Influence of global rotation and Reynolds number on the large-scale features of a turbulent Taylor-Couette flow

0.41 MB·English

by  Florent Ravelet

Influence of global rotation and Reynolds number on the large-scale features of a
turbulent Taylor-Couette flow
F. Ravelet
DynFluid, Arts et M´etiers-ParisTech, 151 Bld de l’Hoˆpital, 75013 Paris, France∗
R. Delfos and J. Westerweel
Laboratory for Aero and Hydrodynamics, Mekelweg 2, 2628 CD Delft, The Netherlands.
(Dated: Submitted toPhys. Fluids: January 26, 2010)
We experimentally study the turbulent flow between two coaxial and independently rotating
0
cylinders. We determined the scaling of the torque with Reynolds numbers at various angular
1
velocity ratios (Rotation numbers), and the behaviour of the wall shear stress when varying the
0
RotationnumberathighReynoldsnumbers. WecomparethecurveswithPIVanalysisofthemean
2
flowandshowthepeculiarroleofperfectcounterrotationfortheemergenceoforganised largescale

n structuresinthemeanpartofthisveryturbulentflowthatappearinasmoothandcontinuousway:
a thetransition resembles a supercritical bifurcation of thesecondary mean flow.
J

5
2 I. INTRODUCTION in astrophysical problems, the basic flow is linearly sta-
bleandcandirectlytransittoturbulenceatasufficiently
]
n high Reynolds number [14].
Turbulentshearflowsarepresentinmanyappliedand
y The structure of the Taylor-Couette flow while it is
fundamentalproblems,rangingfromsmallscales(suchas
d in a turbulent state, is not so well known and only few
- inthecardiovascularsystem)toverylargescales(suchas measurementsareavailable[15]. Theflowmeasurements
u inmeteorology). Oneoftheseveralopenquestionsisthe
reported in [15] and other torque scaling studies only
l
f emergenceofcoherentlarge-scalestructures inturbulent deal with the case where only the inner cylinder rotates
s. flows [1]. Another interesting problem concerns bifur- [12,13]. Inthatprecisecase,recentdirectnumericalsim-
c cations, i.e. transitions in large-scale flow patterns un-
ulations suggest that vortex-like structures still exist at
i
s der parametric influence, such as laminar-turbulent flow high Reynolds number (Re & 104) [16, 17], whereas for
y transitioninpipes,orflowpatternchangewithinthetur-
counter-rotating cylinders, the flows at Reynolds num-
h
bulent regime, such as the dynamo instability of a mag-
p bersaround5000areidentifiedas“featurelessstates”[9].
neticfieldinaconductingfluid[2],ormultistabilityofthe
[ The structure ofthe flow is exemplified with a flow visu-
meanflowinvonK´arm´anorfree-surfaceTaylor–Couette
alisation in Fig. 1 in our experimental set-up for a flow
3 flows [3, 4], leading to hysteresis or non-trivial dynamics
with only the inner cylinder rotating, counter-rotating
v at large scale. In flow simulation of homogeneousturbu-
0 cylinders and only the outer cylinder rotating, respec-
lent shear flow it is observed that there is an important
5 tively.
role for what is called the background rotation, which is
7
2 the rotation of the frame of reference in which the shear
. flowoccurs. Thisbackgroundrotationcanbothsuppress
2
or enhance the turbulence [5, 6]. We will further explicit
1
this in the next section.
7
0 Aflowgeometrythatcangeneratebothmotions,shear
v: and background rotation, at the same time is a Taylor- FIG. 1: Photographs of the flow at Re=3.6×103. Left, A:
i Couette flow, which is the flow produced between dif- only theinner cylinder rotating. Middle, B: counter rotating
X
ferentially rotating coaxial cylinders [7]. When only the cylinders. Right, C: only the outer cylinder rotating. The
r inner cylinder rotates, the first instability, i.e. deviation flow structure is vusualized using microscopic Mica-platelets
a
from laminar flow with circular streamlines, takes the (Pearlessence).
form of toroidal (Taylor) vortices. With two indepen-
dently rotating cylinders, there is a host of interesting In the present paper, we extend the study of torques
secondarybifurcations,extensively studied atintermedi- and flow field for independently rotating cylinders to
ateReynoldsnumbers,followingtheworkofColes[8]and higher Reynolds numbers (up to 105) and address the
Andereck et al. [9]. Moreover,it shares strong analogies question of the transition process between a turbulent
withRayleigh-B´enardconvection[10,11], whichareuse- flow with Taylor-vortices, and this “featureless ”turbu-
ful to explain different torque scalings at high Reynolds lent flow when varying the global rotation while main-
numbers [12, 13]. Finally, for some parameters relevant taining a constant mean shear rate.
