PHYSICS OF FLUIDS
VOLUME 15, NUMBER 2
FEBRUARY 2003
The effect of outer cylinder rotation on Taylor–Couette flow
at small aspect ratio
A. Schulz and G. Pfister
Institut für Experimentelle und Angewandte Physik, Universität Kiel, D-24098 Kiel, Germany
S. J. Tavener
Department of Mathematics, Colorado State University, Fort Collins, Colorado 80623
共Received 25 July 2002; accepted 30 October 2002; published 8 January 2003兲
We present the results of a combined experimental and numerical study of Taylor–Couette flow
where both inner and outer cylinders rotate. Excellent quantitative agreement has been obtained
between finite-element calculations and experimental measurements for a range of aspect ratios and
rotation rates. Counter-rotation was found to enhance the breaking of the reflectional symmetry
about the midplane of the apparatus, corotation to suppress symmetry breaking. For the region of
parameter space explored, increasing the corotation of the outer cylinder had a qualitatively similar
effect to increasing the aspect ratio, and vice versa. © 2003 American Institute of Physics.
关DOI: 10.1063/1.1532340兴
I. INTRODUCTION
set by limiting the geometric size of the apparatus was pioneered by Benjamin14,15 for the study of Taylor–Couette
flow. Subsequent investigations adopting the same philosophy have provided significant insights into hydrodynamic
stability and the route to chaos not only in the Taylor–
Couette system,12,13,16,17 but in other areas such as Rayleigh–
Bénard convection,18,19 and electrohydrodynamic convection
in nematic liquid crystals.20,21
In this paper we examine the effect of rotation of the
outer cylinder in a Taylor–Couette apparatus with radius ratio equal to 0.5, at aspect ratios and Reynolds numbers for
which steady flows with one or two cells occur. Benjamin
and Mullin,22 Lücke et al.,23 Pfister et al.,24 Aitta,25 and Mullin et al.26 have previously studied flows with one and two
cells when the outer cylinder is stationary. We show that
corotation favors flows that are symmetric about the midplane, while counter-rotation promotes the breaking of this
Z 2 symmetry. The symmetry-breaking phenomena encountered here occur at rotation rates at which symmetric cellular
flows are well established and computational techniques are
necessary to determine their behavior. The computational approach is described briefly in Sec. II and the laboratory experiments are discussed in Sec. III. Comparisons of experimental and numerical results over a range of parameter space
are then presented in Sec. IV.
The Taylor–Couette apparatus is a popular vehicle for
testing ideas in hydrodynamic stability and transition to turbulence, and over 2000 papers are dedicated to the subject.
Many excellent reviews exist, of which we mention just two,
written over a decade apart, by Swinney and Gollub1 and
Tagg.2 A number of modern research directions are described
in the volume edited by Egbers and Pfister.3
Coles4 and Andereck et al.5 performed experiments in
which they investigated the effect of rotating the outer cylinder. Their work has attracted considerable attention,6 – 8 due
to the intriguing array of time-dependent flows they observed. Further, by assuming axially periodic boundary conditions, these time-dependent flows and their interactions are
amenable to analysis using ideas of bifurcation in the presence of symmetry, and good qualitative agreement has been
obtained.9,10 The aim of this investigation is to determine the
effect of the rotation of the outer cylinder in a region of
parameter space in which the number of possible solutions is
small, enabling precise quantitative comparisons to be made
between laboratory experiment and finite-element calculations. Symmetry breaking is a feature of many flows and has
important consequences not only for the steady flows that are
observed at low and moderate flow rates, but also for the
dynamics which occurs at larger flow rates. Symmetry breaking can play a role in the development of simply periodic
flows as discussed for example by Mullin and Cliffe,11 but
has also been shown in the Taylor–Couette systems to influence more complex time-dependent phenomena such as
Shilnikov dynamics12 and gluing bifurcations.13 The present
study therefore seeks an accurate quantitative understanding
of the effect of outer cylinder rotation on symmetry-breaking
bifurcation as a foundation for future studies of the complex
dynamics that has been observed at larger Reynolds numbers
and aspect ratios.
