Supercritical transition to turbulence in an inertially-driven von Karman closed flow

0.9 MB·English

by  Florent Ravelet

Supercritical transition to turbulence in an inertially-driven von K´arm´an
closed flow.
Florent Ravelet,1,2 Arnaud Chiffaudel,1 and Franc¸ois Daviaud1
1Service de Physique de l’Etat Condens´e, Direction des Sciences de la Mati`ere, CEA-Saclay, CNRS URA 2464,
91191 Gif-sur-Yvette cedex, France
2Present address: Laboratory for Aero and Hydrodynamics, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands∗
(Dated: accepted in JFM, January 18, 2008)
We study the transition from laminar flow to fullydeveloped turbulence for an inertially-driven
8
von Ka´rma´n flow between two counter-rotating large impellers fitted with curved blades over a
0 widerangeofReynoldsnumber(102−106). Thetransitionisdrivenbythedestabilisationofthe
0 azimuthalshear-layer,i.e.,Kelvin–Helmholtzinstabilitywhichexhibitstravelling/driftingwaves,
2 modulated travelling waves and chaos below the emergence of a turbulent spectrum. A local
quantity—theenergyofthevelocityfluctuations atagivenpoint—andaglobalquantity—the
n
appliedtorque—areusedtomonitorthedynamics. ThelocalquantitydefinesacriticalReynolds
a
J number Rec fortheonset oftime-dependence inthe flow,andanupper threshold/crossover Ret
for the saturation of the energy cascade. The dimensionless drag coefficient, i.e., the turbulent
2 dissipation,reachesaplateauabovethisfiniteRet,asexpectedfora“Kolmogorov”-liketurbulence
2 for Re → ∞. Our observations suggest that the transition to turbulence in this closed flow
is globally supercritical: the energy of the velocity fluctuations can be considered as an order
] parameter characterizing the dynamics from the first laminar time-dependence up to the fully
n
developed turbulence. Spectral analysis in temporal domain moreover reveals that almost all of
y
the fluctuations energy is stored in time-scales one or two orders of magnitude slower than the
d
time-scalebasedonimpellerfrequency.
-
u
l
f
PACSnumbers: 47.20.-k,47.20.Ft,47.27.Cn,47.27.N-
.
s
c
i
s I. INTRODUCTION which it takes place (for instance wakes, jets or closed
y
flows). At finite Re, most turbulent flows could still
h
Hydrodynamic turbulence is a key feature for many keep in average some geometrical or topological prop-
p
[ commonproblems(Lesieur,1990;Tennekes and Lumley, erties of the laminar flow (for example the presence of a
1972). Inafewidealcases,exactsolutionsoftheNavier– B´enard–von K´arm´an street in the wake of a bluff body
1 Stokes equations are available, based on severalassump- whatever Re), which could still influence its statistical
v
tionssuchasauto-similarity,stationariness,orsymmetry properties (La Porta et al., 2001; Ouellette et al., 2006;
7
(for a collection of examples, see Schlichting, 1979). Un- Zocchi et al., 1994).
5
3 fortunately,theyareoftenirrelevantinpractice,because Furthermore,wehaverecentlyshowninavonK´arm´an
3 they are unstable. Two of the simplest examples are flowthataturbulentflowcanexhibitmultistability,first
. the centrifugalinstability of the Taylor–Couetteflowbe- orderbifurcations andcanevenkeeptracesofits history
1
0 tweentwoconcentriccylinders,andtheRayleigh–B´enard atveryhighReynoldsnumber(Ravelet et al.,2004). The
8 convectionbetweentwodifferentiallyheatedplates: once observationofthis turbulentbifurcationleadustostudy
0 the amount of angular momentum or heat is too impor- the transition from the laminar state to turbulence in
: tanttobecarriedbymoleculardiffusion,amoreefficient this inertially driven closed flow.
v
i convective transport arises. Increasing further the con-
X
trolparameterinthesetwoexamples,secondarybifurca-
r tionsoccur,leadingrapidlytotemporalchaos,and/orto A. Overview of the von K´arm´an swirling flow
a
spatio-temporal chaos, then to turbulence.
Several approaches have been carried in parallel con- 1. Instabilities of the von K´arm´an swirling flow between flat
cerningdevelopedturbulence,focusedonstatisticalprop- disks
erties of flow quantities at small scales (Frisch, 1995) or
takingintoaccountthepersistenceofcoherentstructures The disk flow is an example where exact Navier–
in a more deterministic point of view (Lesieur, 1990; Stokes solutions are available. The original problem of
Tennekes and Lumley, 1972). One of the major diffi- the flow of a viscous fluid over an infinite rotating flat
culty concerning a self-consistent statistical treatment disk has been considered by von K´arm´an (1921). Ex-
of turbulence is indeed that it depends on the flow in perimentally, the problem of an infinite disk in an infi-
nite medium is difficult to address. Addition of a sec-
ond coaxial disk has been proposed by Batchelor (1951)