In section II, we present the experimental device and
the measuredquantities. InsectionIII, weintroduce the
specificsetofparametersweusetotakeintoaccountthe
∗Electronicaddress: fl[email protected] globalrotationthrougha“Rotationnumber”andtheim-
2
posed shear througha shear-Reynolds number. We then and a good stability. A LabView program is used to
present torque scalings and typical velocity profiles in control the experiment: the two cylinders are simulta-
turbulent regimes for three particular Rotation numbers neously accelerated or decelerated to the desired rota-
in section IV. We explore the transition between these tion rates, keeping their ratio constant. This ratio can
regimes at high Reynolds number varying the Rotation also be changed while the cylinders rotate, maintaining
number in section V, and discuss the results in section a constant differential velocity.
VI. The torque T on the inner cylinder is measured with
a co-rotating torquemeter (HBM T20WN, 2 N.m). The
signalisrecordedwitha12bitsdataacquisitionboardat
II. EXPERIMENTAL SETUP AND a sample rate of 2 kHz for 180 s. The absolute precision
MEASUREMENT TECHNIQUES on the torque measurements is ±0.01 N.m, and values
below 0.05N.m arerejected. We alsouse the encoderon
the shaft of the torquemeter to record the rotation rate
oftheinnercylinder. Sincethatmatchesexcellentlywith
the demandedrateofrotation,we assumethatthe outer
cylinder rotates at the demanded rate as well.
Sincethetorquemeterismountedintheshaftbetween
driving motor and cylinder, it also records (besides the
intended torque on the wall bounding the gap between
the two cylinders) the contribution of mechanical fric-
tion suchas in the two bearings,and the fluid friction in
the horizontal(Ka´rm´an) gaps between tank bottom and
tank top. While the bearing friction is consiedered to
be marginal (and measured so in an empty i.e. air filled
system), the K´arm´an-gap contribution is much bigger:
during laminar flow, we calculated and measured this to
be of the order of 80% of the gap torque. Therefore,
ω all measured torques were divided by a factor 2, and we
o
shouldconsiderthescalingoftorquewiththeparameters
ω
i definedin§IIIasmoreaccuratethantheexactnumerical
r = 110 mm
i
values of torque.
A constructionally more difficult, but also more accu-
m
m rate,solutionforthetorquemeasurementistoworkwith
20 PIV plane threestackedinnercylindersandonlymeasurethetorque
2
= on the central section, such as is done in the Maryland
h
Taylor-Couette set-up [12], and (under development) in
the Twente Turbulent Taylor-Couette set-up [18].
r = 120 mm We measure the three components of the velocity by
o
stereoscopicPIV[19]inaplane illuminatedbya double-
1.5 mm
pulsed Nd:YAG laser. The plane is vertical (Fig. 2), i.e.
FIG.2: Pictureandsketchwithdimensionsoftheexperimen-
tal setup. One can see the rotating torquemeter (upper part normal to the mean flow: the in-plane components are
of picture), the calibration grid displacement device (on top the radial (u) and axial (v) velocities, while the out-of-
oftheupperplate),oneofthetwocameras(leftside)andthe plane component is the azimuthal component (w). It is
light sheet arrangement (right side). The second camera is observed from both sides with an angle of 60o (in air),
furtherto the right. using two double-frame CCD-cameras on Scheimpflug
mounts. The light-sheet thickness is 0.5 mm. The tracer
The flow is generated between two coaxial cylinders particles are 20 µm fluorescent (rhodamine B) spheres.
(Fig. 2). The inner cylinder has a radius of r = 110± The field of view is 11×25 mm2, corresponding to a res-
i
0.05 mm, and the outer cylinder of r =120±0.05 mm. olutionof 300×1024pixels. Special carehas been taken
o
The gap between the cylinders is thus d = r − r = concerningthecalibrationprocedure,onwhichespecially
o i
10 mm, and the gap ratio is η = r /r = 0.917. The theevaluationofthe plane-normalazimuthalcomponent
i o
system is closed at both ends, with top and bottom lids hevaily relies. As a calibration target we use a thin
rotatingwith the outer cylinder. The lengthofthe inner polyester sheet with lithographically printed crosses on
cylinder is L=220 mm (axial aspect ratio is L/d=22). it, stably attached to a rotating and translating micro-
Both cylinders can rotate independently with the use of traverse. It is first put into the light sheet and traversed
two DC motors (Maxon, 250W). The motors are driven perpendicularly to it. Typically five calibration images
by a home-made regulation device, ensuring a rotation are taken with intervals of 0.5mm. The raw PIV-images
rate up to 10Hz, with an absolute precision of ±0.02 Hz are processed using Davis(cid:13)R 7.2 by Lavision [20]. They
3
1 III. PARAMETER SPACE
The two traditional parameters to describe the flow
0.8
are the inner (resp. outer) Reynolds numbers, Re =
i
s) (r ω d/ν) (resp. Re =(r ω d/ν)), with the inner (resp.
s i i o o o
e
ml 0.6 outer) cylinder rotating at rotation rates ωi (resp. ωo),
di and ν the kinematic viscosity.
y ( We choose to use the set of parameters defined by
cit 0.4
o Dubrulle et al. [21]: a shear Reynolds number Re and
el S
V a “Rotation number” Ro:
0.2
2|ηRe −Re |
o i
Re =
0 S 1+η
0 .2 .4 .6 .8 1 (1)
(r−r)/d Rei+Reo
i Ro=(1−η) .
