic vortex shedding street becomes
unstable to three-dimensional instabilities at Re > 178 (Williamson 1988a, 1996).
This transition was studied using a linear Floquet stability analysis (Barkley &
Henderson 1996). They found that at Re = 188.5, the cylinder wake becomes unstable
to three-dimensional perturbations with a spanwise wavelength of 3.96 diameters (d).
Asecond instability on the two-dimensional base flow was found at Re = 259, with
aspanwise wavelength of 0.822d.These instabilities and their respective spanwise
wavelengths agree closely with experimental observations of the mode A and mode B
wake structures observed experimentally by Williamson (1988b). Three-dimensional
150 G. J. Sheard, M. C. Thompson and K. Hourigan
simulations by Thompson, Hourigan & Sheridan (1994, 1996) captured detailed im-
ages of the saturated three-dimensional streamwise vortical structures corresponding
to these two different bifurcations.
Henderson (1997) performed three-dimensional simulations on the wake of the cir-
cular cylinder through the mode A and mode B transitions. The span of the simula-
tions was varied up to 4 times the spanwise wavelength of the mode A instability. The
interaction between the mode A and mode B instabilities was studied by monitoring
the energy present in the various spanwise Fourier modes of the simulations. Wake
visualizations were captured at Re = 265 showing the coexistence of both mode A
and mode B wake structures. This spontaneous switching between one mode and
the other may explain the presence of two distinct Strouhal frequencies in the wake
in the Reynolds-number range 230 < Re < 260 as observed by Williamson (1988b).
An attempt was made to study the physical mechanism leading to the formation
of streamwise vortical wake structures (Mittal & Balachandar 1995); however, the
computational domain only spanned a single cylinder diameter, resulting in the
artificial suppression of mode A structures. They did, however, observe the formation
of well-defined vortical structures in the braid region of the vortex street, associated
with mode B shedding. A detailed Floquet analysis was performed (Thompson,
Leweke & Williamson 2001b)inanattempttoidentifythe physical mechanism of
the mode A transition showing, although complex, it is consistent with an elliptic
instability of the vortex cores. Evidence suggests that the transition is in fact a
cooperative elliptic instability (Leweke & Williamson 1998), with the elliptic instability
dominant in initiating the growth of the three-dimensional flow structures in the near
wake. Advection then transports some perturbation into the braid regions as the wake
convects downstream.
Ageometric analogy exists between the two-dimensional circular cylinder placed
close to a wall, and the circular cross-section bluff ring at small aspect ratios, where the
ring cross-section lies in the vicinity of the axis. Essentially, both the axis of the ring
and the boundary near the cylinder constrain and deform the resulting wake. A free
surface with a Froude number (Fr =0) dominated by gravity is essentially a boundary
with zero tangential stresses. Hourigan, Reichl & Thompson (2002) modelled such
a case with numerical simulations at a Reynolds number Re = 180. They showed
that as the cylinder approached the free surface, the Strouhal number for the vortex
shedding street increased by 10% from the reference cylinder with no boundaries in
its vicinity. This maximum shedding frequency occurred where the gap between the
cylinder and the wall was 0.7 times the diameter of the cylinder (0.7d). A further
reduction in this gap saw a rapid drop in frequency, until for gap ratios less than 0.1d
no vortex shedding was observed. The numerical bluff ring study by Sheard et al.
(2001) presents Strouhal-number profiles showing a similar reduction in Strouhal
number with decreasing aspect ratio (i.e. a decreasing gap between the axis and
the circular ring cross-section) to the work of Hourigan et al. (2002). Despite this,
no Strouhal-number rise with decreasing aspect ratio can be found for bluff rings,
although a small rise in Strouhal number as the gap ratio approached 0.7d was
observed for the cylinder near a free surface.
1.3. Square cylinder wake
When studying the three-dimensional stability of the square cylinder, Robichaux et al.
(1999) found an intermediate wavelength mode (2.8d)thattheydesignated mode S.
This mode was periodic over two shedding cycles of the base flow (2T symmetry). This
symmetry is in contrast to modes A and B, which are periodic over a single period of
Spheres to circular cylinders 151
Sphere transition type Reynolds number
Boundary-layer separation 20
Regular asymmetric transition – regular bifurcation 210 to 212
Hopf transition – Hopf bifurcation 270 to 280
Tab le 1. Transition Reynolds numbers for the wake around a sphere (Johnson & Patel 1999;
Tomboulides & Orszag 2000).
Circular cylinder transition type Reynolds number
Boundary-layer separation 4 to 5
Hopf transition – Hopf bifurcation 47
Three-dimensional transition – Hopf bifurcation 188.5
Tab le 2. Transition Reynolds numbers for the wake around a circular cylinder (Noak &
Eckelmann 1994b;Williamson 1988a;Barkley & Henderson 1996).
the base flow (1T symmetry). Their study also found analogous mode A and mode B
instabilities in the wake; however, the spanwise wavelengths of both modes were about
40% larger than those for a circular cylinder. This is consistent with the diagonal
length being the dimension controlling the scaling of the spanwise wavelengths of
the instability modes for this particular geometry. The mode S topology and periodi-
city is identical to the mode C instability observed for the circular cylinder wake
(Zhang et al. 1995) when a tripwire is placed close to the body. The wavelength of
the mode C instability was predicted to be 2d.Thecircular cylinder wake has not
been found to exhibit this mode C type instability without artificial forcing.
Asummary of the sphere and cylinder transitions is presented in tables 1 and 2,
respectively.
1.4. Wake of a toroidal body
Abluff-body geometry that spans the extremes of wake behaviour shown by the
sphere and circular cylinder as a single geometric parameter is varied, is the torus
(or ring) with its axis placed parallel to the flow. This particular geometry has been
studied previously by Leweke & Provansal (1995), with the main motivation to remove
the end effects that experimentally hinder circular cylinder wake studies.
The parameters specifying the bluff ring geometry are defined as in Leweke &
Provansal (1995), consistent with earlier work by the current authors. We define the
aspect ratio as Ar = D/d,whereD is the major diameter of the circular centreline of
the ring cross-section, and d is the minor diameter of the cross-section of the ring.
The geometry is represented schematically in figure 1. By varying the single geometric
parameter Ar,auniformaxisymmetric body is described varying from a sphere at
Ar =0, to a straight cylinder in the limit Ar →∞.Thehole in the centre of the ring
first appears at the axis at Ar =1.
The Reynolds number is based on the uniform free-stream velocity, U ,thelength
dimension d,andthekinematic viscosity, ν,consistent with the previous definitions
for both the sphere and cylinder.
The flow around a bluff ring has been afforded limited attention in the literature.
Roshko (1953) showed experimentally that laminar vortex shedding from rings occur-
red at frequencies lower than for the circular cylinder by up to a few per cent. This
behaviour was quantified experimentally by Leweke & Provansal (1995). They defined
aStrouhal–Reynolds-curvature relationship for laminar shedding for a ring diameter
