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Florent RAVELET, René DELFOS, Jerry WESTERWEEL - Influence of global rotation and
Reynolds number on the large-scale features of a turbulent Taylor–Couette flow - Physics of
Fluids p.1-8 - 2010
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Influence of global rotation and Reynolds number on the large-scale
features of a turbulent Taylor–Couette flow
F. Ravelet,
1,a兲
R. Delfos,
2
and J. Westerweel
2
1
DynFluid, Arts et Métiers-ParisTech, 151 Blvd. de l’Hôpital, 75013 Paris, France
2
Laboratory for Aero and Hydrodynamics, Mekelweg 2, 2628 CD Delft, The Netherlands
共Received 25 January 2010; accepted 27 February 2010; published online 7 May 2010兲
We experimentally study the turbulent flow between two coaxial and independently rotating
cylinders. We determined the scaling of the torque with Reynolds numbers at various angular
velocity ratios 共Rotation numbers兲 and the behavior of the wall shear stress when varying the
Rotation number at high Reynolds numbers. We compare the curves with particle image velocimetry
analysis of the mean flow and show the peculiar role of perfect counter-rotation for the emergence
of organized large scale structures in the mean part of this very turbulent flow that appear in a
smooth and continuous way: the transition resembles a supercritical bifurcation of the secondary
mean flow. © 2010 American Institute of Physics. 关doi:10.1063/1.3392773兴
I. INTRODUCTION
Turbulent shear flows are present in many applied and
fundamental problems, ranging from small scales 共such as in
the cardiovascular system兲 to very large scales 共such as in
meteorology兲. One of the several open questions is the emer-
gence of coherent large-scale structures in turbulent flows.
1
Another interesting problem concerns bifurcations, i.e., tran-
sitions in large-scale flow patterns under parametric influ-
ence, such as laminar-turbulent flow transition in pipes, or
flow pattern change within the turbulent regime, such as the
dynamo instability of a magnetic field in a conducting fluid,
2
or multistability of the mean flow in von Kármán or free-
surface Taylor–Couette flows,
3,4
leading to hysteresis or non-
trivial dynamics at large scale. In flow simulation of homo-
geneous turbulent shear flow it is observed that there is an
important role for what is called the background rotation,
which is the rotation of the frame of reference in which the
shear flow occurs. This background rotation can both sup-
press or enhance the turbulence.
5,6
We will further explicit
this in Sec. III.
A flow geometry that can generate both motions—shear
and background rotation—at the same time is the Taylor–
Couette flow which is the flow produced between differen-
tially rotating coaxial cylinders.
7
When only the inner cylin-
der rotates, the first instability, i.e., deviation from laminar
flow with circular streamlines, takes the form of toroidal
共Taylor兲 vortices. With two independently rotating cylinders,
there is a host of interesting secondary bifurcations, exten-
sively studied at intermediate Reynolds numbers, following
the work of Coles
8
and Andereck et al.
9
Moreover, it shares
strong analogies with Rayleigh–Bénard convection,
10,11
which are useful to explain different torque scalings at high
Reynolds numbers.
12,13
Finally, for some parameters relevant
in astrophysical problems, the basic flow is linearly stable
and can directly transit to turbulence at a sufficiently high
Reynolds number.
14
The structure of the Taylor–Couette flow, while it is in a
turbulent state, is not so well known and only few measure-
ments are available.
15
The flow measurements reported in
Ref. 15 and other torque scaling studies only deal with the
case where only the inner cylinder rotates.
12,13
In that precise
case, recent direct numerical simulations suggest that
vortexlike structures still exist at high Reynolds number
共Reⲏ10
4
兲,
16,17
whereas for counter-rotating cylinders, the
flows at Reynolds numbers around 5000 are identified as
“featureless states.”
9
The structure of the flow is exemplified
with a flow visualization in Fig. 1 in our experimental setup
for a flow with only the inner cylinder rotating, counter-
rotating cylinders, and only the outer cylinder rotating,
respectively.
In the present paper, we extend the study of torques and
flow field for independently rotating cylinders to higher
Reynolds numbers 共up to 10
5
兲 and address the question of
the transition process between a turbulent flow with Taylor
vortices, and this “featureless” turbulent flow when varying
the global rotation while maintaining a constant mean shear
rate.
In Sec. II, we present the experimental device and the
measured quantities. In Sec. III, we introduce the specific set
of parameters we use to take into account the global rotation
through a “Rotation number” and the imposed shear through
a shear-Reynolds number. We then present torque scalings
and typical velocity profiles in turbulent regimes for three
particular Rotation numbers in Sec. IV. We explore the tran-
sition between these regimes at high Reynolds number vary-
ing the Rotation number in Sec. V and discuss the results in
Sec. VI.