ηRe −Re
FIG. 3: Dimensionless azimuthal velocity profile (w/(r ω ) o i
i i
vs. (r−r )/d) for Ro=Ro ,at Re =90 (see section IIIfor
i i S With this choice, Re is based on the laminar shear
the definition of the parameters). Solid line: measured mean S
rateS: Re =Sd2/ν. Forinstancewitha20Hzvelocity
azimuthal velocity. Dotted line: theoretical profile. Dashed S
line: fit of the form w = ar+b/r. The radial component u difference in counter-rotation, the shear rate is around
which should be zero is also shown as a thin solid line. 1400 s−1 and ReS ≃ 1.4 × 105 for water at 20oC. A
constantshearReynoldsnumber correspondsto aline of
slopeηinthe{Re ; Re }coordinatesystem(seeFig.4).
o i
The Rotationnumber Ro compares the meanrotation
to the shear and is the inverse of a Rossby number. Its
are first mapped to world coordinates, then they are fil- sign defines cyclonic (Ro > 0) or anti-cyclonic (Ro < 0)
tered with a min-max filter, then PIV processed using flows. The Rotation number is zero in case of perfect
a multi-pass algorithm, with a last interrogation area of counter-rotation (r ω = −r ω ). Two other relevant
i i o o
32×32pixelswith50%overlap,andnormalisedusingme- values ofthe Rotationnumber areRo =η−1≃−0.083
i
dian filtering as post-processing. Then the three compo- and Ro = (1−η)/η ≃ 0.091 for respectively inner and
o
nent are reconstructed from the two camera views. The outer cylinder rotating alone. Finally, a further choice
mapping function is a third-order polynomial, and the that we made in our experiment was the value of η =
interpolations are bilinear. The PIV data acquisition is r /r , which we have chosen as relatively close to unity,
i o
triggered with the outer cylinder when it rotates, in or- i.e. η = 110/120 ≃ 0.91, which is considered a narrow-
der to take the pictures at the same angular position as gap, and is the most common in reported experiments,
used during the calibration. such as [8, 9, 22, 23], though a value as low as 0.128 is
described as well [4]. A high η, i.e. (1 − η) ≪ 1, is
special in the sense that for η →1 a plane Couette flow
To check the reliability of the stereoscopic velocity
measurement method, we performed a measurement for with background rotation; at high η, the flow is linearly
a laminar flow when only the inner cylinder rotates at unstable for −1<Ro<Roo [5, 21, 24].
a Reynolds number as low as Re = 90, using a 86% Inthepresentstudyweexperimentallyexploreregions
S
glycerol-water mixture. In that case, the analytical ve- of the parameter space that, to our knowledge, have not
locity field is known: the radial and axial velocities are been reported before. We present in Fig. 4 the parame-
zero,andtheazimuthalvelocityw shouldbeaxisymmet- terspacein{Reo ; Rei}coordinateswithasketchofthe
ric with no axial dependance, and a radial profile in the flowstatesidentifiedbyAnderecketal. [9],andtheloca-
form w(r) =Ωr η(r /r−r/r )/(1−η2) [8]. The results tion of the data discussed in the present paper. One can
i o o
are plotted in Fig. 3. The measured profile (solid line) notice that the present range of Reynolds numbers is far
hardlydiffersfromthe theoreticalprofile(dottedline)in beyondthatofAndereck,andthatwiththePIV-datawe
the bulk of the flow (0.1 . (r−r )/d . 0.7). The dis- mainly explore the zone between perfect counterrotation
i
crepancyishoweverquite strongclosetotheoutercylin- and only the inner cylinder rotating.