The strategy of limiting the multiplicity of the solution
1070-6631/2003/15(2)/417/9/$20.00
II. FINITE-ELEMENT COMPUTATIONS
Let r 1* and r *
2 be the radii of the inner and outer cylinders, respectively, l * be the height of the cylinders, and ⍀ 1
be the rotation rate of the inner cylinder. In order to perform
finite-element computations, the steady, axisymmetric
Navier–Stokes equations were nondimensionalized, using
r⫽
417
r*
d*
⫺␤,
z⫽
z*
l*
,
u⫽
1
r 1* ⍀ 1
冉
u r* ,u ␪* ,
u z*
⌫
冊
,
© 2003 American Institute of Physics
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418
Phys. Fluids, Vol. 15, No. 2, February 2003
Schulz, Pfister, and Tavener
FIG. 1. The Taylor–Couette experimental apparatus.
p⫽
d*p*
␮ r 1* ⍀ 1
,
where d * ⫽r *
2 ⫺r 1* , ␤ ⫽r 1* /d * , ⌫⫽l * /d * , ␮ is the molecular viscosity, and * denotes dimensional lengths, velocities,
or pressures. Defining ␩ ⫽r 1* /r 2* , we see that ␤ ⫽ ␩ /(1
⫺ ␩ ). The resulting nondimensional equations are
冉
R ur
⳵ur
⳵r
⫻
冉
⫹u z
冋
⳵ur
⳵z
⫺
u 2␪
共 r⫹ ␤ 兲
册
冊 冉
⫹
⳵p
1
⫺
⳵r
共 r⫹ ␤ 兲
⳵u␪
⳵u␪
u ru ␪
冊冉
1
⫹u z
⫹
⫺
R ur
共 r⫹ ␤ 兲
⳵r
⳵z
共 r⫹ ␤ 兲
⫻
冉
R ur
冋
册
⳵r
⫹u z
冋
⳵uz
⳵z
冊
⫹
冉
共1兲
u ␪⫽
再
0
on z⫽⫾1/2,
1
on r⫽0,
␦
on r⫽1.
Note that ␦ is the ratio of the azimuthal velocities of the
冊
1 ⳵ 2u ␪
u␪
⳵
⳵u␪
⫹ 2
⫺
⫽0,
共 r⫹ ␤ 兲
2
⳵r
⳵r
⌫ ⳵z
共 r⫹ ␤ 兲 2
⳵uz
⫻
冊
⳵
⳵ur
1 ⳵ 2u r
ur
⫹ 2
⫺
⫽0,
共 r⫹ ␤ 兲
2
⳵r
⳵r
⌫ ⳵z
共 r⫹ ␤ 兲 2
FIG. 2. One of the two stable one-cell flows and the stable two-cell flow at
R⫽350, ⌫⫽1.2, ␩⫽0.5, and ␦⫽0. Flow visualization photographs and the
computed streamfunction at 20 equally spaced values between 共a兲
⫺5.25E-02 and 4.16E-03; 共b兲 ⫺4.02E-02 and 4.02E-02.
共2兲
1 ⳵p
1
⫺
2 ⳵z
共 r⫹ ␤ 兲
⌫
册
冊
⳵
⳵uz
1 ⳵ 2u z
r⫹
␤
⫹
⫽0,
兲
共
⳵r
⳵r
⌫2 ⳵z2
⳵
⳵uz
1
⫽0,
关共 r⫹ ␤ 兲 u r 兴 ⫹
共 r⫹ ␤ 兲 ⳵ r
⳵z
共3兲
共4兲
which were solved in a region
D⫽ 兵 共 r,z 兲 ,0⭐r⭐1,⫺1/2⭐z⭐1/2其 ,
subject to the boundary conditions
u r ⫽u z ⫽0
on ⳵ D
共the boundary of D),
FIG. 3. Symmetry breaking 共solid line兲 and limit points 共dashed line兲 associated with the change of primary flow from a one-cell flow to a two-cell
flow as the aspect ratio is increased at ␦⫽0 and ␩⫽0.5. Q is a quartic
bifurcation point and C is a coalescence point.
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Phys. Fluids, Vol. 15, No. 2, February 2003
Effect of outer cylinder rotation
419
FIG. 5. LDV measurements for ⌫⫽1.04 at different fixed values of the outer
cylinder Reynolds number, R 0 ⫽⫺47, ⫺33, ⫺18, 0, ⫹18, ⫹33, ⫹47, ⫹62.
all four corners of the domain, and the discontinuities in the
azimuthal velocity that exist at the corners were smoothed by
quadratically interpolating the azimuthal velocity over the
element nearest the corner. Numerical bifurcation techniques
are implemented within the finite-element code ENTWIFE,27
which was used to obtain all numerical results reported here.