II. EXPERIMENTAL SETUP AND MEASUREMENT
TECHNIQUES
The flow is generated between two coaxial cylinders
共Fig. 2兲. The inner cylinder has a radius of r
i
=110
⫾0.05 mm and the outer cylinder of r
o
=120⫾0.05 mm.
The gap between the cylinders is thus d = r
o
−r
i
=10 mm and
a兲
Electronic mail: florent.ravelet@ensta.org.
PHYSICS OF FLUIDS 22, 055103 共2010兲
1070-6631/2010/22共5兲/055103/8/$30.00 © 2010 American Institute of Physics22, 055103-1
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the gap ratio is
=r
i
/ r
o
=0.917. The system is closed at both
ends, with top and bottom lids rotating with the outer cylin-
der. The length of the inner cylinder is L = 220 mm 共axial
aspect ratio is L/ d=22兲. Both cylinders can rotate indepen-
dently with the use of two dc motors 共Maxon, 250 W兲. The
motors are driven by a homemade regulation device, ensur-
ing a rotation rate up to 10 Hz, with an absolute precision of
⫾0.02 Hz and a good stability. A
LABVIEW program is used
to control the experiment: the two cylinders are simulta-
neously accelerated or decelerated to the desired rotation
rates, keeping their ratio constant. This ratio can also be
changed while the cylinders rotate, maintaining a constant
differential velocity.
The torque T on the inner cylinder is measured with a
corotating torque meter 共HBM T20WN, 2 N m兲. The signal
is recorded with a 12 bit data acquisition board at a sample
rate of 2 kHz for 180 s. The absolute precision on the torque
measurements is ⫾0.01 N m, and values below 0.05 N m are
rejected. We also use the encoder on the shaft of the torque
meter to record the rotation rate of the inner cylinder. Since
that matches excellently with the demanded rate of rotation,
we assume that the outer cylinder rotates at the demanded
rate as well.
Since the torque meter is mounted in the shaft between
driving motor and cylinder, it also records 共besides the in-
tended torque on the wall bounding the gap between the two
cylinders兲 the contribution of mechanical friction such as in
the two bearings, and the fluid friction in the horizontal
共Kármán兲 gaps between tank bottom and tank top. While the
bearing friction is considered to be marginal 共and measured
so in an empty, i.e., air filled system兲, the Kármán-gap con-
tribution is much bigger: during laminar flow, we calculated
and measured this to be of the order of 80% of the gap
torque. Therefore, all measured torques were divided by a
factor of 2, and we should consider the scaling of torque with
the parameters defined in Sec. III as more accurate than the
exact numerical values of torque.
A constructionally more difficult, but also more accurate,
solution for the torque measurement is to work with three
stacked inner cylinders and only measure the torque on the
central section, such as is done in the Maryland Taylor–
Couette setup,
12
and 共under development兲 in the Twente
Turbulent Taylor–Couette setup.
18
We measure the three components of the velocity by
stereoscopic particle image velocimetry 共PIV兲
19
in a plane
illuminated by a double-pulsed Nd:yttrium aluminum garnet
laser. The plane is vertical 共Fig. 2兲, i.e., normal to the mean
flow: the in-plane components are the radial 共u兲 and axial 共v兲
velocities, while the out-of-plane component is the azimuthal
component 共w兲. It is observed from both sides with an angle
of 60° 共in air兲 using two double-frame charge coupled device
cameras on Scheimpflug mounts. The light-sheet thickness is
0.5 mm. The tracer particles are 20
m fluorescent
共rhodamine B兲 spheres. The field of view is 11⫻25 mm
2
,
corresponding to a resolution of 300 ⫻1024 pixels. Special
care has been taken concerning the calibration procedure, on
which especially the evaluation of the plane-normal azi-
muthal component heavily relies. As a calibration target we
use a thin polyester sheet with lithographically printed
crosses on it, stably attached to a rotating and translating
microtraverse. It is first put into the light sheet and traversed
perpendicularly to it. Typically, five calibration images are
taken with intervals of 0.5 mm. The raw PIV images are
processed using
DAVIS 7.2 by Lavision. They are first mapped
to world coordinates, then they are filtered with a min-max
filter, then PIV processed using a multipass algorithm, with a
last interrogation area of 32⫻32 pixels with 50% overlap,
and normalized using median filtering as postprocessing.
Then, the three components are reconstructed from the two
camera views. The mapping function is a third-order polyno-
mial, and the interpolations are bilinear. The PIV data acqui-
sition is triggered with the outer cylinder when it rotates in
order to take the pictures at the same angular position as used
during the calibration.
FIG. 1. Photographs of the flow at Re=3.6⫻10
3
. Left, A: only the inner cylinder rotating. Middle, B: counter-rotating cylinders. Right, C: only the outer
cylinder rotating. The flow structure is visualized using microscopic mica platelets 共Pearlessence兲.