der ((r −r )/d = 1). The in-plane components which
i
shouldbezerodonotexceed1%oftheinnercylinderve-
locity everywhere. In conclusion, the measurements are IV. STUDY OF THREE PARTICULAR
verysatisfyingin the bulk. Further improvementsto the ROTATION NUMBERS
technique have been made since this first PIV test, in
particular a new outer cylinder of improved roundness, In the experiments reported in this section, we main-
and the measurements performed in water for turbulent tain the Rotation number at constant values and vary
cases are reliable in the range (0.1.(r−r )/d.0.85). the shear Reynolds number. We compare three partic-
i
4
40000
ei
R
Ro<−0.083
20000
Ro=0
10000
0<Ro<0.091
5000
0
−20000 −10000 −5000 Re 0 5000
o
FIG. 4: Parameter space in {Re ; Re } coordinates. The vertical axis Re = 0 corresponds to Ro = Ro = −0.083,
o i o i
and has been widely studied [12, 13, 16, 17]. The horizontal axis Re = 0 corresponds to Ro = Ro = 0.091. The line
i o
Re =−Re correspondstocounter-rotation, i.e. Ro=Ro =0. ThePIVdatataken at aconstant shearReynoldsnumberof
i o c
Re =1.4×104areplottedwith(◦). TorquedatawithvaryingRoatconstantshearforvariousRe rangingfromRe =3×103
S S S
to Re = 4.7×104 are plotted as blue lines. We also plot the states identified at much lower Re by Andereck et al. [9] as
S S
color patches: red corresponds to laminar Couette flow, green to“spiral turbulence”,grey to“featureless turbulence”andblue
to an “unexplored”zone.
ular Rotation numbers. Roi, Roc and Roo, correspond- −1 2
10
ing to rotation of the inner cylinder only, exact counter-
rotationand rotationof the outer cylinder rotating only,
1.5
respectively. In section A we report torque scaling mea- α
surements for a wide range of Reynolds numbers —from
1
base laminarflowto highly turbulentflows—andinsec-
−2
tion B we present typical velocity profiles in turbulent 10 1 2 3 4 5 6
10 10 10 10 10 10
conditions. Re
Cf
A. Torque scaling measurements
−3
10
We present in Fig. 5 the friction factor c =
f
2 2 2
T/(2πρr LU ) ∝ G/Re , with U = Sd and G =
i
T/(ρLν2), as a function of Re for the three Rotation
S
numbers. A common definition for the scaling exponent 1 2 3 4 5 6
10 10 10 10 10 10
α of the dimensionless torque is based on G: G ∝ Reα. Re
S
We keep this definition and present the local exponent FIG. 5: Friction factor c vs. Re for Ro =η−1 (black ◦),
f S i
α in the inset in Fig. 5. We compute α by means of a Ro =0(blue(cid:3))andRo =(1−η)/η (red⋄). Relativeerror
c o
logarithmic derivative, α=2+dlog(cf)/dlog(ReS). on ReS: ±5%, absolute error on torque: ±0.01 Nm. Inset:
At low Re, the three curves collapse on a Re−1 curve. local exponent α such that Cf ∝ ReSα−2, computed as 2+
This characterizes the laminar regime where the torque dlog(Cf)/dlog(ReS),forRoi=η−1(black),Roc =0(blue)
is proportional to the shear rate on which the Reynolds and Roo = (1−η)/η (red). Solid green line: Lewis’ data,
(Ref. [13] eq. 3), for Ro and η = 0.724. Solid magenta line:
number is based. i
Racina’s data (Ref. [23], eq. 10). Solid black line: laminar
For Ro = η−1, one can notice a transition to a dif-
i friction factor c =1/(ηRe).
ferent regime at Re ≃140 (the theoretical threshold is f
ci
computed as Re = 150 [24]). This corresponds to the
linear instability of the basic flow, leading in this case to
the growth of laminar Taylor vortices. The friction fac-
tor is then supposed to scale as c ∝ Re−1/2 (α = 3/2),
f
5
which is the case here (see inset in Fig. 5). For exact
−0.2
counter-rotation(Ro =0),thefirstinstabilitythreshold
c
is Re ≃400. This is somewhat lower than the theoret-
cc
ical prediction Re =515 [24], which is probably due to
cc
2
our finite aspect-ratio. Finally, the Taylor-Couette flow
with only the outer cylinder rotating (Ro = (1−η)/η) −0.3
o
is linearly stable whatever Re. We observe the experi-
mental flow to be still laminar up to high Re; then in a 1.5
rather short range of Re-numbers, the flow transits to a
turbulent state at 4000.Reto ≃5000. z/d −0.4
Further increase of the shear Reynolds number also
1
increases the local exponent (see inset in Fig. 5). For
Ro =η−1, it gradually rises from α≃1.5 at Re≃200
i
to α ≃ 1.8 at Re ≃ 105. The order of magnitude of
−0.5
these values agree with the results of Lewis et al [13], 0.5
though a direct comparison is difficult, owing to the dif-
ferent gap ratios of the experiments. The local expo-
nent is supposed to approach a value of 2 for increas-
0
ing gap ratio. Dubrulle & Hersant [10] attribute the in- −0.6
0 0.5 1
crease of α to logarithmic corrections,whereas Eckhardt
(r−r)/d
et al [11] attribute the increase of α to a balance be- i
tween a boundary-layer/hairpin contribution (scaling as FIG. 6: Secondary flow for Ro = Roi at Re = 1.4×104.