A recent survey of numerical bifurcation techniques with
particular application to the Navier–Stokes equations appears in Cliffe et al.28 A computer algebra package was used
to write the subroutines to evaluate the derivatives required
for the extended systems employed. A discussion of these
implementation details appears in Cliffe and Tavener.29
III. EXPERIMENTS
A schematic diagram of the experimental apparatus is
shown in Fig. 1. The inner cylinder was machined from
FIG. 4. Bifurcation diagrams at 共a兲 ⌫⫽1.2; 共b兲 ⌫⫽1.28; 共c兲 ⌫⫽1.3, and ␦⫽0
and ␩⫽0.5. The ordinate axis measures the axial velocity u z at (r,z)
⫽(1/2,0). A and B are symmetry-breaking bifurcation points and C are limit
points. Solution branches that are stable or unstable with respect to axisymmetric steady disturbances are labeled ‘‘s’’ or ‘‘u,’’ respectively.
outer and inner cylinders. The nondimensional dynamical parameters are the Reynolds numbers of the inner (R) and
outer cylinder (R 0 ), and are defined as
R⫽ 共 ⍀ 1 r 1* 兲 d * / ␯ ,
R 0 ⫽ 共 ⍀ 2 r 2* 兲 d * / ␯ ,
where ␯ is the kinematic viscosity and ⍀ 2 is the rotation rate
of the outer cylinder.
This boundary value problem was solved using the
finite-element method using quadrilateral elements with biquadratic velocity interpolation and discontinuous linear
pressure interpolation. The pressure was normalized to be
zero at one interior node. Mesh refinement was performed in
FIG. 6. Symmetry breaking 共solid line兲 and limit points 共dashed line兲 associated with the change of primary flow from a one-cell flow to a two-cell
flow as a function of ␦ at ⌫⫽1.0 and ␩⫽0.5. Q is a quartic bifurcation point
and C is a coalescence point.
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420
Phys. Fluids, Vol. 15, No. 2, February 2003
FIG. 7. Bifurcation diagram at 共a兲 ␦⫽0.14; 共b兲 ␦⫽0.15; 共c兲 ␦⫽0.17, ⌫⫽1.0,
and ␩⫽0.5. The ordinate axis measures the axial velocity u z at (r,z)
⫽(1/2,0). A and B are symmetry-breaking bifurcation points and C are limit
points. Solution branches that are stable or unstable with respect to axisymmetric steady disturbances are labeled ‘‘s’’ or ‘‘u,’’ respectively.
stainless steel with radius r 1* ⫽12.5 mm, and the outer cylinder was constructed of high-quality glass 共ground and polished兲 with an inner radius r 2* ⫽25.0 mm. A tolerance of
better than 0.01 mm was achieved over the lengths of the
cylinders. The radius ratio ␩ was thereby fixed at 0.5, while
the aspect ratio ⌫ was continuously adjustable between 0 and
16. The angular rotation rates of the inner and outer cylinders
could be varied independently with an accuracy of better
than one part in 10 000, and the top and bottom plates were
held at rest. Experiments were performed by first fixing the
speed of the outer cylinder, then varying the rotation rate of
the inner cylinder in small steps. These changes in Reynolds
number were small enough, and the settling times between
speed changes long enough, to ensure repeatable observa-
Schulz, Pfister, and Tavener
FIG. 8. Symmetry-breaking bifurcation at 共a兲 ⌫⫽0.56; 共b兲 ⌫⫽0.71; 共c兲
⌫⫽0.86. The solid line is the computed locus of symmetry-breaking bifurcation points. The upward-pointing triangles indicate symmetry breaking in
the experiment with increasing Reynolds number.
tions. Note that during such an experiment the outer cylinder
Reynolds number R 0 remains fixed while the inner Reynolds
number R and ␦ change simultaneously. Silicon oils with
kinematic viscosities ␯ ⬇11– 12 mm2/s were used as working fluids. The temperature of the working fluid was maintained at 共21.00⫾0.01兲 °C by circulating fluid through a surrounding square box. An adjustable LDV system with a
spatial resolution of the measurement volume of 1/2 mm
共diameter兲⫻1 mm 共length兲 was used to measure a single velocity component at any chosen point in the flow. Standard
techniques were used to visualize the flow. Further details of
the experimental apparatus can be found in Schulz and
Pfister7 and Stamm et al.30
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Phys. Fluids, Vol. 15, No. 2, February 2003
Effect of outer cylinder rotation
421
FIG. 9. Symmetry-breaking bifurcation at 共a兲 ⌫⫽0.96; 共b兲 ⌫⫽1.04; 共c兲
⌫⫽1.08; 共d兲 ⌫⫽1.12; 共e兲 ⌫⫽1.2; 共f兲
⌫⫽1.24. The solid line is the computed locus of symmetry-breaking bifurcation points. The dashed line is the
computed locus of limit points. The
upward-pointing triangles indicate
symmetry breaking in the experiment
with increasing Reynolds number. The
downward-pointing triangles indicate
symmetry breaking in the experiment
with decreasing Reynolds number.