ω
ω
r = 120 mm
o
i
o
r = 110 mm
i
h = 220 mm
PIV plane
1
.5 mm
(b)(a)
FIG. 2. Picture and sketch with dimensions of the experimental setup. One
can see the rotating torque meter 共upper part of the picture兲, the calibration
grid displacement device 共on top of the upper plate兲, one of the two cameras
共left side兲, and the light sheet arrangement 共right side兲. The second camera is
further to the right.
055103-2 Ravelet, Delfos, and Westerweel Phys. Fluids 22, 055103 共2010兲
Downloaded 09 May 2010 to 145.94.113.34. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
To check the reliability of the stereoscopic velocity
measurement method, we performed a measurement for a
laminar flow when only the inner cylinder rotates at a
Reynolds number as low as Re
S
=90 using an 86% glycerol-
water mixture. In that case, the analytical velocity field is
known: the radial and axial velocities are zero, and the azi-
muthal velocity w should be axisymmetric with no axial de-
pendence, and a radial profile in the form w共r兲=⍀r
i
共r
o
/ r
−r/ r
o
兲/ 共1−
2
兲.
8
The results are plotted in Fig. 3. The mea-
sured profile 共solid line兲 hardly differs from the theoretical
profile 共dashed line兲 in the bulk of the flow 关0.1ⱗ共r − r
i
兲/ d
ⱗ0.7兴. The discrepancy is, however, quite strong close to the
outer cylinder 关共r − r
i
兲/ d=1兴, which may be due to the refrac-
tion close to the curved wall that causes measurement errors.
The in-plane components, which should be zero, do not ex-
ceed 1% of the inner cylinder velocity everywhere. In con-
clusion, the measurements are very satisfying in the bulk.
Further improvements to the technique have been made since
this first PIV test, in particular a new outer cylinder of im-
proved roundness, and the measurements performed in water
for turbulent cases are reliable in the range 关0.1ⱗ共r − r
i
兲/ d
ⱗ0.85兴. The PIV measurements close to the wall could be
further improved by using an outer cylinder that is machined
in a block with flat outer interfaces normal to the cameras.
III. PARAMETER SPACE
The two traditional parameters to describe the flow are
the inner 共respectively, outer兲 Reynolds numbers Re
i
=共r
i
i
d/
兲关respectively, Re
o
=共r
o
o
d/
兲兴, with the inner
共respectively, outer兲 cylinder rotating at rotation rates
i
共re-
spectively,
o
兲, and
the kinematic viscosity.
We choose to use the set of parameters defined by Du-
brulle et al.:
20
a shear Reynolds number Re
S
and a Rotation
number Ro,
Re
S
=
2兩
Re
o
−Re
i
兩
1+
Ro = 共1−
兲
Re
i
+Re
o
Re
o
−Re
i
. 共1兲
With this choice, Re
S
is based on the laminar shear rate
S:Re
S
=Sd
2
/
. For instance, with a 20 Hz velocity difference
in counter-rotation, the shear rate is around 1400 s
−1
and
Re
S
⯝1.4⫻10
5
for water at 20 °C. A constant shear
Reynolds number corresponds to a line of slope
in the
兵Re
o
;Re
i
其 coordinate system 共see Fig. 4兲.
The Rotation number Ro compares the mean rotation
with the shear and is the inverse of a Rossby number. Its sign
defines cyclonic 共Ro⬎0兲 or anticyclonic 共Ro⬍0兲 flows. The
Rotation number is zero in case of perfect counter-rotation
共r
i
i
=−r
o
o
兲. Two other relevant values of the Rotation
number are Ro
i
=
−1⯝−0.083 and Ro
o
=共1−
兲/
⯝0.091
for, respectively, inner and outer cylinders rotating alone.
Finally, a further choice that we made in our experiment was
the value of
=r
i
/ r
o
, which we have chosen as relatively
close to unity, i.e.,
=110/ 120⯝0.91, which is considered a
narrow gap, and is the most common in reported experi-
ments, such as Refs. 8, 9, 21, and 22, although a value as low
as 0.128 is described as well.
4
A high
, i.e., 共1−
兲Ⰶ1, is
special in the sense that for
→ 1 the flow is equivalent to a
plane Couette flow with background rotation; at high
, the
flow is linearly unstable for −1⬍ Ro⬍ Ro
o
.
5,20,23
In the present study we experimentally explore regions
of the parameter space that, to our knowledge, have not been
reported before. We present in Fig. 4 the parameter space in
兵Re
o
;Re
i
其 coordinates with a sketch of the flow states iden-
tified by Andereck et al.