∝ Re3/2) and a bulk contribution (scaling as ∝ Re2). Arrows indicate radial and axial velocity, color indicates az-
imuthalvelocity (normalized to innerwall velocity).
The case of perfect counter-rotation shows a plateau at
α ≃ 1.5 and a sharp increase of the local exponent to
time-averagecoherentstructures,ratherthanbythecor-
α ≃ 1.75 at Re ≃ 3200, possibly tracing back to a
tc related fluctuations as in regular shear flow.
secondary transition. The local exponent then seems to
increase gradually. Finally, for outer cylinder rotating
1
alone(Ro )the transitionis verysharpandthe localex-
o
ponent is already around α = 1.77 at Re & 5000. Note
0.8
thatthedimensionalvaluesofthetorqueatRo arevery
o
small and difficult to measure accurately, and that these 0.6
may become smaller than the contributions by the two
Ro
Ka´rm´anlayers(end-effects) that we simply take into ac- 0.4 o
s)
count by dividing by 2 as described in § II. One can fi- s
e 0.2
nallynoticethatatthesameshearReynoldsnumber,for ml
Re≥104 thelocalexponentsforthethreerotationnum- y (di 0
bers are equal within ±0.1 and that the torque with the cit Ro
inner cylinder rotating only is greaterthan the torque in elo −0.2 c
V
counter-rotation,thelatterbeinggreaterthanthetorque
−0.4
for only the outer cylinder rotating.
Ro
−0.6 i
−0.8
B. Velocity profiles at a high shear-Reynolds
number −1
0 0.2 0.4 0.6 0.8 1
(r−r)/d
The presence of vortex-like structures at high shear- i
Reynolds number (Re & 104) in turbulent Taylor- FIG. 7: Profiles of the mean azimuthal velocity component
S
for three Rotation numbers corresponding to only the inner
Couetteflowwiththeinnercylinderrotatingaloneiscon-
cylinderrotating(×,black),perfectcounterrotation(◦,blue)
firmedin ourexperiment throughstereoscopicPIVmea-
and only the outer cylinder rotating ⋆, red), at Re = 1.4×
surements [25]. As shown in Fig. 6, the time-averaged 104. Thin line (·, black) : axial velocity v (for Ro = Ro ),
flow shows a strong secondary mean flow in the form i
averagedoverhalfaperiod. Thevelocitiesarepresentedina
of counter-rotating vortices, and their role in advecting
dimensionless form : w/(Sd)with Sd=2r (ω −ω )/(1+η).
i o i
angularmomentum(asvisibleinthecolouringbytheaz-
imuthalvelocity)isclearlyvisibleaswell. Theazimuthal We then measured the counter rotating flow, at the
velocity profile averaged over both time and axial posi- sameRe . The measurementsaretriggeredonthe outer
S
tion, w,asshowninFig.7,isalmostflat,indicatingthat cylinder position, and are averaged over 500 images. In
the transportofangularmomentumisdue mainlytothe the counter rotating case, for this large gap ratio and at
6
this value of the shear-Reynolds number, the instanta- cylinder rotating alone (Ro ) is approximately 50% of
o
neous velocity field is really desorganised and does not c (Ro ). These values compare well with the few avail-
f i
contain obvious structures like Taylor-vortices, in con- able data, compiled by Dubrulle et al [21]. The curve
trast with other situations [26]. No peaks are present showsaplateauofconstanttorqueespeciallyatthelarger
in the time spectra, and there is no axial-dependency Re from Ro=−0.2, i.e. when both cylinders rotate in
S
of the time-averaged velocity field. We thus average in thesamedirectionwiththeinnercylinderrotatingfaster
the axial direction the different radial profiles; the az- than the outer cylinder, to Ro ≃ −0.035, i.e. with a
imuthal component w is presented in Fig. 7 as well. In smallamountofcounter-rotationwiththe inner cylinder
thebulkitislow,i.e. itsmagnitudeisbelow0.1between still rotating faster than the outer cylinder. The torque
0.15 . (r−r )/d . 0.85 that is 75% of the gap width. then monotonically decreases when increasing the angu-
i
The two other components are zero within 0.002. lar speed of the outer cylinder, with an inflexion point
We finally address the outer cylinder rotating alone, close or equal to Ro , It is observed that the transition
c
again at the same Re . These measurements are done is continuous and smooth everywhere, and without hys-
S
much in the same way as the counter-rotating ones, i.e. teresis.