IV. RESULTS
A. One-cellÕtwo-cell exchange with ⌫ at ␦Ä0
We begin by briefly summarizing the pertinent results of
Pfister et al.24 in order to compare and contrast them with the
new results presented here. These authors performed a study
in a small aspect ratio Taylor–Couette apparatus with a stationary outer cylinder and a radius ratio of 0.5. Consistent
with earlier work, we define the primary flow to be the flow
which develops at fixed aspect ratio with slowly increasing
共inner兲 rotation rate from rest.
At the aspect ratios investigated here, two different types
of flow exist. The flow may be symmetric about the midplane as shown in Fig. 2共b兲, which we call a two-cell flow. At
other values of the parameters, the flow may be asymmetric
about the midplane, with a single dominant cell and a second, much smaller recirculation as shown in Fig. 2共a兲, which
we call a one-cell flow. Smaller vortices can be found in each
corner. They are hardly visible and will not be discussed in
this paper.
Both of the flows shown in Fig. 2 occur for identical
values of the parameters, but obviously both cannot be primary. Multiplicity of the solution set occurs as explained
below.
At an aspect ratio of approximately 1.3, there is an exchange of stability between a one-cell and a two-cell primary
flow. The paths of symmetry-breaking bifurcation and limit
points in the (⌫,R) plane are shown in Fig. 3. The corresponding sequence of bifurcation diagrams was first reported
by Cliffe31 and appears in Fig. 4. The ordinate in the bifurcation diagrams measures the axial velocity at the center of a
radial slice through the flow, i.e., u z at (r,z)⫽(1/2,0). This
is zero for symmetric two-cell flows. Solution branches that
are stable or unstable with respect to axisymmetric steady
disturbances are indicated by ‘‘s’’ or ‘‘u,’’ respectively.
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422
Phys. Fluids, Vol. 15, No. 2, February 2003
Schulz, Pfister, and Tavener
FIG. 10. Experimental velocity measurements at ⌫⫽1.24, for four different values of R 0 which increases from
共a兲 to 共d兲; 共a兲 R 0 ⫽⫹4.1; 共b兲 R 0
⫽⫹6.1; 共c兲 R 0 ⫽⫹8.1; 共d兲 R 0
⫽⫹20.4.
Both symmetry-breaking bifurcation points A and B are
supercritical at ⌫⫽1.2 as shown in Fig. 4共a兲. With increasing
aspect ratio, the supercritical symmetry-breaking point A becomes subcritical at a quartic bifurcation point, and the bifurcation diagram at ⌫⫽1.28 is shown in Fig. 4共b兲. With
further increase in aspect ratio the subcritical bifurcation
point A and the supercritical bifurcation point B coincide at a
coalescence point. Beyond the coalescence point 共in terms of
increasing aspect ratio兲, the asymmetric branches reattach in
the opposite manner as shown in Fig. 4共c兲 at ⌫⫽1.3. The
symmetric two-cell flow remains stable for all Reynolds
numbers shown and a pair of stable asymmetric flows exists
as disconnected solutions. The asymmetric branches appear
to intersect in the bifurcation diagrams at ⌫⫽1.28 and 1.3,
but this is simply a consequence of the projection.
FIG. 11. Detail of the hysteretic region at ⌫⫽1.24. The solid line is the
computed locus of symmetry-breaking bifurcation points. The dashed line is
the computed locus of limit points. Q is a quartic bifurcation point and C is
a coalescence point. The upward-pointing triangles indicate transitions in the
experiment observed with increasing Reynolds number. The downwardpointing triangles indicate transitions in the experiment observed with decreasing Reynolds number. The dotted line indicates R 0 ⫽⫹4.1, the chained
line R 0 ⫽⫹6.1, and the double-chained line R 0 ⫽⫹8.1.