9
and the location of the data dis-
cussed in the present paper. One can notice that the present
range of Reynolds numbers is far beyond that of Andereck,
and that with the PIV data we mainly explore the zone be-
tween perfect counter-rotation and only the inner cylinder
rotating.
IV. STUDY OF THREE PARTICULAR
ROTATION NUMBERS
In the experiments reported in this section, we maintain
the Rotation number at constant values and vary the shear
Reynolds number. We compare three particular Rotation
numbers, Ro
i
,Ro
c
, and Ro
o
, corresponding to rotation of the
inner cylinder only, exact counter-rotation, and rotation of
the outer cylinder rotating only, respectively. In Sec. IV A,
we report torque scaling measurements for a wide range of
Reynolds numbers—from base laminar flow to highly turbu-
lent flows—and in Sec. IV B, we present typical velocity
profiles in turbulent conditions.
A. Torque scaling measurements
We present in Fig. 5 the friction factor c
f
=T/ 共2
r
i
2
LU
2
兲⬀G/ Re
2
, with U = Sd and G = T / 共
L
2
兲,asa
function of Re
S
for the three Rotation numbers. A common
definition for the scaling exponent
␣
of the dimensionless
torque is based on G: G ⬀ Re
S
␣
. We keep this definition and
present the local exponent
␣
in the inset in Fig. 5.We
compute
␣
by means of a logarithmic derivative,
␣
=2
+d log共c
f
兲/ d log共Re
S
兲.
At low Re, the three curves collapse on a Re
−1
curve.
This characterizes the laminar regime where the torque is
proportional to the shear rate on which the Reynolds number
is based.
0 .2 .4 .6 .8
1
0
0.2
0.4
0.6
0.8
1
(r−r
i
)/d
Velocity
(
dimless
)
FIG. 3. Dimensionless azimuthal velocity profile 关w / 共r
i
i
兲 vs 共r −r
i
兲/ d兴 for
Ro=Ro
i
at Re
S
=90 共see Sec. III for the definition of the parameters兲. Solid
line: measured mean azimuthal velocity. Dashed line: theoretical profile.
The radial component u, which should be zero, is also shown as a thin solid
line.
055103-3 Influence of global rotation and Reynolds number Phys. Fluids 22, 055103 共2010兲
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FIG. 4. 共Color online兲 Parameter space in 兵Re
o
;Re
i
其 coordinates. The vertical axis Re
o
=0 corresponds to Ro = Ro
i
=−0.083 and has been widely studied 共Refs.
12, 13, 16, and 17兲. The horizontal axis Re
i
=0 corresponds to Ro=Ro
o
=0.091. The line Re
i
=−Re
o
corresponds to counter-rotation, i.e., Ro=Ro
c
=0. The PIV
data taken at a constant shear Reynolds number of Re
S
=1.4⫻10
4
are plotted with 共䊊兲. The torque data with varying Ro at constant shear for various Re
S
ranging from Re
S
=3⫻10
3
to Re
S
=4.7⫻10
4
are plotted as gray 共blue online兲 lines. They are discussed in Fig. 8. We also plot the states identified at much
lower Re
S
by Andereck et al. 共Ref. 9兲 as patches: black corresponds to laminar Couette flow, gray 共green online兲 to “spiral turbulence,” dotted zone to
“featureless turbulence,” and vertical stripes to an “unexplored” zone.
10
1
10
2
10
3
10
4
10
5
10
6
10
−3
10
−2
10
−1
R
e
C
f
10
1
10
2
10
3
10
4
10
5
10
6
1
1.5
2
Re
α
FIG. 5. 共Color online兲 Friction factor c
f
vs Re
S
for Ro
i
=
−1 共black 䊊兲,Ro
c
=0 共blue 䊐兲, and Ro
o
=共1−
兲/
共red 〫兲. Relative error on Re
S
: ⫾5%; absolute
error on torque: ⫾0.01 N m. Dashed 共green online兲 line: Lewis’ data 关Ref. 13,Eq.共3兲兴, for Ro
i
and
=0.724. Dash-dotted 共magenta online兲 line: Racina’s data
关Ref. 22,Eq.共10兲兴. Solid thin black line: laminar friction factor c
f
=1/ 共
Re兲. Inset: local exponent
␣
such that C
f
⬀Re
S
␣
−2
, computed as 2
+d log共C
f
兲/ d log共Re
S
兲, for Ro
i
=
−1 共black 䊊兲,Ro
c
=0 共blue 䊐兲, and Ro
o
=共1−
兲/
共red 〫兲. Dashed 共green online兲 line: Lewis’ data 关Ref. 13, Eq. 共3兲兴 for
Ro
i
and
=0.724.
055103-4 Ravelet, Delfos, and Westerweel Phys. Fluids 22, 055103 共2010兲
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