again the PIV system is triggered by the outer cylin- Wenowaddressthequestionofthetransitionbetween
der. As in the counter-rotating flow, this flow does not the different torque regimes by considering the changes
show any large scale structures. The gradient in the av- observed in the mean flow. To extract quantitative data
erage azimuthal velocity, again shown in Fig. 7, is much fromthe PIVmeasurements, we use the following model
steeper than in the counter-rotating case, which can be for the stream function Ψ of the secondary flow:
attributed to the much lower turbulence, as it also man-
ifests itself in the low c value for Ro .
f o π(r−r )
i
Ψ=sin ×
(cid:18) d (cid:19)
(2)
V. INFLUENCE OF ROTATION ON THE π(z−z0) 3π(z−z0)
EMERGENCE AND STRUCTURE OF THE ×(cid:20)A1 sin(cid:18) ℓ (cid:19)+A3 sin(cid:18) ℓ ](cid:19)(cid:21)
TURBULENT TAYLOR VORTICES
with as free parameters A1,A3,ℓ and z0. This model
comprises of a flow that fulfills the kinematic boundary
conditionattheinnerandouterwall,r ,r +d,andinbe-
i i
tween forms in the axial direction alternating rolls, with
2.2
Ro Ro Ro a roll height of ℓ. In this model, the maximum radial
i c o
2.0
velocity is formed by the two amplitudes and given by
1.8
1.6 ur,Max = (∂Ψ/∂z)Max = π(A1/ℓ + 3A3/ℓ). It is im-
plicitly assumed that the flow is developed sufficiently
Cf 1.4
× 1.2 to restore the axisymmetry, which is checked a posteri-
30 1.0 ori. Our fitting model comprises of a sinusoidal (funda-
1
mental) mode, and its first symmetric harmonic (third
0.8
0.6 mode), the latterwhichappearstoconsiderablyimprove
0.4 the matching between the model and the actual average
0.2 velocity fields, especially close to Roi (see Fig. 10).
0 We first discuss the case Ro = Roi. A sequence of
−0.1 −0.05 0 0.05 0.1 4,000 PIV images at a data rate of 3.7 Hz is taken, and
Ro 20 consecutive PIV images, i.e. approximately 11 cylin-
FIG. 8: The friction factor cf as a fucntion of Ro at various der revolutions, are sufficient to obtain a reliable esti-
constant shear Reynolds numbers: (blue) (cid:3) Re= 1.1×104, mate of the mean flow [17]. It is known that for the
(red) ⋄ Re = 1.4×104, (green) ◦ Re = 1.7×104, (black)
firsttransitiontheobservedflowstatecandependonthe
⋆ Re = 2.9×104, (magenta) × Re = 3.6×104, (cyan) ▽
Re=4.7×104. initial conditions [8]. When starting the inner cylinder
fromrestandacceleratingitto 2 Hz in 20s, the vortices
To characterize the transition between the three flow grow very fast, reach a value with a velocity amplitude
regimes, we first consider the global torque measure- of 0.08ms−1, and then decay to become stabilized at a
ments. We plot in Fig. 8 the friction factor or dimen- value around 0.074ms−1 after 400 seconds. Transients
sionless torque as a function of Rotation number Ro are thus also very long in turbulent Taylor-vortex flows.
at six different shear Reynolds numbers, Re , as indi- Forsloweracceleration,the vorticesthat appearfirstare
S
cated in Fig. 4. We show three series centered around muchweakerandhavealargerlengthscale,beforereach-
Re =1.4×104, and three around Re =3.8×104. As ing the same final state. The final length scale ℓ of the
S S
already seen in Fig. 5, the friction factor reduces with vorticesfor Ro is about1.2times the gapwidth, consis-
i
increasing Re . More interesting is the behavior of c tent with data from Bilson et al. [16].
S f
with Ro: the torque in counter-rotation (Ro ) is ap- In a subsequent measurement we start from Ro=Ro
c i
proximately 80% of c (Ro ), and the torque with outer and vary the rotationnumber in small increments, while
f i
7
0.08
Ro Ro
i 0.03 c
0.07
0.02
2
0.06
0.01
e
ud 0.05 0
plit −0.03 −0.02 −0.01 0 0.01
1.5 m 0.04
A
el. 0.03
d V
z/ 0.02
1
0.01
0
−0.1 −0.05 0 0.05
0.5 Ro
FIG.10: Secondaryflowamplitudevs. rotationnumber(Ro)
at constant shear rate. Black (⋄): model with fundamental
modeonly,andred(◦): completemodelwiththirdharmonic.
0
Solid line is a fit of the form A = a(−Ro)1/2. Inset: zoom
0 0.5 1
close to counter-rotation, combined with results from a con-
(r−r)/d
i tinuoustransient experiment (see text).