At an aspect ratio of 1.2 and a Reynolds number of 350,
three steady solutions are stable, two of which are asymmetric and one which is symmetric. Flow visualization photographs of the symmetric flow and one of the asymmetric
flows are shown in Fig. 2 and compared with computed
streamlines.
B. One-cellÕtwo-cell exchange with ␦ at ⌫Ä1
The relative azimuthal velocity of the inner and outer
cylinders provides an additional parameter which alters the
bifurcation structure and can change the primary flow from a
single-cell flow to a two-cell flow. It does so in a surprisingly
similar manner to the aspect ratio.
The effect of outer cylinder rotation on the location of
the first symmetry-breaking bifurcation point is demonstrated in Fig. 5, which shows the result of seven experiments performed at different values of R 0 . In each of these
experiments the outer cylinder rotation rate was fixed and the
Reynolds number of the inner cylinder was slowly 共quasistatically兲 increased. The resulting axial velocity component
in the middle of the gap height and 4 mm away from the
inner cylinder is plotted against the Reynolds number. The
solid line indicates velocity measurements taken when the
outer cylinder was at rest and the symmetric two-cell flow
becomes unstable to a single-cell flow 共via a supercritical
bifurcation兲 at R⬇127.2.
Velocity measurements taken when the outer cylinder
was made to corotate with the inner cylinder at a fixed speed
(R 0 ⬎0) are indicated by ‘‘⫻.’’ Corotation is seen to move
the transition to a one-cell flow towards larger Reynolds
number. Indeed, for R 0 ⫽⫹62, the flow remained in the symmetric two-cell state for all Reynolds numbers shown in Fig.
5. Velocity measurements taken with a counter-rotating outer
cylinder (R 0 ⬍0) are indicated by ‘‘⫹.’’ Counter-rotation is
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Phys. Fluids, Vol. 15, No. 2, February 2003
Effect of outer cylinder rotation
423
FIG. 12. 共a兲 Aspect ratio and 共b兲 Reynolds number at the coalescence point
near the one-cell, two-cell exchange as
functions of ␦.
seen to destabilize the two-cell flow and move the transition
to a one-cell flow to smaller Reynolds numbers. The effect of
counter-rotating cylinders on the transition points is less than
it is for corotating cylinders.
The calculated locus of singular points varying the aspect ratio and holding an outer cylinder at rest 共␦⫽0兲 is
shown in Fig. 3. Investigating the effect of different rotation
rates of the outer cylinder at a fixed aspect ratio, it is observed that the locus of singular points in the ( ␦ ,R) plane
shown in Fig. 6 is qualitatively similar to the locus of singular points in the (⌫,R) plane in Fig. 3. Consequently, the
corresponding sequence of bifurcation diagrams shown in
Fig. 7 is qualitatively similar to those in Fig. 4. Corotation
appears to promote symmetric solutions and reduce the tendency for symmetry breaking. Counter-rotation has the opposite effect. Thus, the effect of increasing corotation is seen
to be qualitatively similar to the effect of increasing aspect
ratio. We note once again that at the Reynolds numbers at
which symmetry breaking is occurring, the two-cell flow is
already well established.
C. Experimental and numerical comparisons for
aspect ratios 0.56Ï⌫Ï1.24
Evidence that counter-rotation promotes symmetric solutions and corotation has the opposite effect is amply provided in Figs. 8 and 9, in which computed paths of
symmetry-breaking bifurcation points are compared with experimental observations at aspect ratios 0.56, 0.71, 0.86,
0.96, 1.04, 1.08, 1.12, 1.2, and 1.24.
At aspect ratios 0.56, 0.71, and 0.86 关Figs. 8共a兲– 8共c兲兴,
corotation is seen to delay the onset of symmetry breaking
and counter-rotation to promote symmetry breaking. At these
small aspect ratios it is not reasonable to corotate the outer
cylinder sufficiently rapidly to eliminate the symmetry
breaking altogether, and a coalescence point is not encountered within the range of ␦ examined. The differences between the measured and calculated locations of the
symmetry-breaking bifurcation points are due to imperfections of the experimental apparatus as reported earlier by
Pfister et al.16 共but note the scale兲. To determine all the transition points we fitted a square-root function to the velocity
measurements 共see Fig. 5兲. For aspect ratios equal to or
greater than 0.96, moderate corotation of the outer cylinder
can force the two-cell flow to remain stable, and Figs. 9共a兲–
9共f兲 include the coalescence point.