FIG. 9: Overlay of measured time-average velocity field at
Ro = Ro (in red), and the best-fit model velocity field (in fromRo=0.004to Ro=–0.0250in 3000s,alwayskeeping
i
black). The large third harmonic makes the radial flow be- the Reynolds number constant at Re =1.4×104. The
S
ing concentrated in narrow bands, rather than sinusoidally amplitude of the mean secondary flow, computed on se-
distributed. quences of 20 images, is plotted in the inset of Fig. 10.
The curve follows the static experiments (given by the
maintaining a constant shear rate. We allow the system
singlepoints),butsomedownwardpeakscanbenoticed.
to spend 20 minutes in each state before acquiring PIV
Wecheckedthatthesearenottheresultofafittingerror,
data. We verify that the fit parameters are stationary,
andindeedcorrespondtotheoccasionaldisappearanceof
andcomputethemusingtheaverageofthefullPIVdata
the vortices. Still, the measurements are done at a fixed
setateachRo. TheresultsareplottedinFig.10. Please
position in space. Though the very long time-averaged
note that Ro has been varied both with increasing and
series leads to well-established stationary axisymmetric
decreasing values, to check for a possible hysteresis. All
states, it is possible that the instantaneous whole flow
pointsfallonasinglecurve;thetransitionissmoothand
consists of different regions. Further investigation in-
without hysteresis. For Ro ≥ 0, the fitted modes have
cluding time-resolved single-point measurements or flow
zero or negligible amplitudes, since there are no struc-
visualizations need to be done to verify this possibility.
tures in the time-average field [25]. One can notice that
as soon as Ro < 0, i.e. as soon as the inner cylinder
wall starts to rotate faster than the outer cylinder wall,
VI. CONCLUSION
vortices begin to grow. We plot in Fig. 10 the veloc-
ity amplitude associated with the simple model (single
mode ⋄), and with the complete model (modes 1 and 3, The net system rotation as expressed in the Rota-
◦). ClosetoRo=0,thetwomodelscoincide: A3 ≃0and tion number Ro obviously has strong effects on the
the mean secondary flow is well described by pure sinu- torque scaling. Whereas the local exponent evolves in
soidal structures. For Ro . −0.04, the vortices start to a smooth way for inner cylinder rotating alone, the
haveelongatedshapes,withlargecoresandsmallregions counter-rotatingcaseexhibitstwosharptransitions,from
of large radial motions in between adjacent vortices; the α = 1 to α ≃ 1.5 and then to α ≃ 1.75. We also notice
third mode is then necessary to adequately describe the that the second transition for counter-rotation Re is
tc
secondary flow. The first mode becomes saturated (i.e. close to the threshold Re of turbulence onset for outer
to
it does not grow in magnitude) in this region. Finally, cylinder rotating alone.
we give in Fig. 10 a fit ofthe amplitudes close to Ro=0 The rotation number Ro is thus a secondary control
of the form: A= a(−Ro)1/2. The velocity amplitude of parameter. Itisverytemptingtousetheclassicalformal-
thevortexbehaveslikethesquarerootofthe distanceto ism of bifurcations and instabilities to study the transi-
Ro = 0, a situation reminiscent to a classical supercrit- tionbetweenfeaturelessturbulenceandturbulentTaylor-
ical bifurcation, with A as order parameter, and Ro as vortex flow at constant Re , which seems to be super-
S
control parameter. critical;thethresholdfortheonsetofcoherentstructures
We alsoperformeda continuous transient experiment, inthe meanflowis Ro . Foranticyclonicflows(Ro<0),
c
in which we varied the rotation number quasi-statically the transportisdominatedby largescalecoherentstruc-
8
tures, whereas for cyclonic flows (Ro > 0), it is domi- for PIV does qualitatively not change, these measure-
nated by correlated fluctuations reminiscent to those in ments suggests that the large scale vortices are not only
plane Couette flow. persistent in the flow at higher Re , but that they also
S
InaconsiderablerangeofRe ,counter-rotation(Ro ) dominate the dynamics of the flow. An answer to the
S c
is also close or equal to an inflexion point in the torque persistence may be obtained from either more detailed
curve; this may be relatedto the cross-overpoint, where analysis of instantaneous velocity data or from torque
the role of the correlated fluctuations is taken over by scaling measurements at still higher Reynolds numbers
the large scale vortical structures. The mean azimuthal inTaylor-Couettesystemssuchasareunderdevelopment
velocity profiles show there is only a marginal viscous [18].
contribution for Ro ≤ 0 but of order 10% at Ro [25]. WeareparticularlyindebtedtoJ.R.Bodde,C.Gerrit-
o
Theroleofturbulentvs.large-scaletransport(ofangular sen and W. Tax for building up and piloting the experi-
momentum) should be further investigated from (exist- ment. Wehavebenefitedofveryfruitfuldiscussionswith
ing)numericalorPIVvelocitydata. Sincetorquescaling A.Chiffaudel,F.Daviaud,B.Dubrulle,B.Eckhardtand
withRoas measuredatmuchhigher Re thatthatused D. Lohse.
S
[1] P.J.Holmes,J.L.Lumley,andG.Berkooz. Turbulence, number Couette-Taylor flow. Phys. Rev. E, 59:5457,
Coherent Structures, Dynamical Systems and Symmetry. 1999.
Cambridge University Press, 1996. [14] F. Hersant, B. Dubrulle, and J.-M. Hur´e. Turbulence in
[2] R. Monchaux et al. Generation of magnetic field by dy- circumstellar disks. A&A, 429:531, 2005.
namo action in a turbulent flow of liquid sodium. Phys. [15] S.T.WereleyandR.M.Lueptow. Spatio-temporalchar-
Rev. Lett., 98:044502, 2007. acterofnon-wavyandwavyTaylor-Couetteflow.J.Fluid
[3] F. Ravelet, L. Mari´e, A. Chiffaudel, and F. Daviaud. Mech., 364:59, 1998.
Multistability and Memory Effect in aHighly Turbulent [16] M. Bilson and K. Bremhorst. Direct numerical simula-
Flow: Experimental Evidence for a Global Bifurcation. tion of turbulent Taylor-Couette flow. J. Fluid Mech.,
Phys. Rev. Lett., 93:164501, 2004. 579:227, 2007.
[4] N. Mujica and D. P. Lathrop. Hysteretic gravity-wave [17] S.Dong.DirectnumericalsimulationofturbulentTaylor-
bifurcation in a highly turbulent swirling flow. J. Fluid Couette flow. J. Fluid Mech., 587:373, 2007.
Mech., 551:49, 2006. [18] D.Lohse.TheTwenteTaylorCouetteFacility.Presented
[5] D.J. Tritton. Stabilization and destabilization of tur- at12thEuropeanTurbulenceConference,September7-10
bulent shear flow in a rotating fluid. J. Fluid Mech., 2009, Marburg, Germany, 2009.
241:503–523, 1992. [19] A. K. Prasad. Stereoscopic particle image velocimetry.
[6] G.Brethouwer. Theeffectofrotationonrapidlysheared Exp. Fluids, 29:103, 2000.
homogeneous turbulence and passive scalar transport. [20] Davis(cid:13)R 7.2 , byLavision GmbH. 2006.
linear theory and direct numerical simulation. J. Fluid [21] B. Dubrulle et al. Stability and turbulent transport in
Mech., 542:305–342, 2005. Taylor-Couette flow from analysis of experimental data.
[7] M. Couette. Etude sur le frottement des liquids. Ann. Phys. Fluids, 17:095103, 2005.
Chim. Phys., 21:433, 1890. [22] F. Wendt. Turbulente Str¨omungen zwischen zwei
[8] D. Coles. Transition in circular Couette flow. J. Fluid rotierenden konaxialen Zylindern. Arch. Appl. Mech.,
Mech., 21:385, 1965. 4:577–595, 1933.
[9] C. D. Andereck, S. S. Liu, and H. L. Swinney. Flow [23] A. Racina and M. Kind. Specific power input and lo-
regimesinacircularCouettesystemwith independently cal micromixing times in turbulent Taylor-Couette flow.
rotating cylinders. J. Fluid Mech., 164:155, 1986. Exp. Fluids, 41:513, 2006.
[10] B. Dubrulle and F. Hersant. Momentum transport and [24] A. Esser and S. Grossmann. Analytic expression for
torque scaling in Taylor-Couette flow from an analogy Taylor-Couettestability boundary. Phys. Fluids,8:1814,
with turbulent convection. Euro. Phys. J. B, 26:379, 1996.
2002. [25] F. Ravelet, R. Delfos, and J. Westerweel. Ex-
[11] B. Eckhardt, S. Grossmann, and D. Lohse. Torque scal- perimental studies of turbulent Taylor-Couette
ingin Taylor-Couetteflowbetween independentlyrotat- flows. In Proc. 5th Int. Symp. on Turbulence and
ing cylinders. J. Fluid Mech., 581:221, 2007. Shear Flow Phenomena, Munich, page 1211, 2007.
[12] D.P.Lathrop,J.Fineberg,andH.L.Swinney.Transition http://arxiv.org/abs/0707.1414.
toshear-driventurbulenceinCouette-Taylorflow. Phys. [26] L. Wang, M. G. Olsen, and R. D. Vigil. Reappearance
Rev. A, 46:6390, 1992. of azimuthal waves in turbulent Taylor-Couette flow at
[13] G. S. Lewis and Harry L. Swinney. Velocity struc- large aspect ratio. Chem. Eng. Sci., 60:5555, 2005.
turefunctions,scaling,andtransitionsinhigh-Reynolds-