A sequence of velocity measurements is shown in Fig.
10 for four different rotation rates of the outer cylinder 关increasing from 共a兲 to 共d兲兴 and constant aspect ratio ⌫⫽1.24. In
Fig. 10共a兲 the two-cell symmetric flow 共denoted by ‘‘2’’兲
loses stability with increasing Reynolds number to a one-cell
flow 共denoted by ‘‘1’’兲 at a supercritical pitchfork bifurcation
point. The two-cell flow which regains stability at larger
Reynolds number is denoted by ‘‘2’’ in Fig. 10共a兲 and can be
obtained by sudden starts within a narrow band of rotation
rates. This two-cell flow loses stability with decreasing Reynolds number at the upper supercritical pitchfork bifurcation
point to a well-established one-cell flow. As the rate of corotation of the outer cylinder increases and the bifurcation at
lower Reynolds number becomes subcritical, as shown in
Figs. 10共b兲 and 10共c兲, the Reynolds number at the transition
becomes more obvious. Finally, in Fig. 10共d兲 the two-cell
state remains stable for all Reynolds numbers shown and
asymmetric flows exist only as disconnected states. These
asymmetric flows are not shown in Fig. 10共d兲.
Details of the hysteretic region at ⌫⫽1.24, obtained from
the three experiments shown in Figs. 10共a兲–10共c兲, appear in
Fig. 11. The direction of the triangles indicates whether the
observed transition occurred with increasing or decreasing
Reynolds number.
We can most efficiently examine the effect of counter- or
corotating the outer cylinder by computing the location of
the coalescence point as a function of ␦. The locus of coalescence points is plotted in the ( ␦ ,⌫) and ( ␦ ,R) planes in
Fig. 12. Corotating the outer cylinder with the inner cylinder
moves the coalescence point to smaller values of aspect ratio
and to larger values of the Reynolds number. Thus, corotation apparently favors symmetric flows.
D. Role of radius ratio
While difficult to study experimentally, we can examine
the effect of varying the radius ratio computationally by calculating the location of the coalescence point as a function of
radius ratio. The aspect ratio and Reynolds number at the
coalescence point are plotted as functions of radius ratio for
a stationary outer cylinder in Figs. 13共a兲 and 13共b兲. The azimuthal velocity ratio ␦ and Reynolds number at the coalescence point are plotted as functions of radius ratio at a fixed
value of the aspect ratio in Figs. 13共c兲 and 13共d兲. A qualitative similarity can be observed between these two figures,
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424
Phys. Fluids, Vol. 15, No. 2, February 2003
Schulz, Pfister, and Tavener
FIG. 13. 共a兲 Aspect ratio and 共b兲 Reynolds number at the coalescence point
near the one-cell, two-cell exchange as
functions of radius ratio at ␦⫽0. 共c兲
Relative azimuthal velocity and 共d兲
Reynolds number at the coalescence
point near the one-cell, two-cell exchange as functions of radius ratio at
⌫⫽1.29.
and once again the effect of increasing corotation of the outer
cylinder is similar to the effect of increasing aspect ratio.
Increasing the radius ratio above 0.5 moves the coalescence
point to larger Reynolds number and smaller aspect ratio,
and to larger Reynolds number and more negative counterrotation. Similarly, as the radius ratio approaches 0.2, the
coalescence point moves to larger Reynolds number and
smaller aspect ratio, and to larger Reynolds number and
more negative counter-rotation.
V. CONCLUSIONS
We have investigated the effect of outer cylinder rotation
in Taylor–Couette flow and observed excellent quantitative
agreement between experiment and finite-element solutions
of the Navier–Stokes equations. For the range of Reynolds
numbers, aspect ratios, and radius ratios considered here, increasing the degree of corotation of the outer cylinder has the
same qualitative effect as increasing the aspect ratio. Corotation stabilizes symmetric flows and suppresses symmetry
breaking. This study illustrates that the case when the outer
cylinder is at rest is not particularly special, but that the
bifurcation structure is robust.
ACKNOWLEDGMENT
The authors would like to thank Professor Tom Mullin
for his critical reading of the manuscript.
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1999兲.
4
D. Coles, ‘‘Transition in circular Couette flow,’’ J. Fluid Mech. 21, 385
共1965兲.
5
C. D. Andereck, S. S. Liu, and H. L. Swinney, ‘‘Flow regimes in a circular
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