and Stewartson (1953). A cylindrical housing to the
∗Electronicaddress: fl[e-mail] flow can also be added. Instabilities and transitions
2
havebeenextensivelystudiedinthissystem(forinstance f > 0
in Escudier, 1984; Gauthier et al., 1999; Gelfgat et al.,
1996; Harriott and Brown, 1984; Mellor et al., 1968;
+
Nore et al., 2005, 2004, 2003; Schouveiler et al., 2001;
Sørensen and Christensen, 1995; Spohn et al., 1998).
The basic principle of this flow is the following: a layer
of fluid is carriednear the disk by viscous friction and is −
thrownoutwardsbycentrifugation. Byincompressibility
oftheflow,fluidispumpedtowardthecentreofthedisk. z
SincethereviewofZandbergen and Dijkstra(1987),this
H O r
family of flow is called “von K´arm´an swirling flow”. In
allcases, it deals with the flow betweensmooth disks, at
low-Reynoldsnumbers,enclosedornotinto a cylindrical Rc R
container. (b)
2. The “French washing machine”: an inertially-driven, highly
turbulentvon K´arm´an swirling flow.
Experimentally, the so-called “French washing-
f > 0
machine” has been a basis for extensive studies
of very high-Reynolds number turbulence in the (a)
last decade (Bourgoin et al., 2002; Cadot et al.,
1997, 1995; Douady et al., 1991; Fauve et al., FIG. 1 (a) Sketch of the experiment. The flow volume be-
1993; La Porta et al., 2001; Labb´e et al., 1996; tween the impellers is of height H = 1.8 Rc. (b) Impellers
Leprovostet al., 2004; Mari´e and Daviaud, 2004; used in this article. The disks radius is R = 0.925 Rc and
theyarefittedwith16curvedblades: thetwodifferentdirec-
Moisy et al., 2001; Ravelet et al., 2004; Tabeling et al.,
tions of rotation defined here are not equivalent. This model
1996; Titon and Cadot, 2003; Zocchi et al., 1994). To
of impellers has been used in the VKS1 sodium experiment
reach a Kolmogorov regime in these studies, a von
(Bourgoin et al., 2002) and is called TM60.
K´arm´an flow is inertially-driven between two disks
fitted with blades, at a very high-Reynolds number
(105 .Re.107). Due to the inertial stirring, very high
turbulence levelscan be reached,with fluctuations up to
the measurement techniques in §II. The main data pre-
50% of the blades velocity, as we shall see in this article.
sented in this article are gathered by driving our experi-
Most of the inertially-driven von K´arm´an setups
mentcontinuouslyfromlaminarto turbulentregimesfor
studied in the past dealt with straight blades. Von
this peculiar direction of rotation, covering a wide range
K´arm´anflowswithcurved-blades-impellerswerefirstde-
of Reynolds numbers. In §III and §IV we characterise
signed by the VKS-team for dynamo action in liquid
the basic flow and describe the transition from the lam-
sodium (Bourgoin et al., 2002; Monchaux et al., 2007).
inar regime to turbulence through quasi-periodicity and
With curved blades, the directions of rotation are no
chaosandexplorethe constructionofthe temporalspec-
longer equivalent. One sign of the curvature —i.e.,
trum of velocity fluctuations. The continuity and global
with the convex face of the blades forward, direc-
supercriticality of the transition to turbulence is a main
tion (+)— has been shown to be the most favourable
result of this article.
to dynamo action (Mari´e et al., 2003; Monchaux et al.,
In §V we obtain complementary data by comparing
2007; Ravelet et al., 2005). The turbulent bifurcation
the twodifferent directions ofrotationandthe case with
(Ravelet et al., 2004) has been obtained with the con-
smooth disk. We show how inertial effects clearly dis-
cave face of the blades forward, direction (−). In this
criminate both directions of rotation at high-Reynolds
last work, we discussed about the respective role of the
numbers.
turbulent fluctuations and of the changes in the mean-
We then summarize and discuss the main results
flow with increasing the Reynolds number on the multi-
in §VI.
stability.
B. Outline of the present article II. EXPERIMENTAL SETUP
Our initial motivation to the present study was thus A. Dimensions, symmetries and control parameter
to get an overview of the transition to turbulence and
to check the range where multistability exists. We first The cylinder radius and height are, respectively, R =
c
describethe experimentalsetup,the fluidpropertiesand 100 mm and H = 500 mm. A sketch of the experiment
c
3
C µ at 15oC µ at 30oC ρ Rerange rotation frequency f. The integral Reynolds number Re
99% 1700 580 1260 50−2,000 is thus defined as Re=2πfRc2ν−1 with ν the kinematic
viscosity of the fluid.
93% 590 210 1240 130−5,600
As in previous works (Mari´e and Daviaud, 2004;
85% 140 60 1220 550−19,000
Ravelet et al., 2005, 2004), we use water at 20− 30oC
81% 90 41 1210 840−28,000
as working fluid to get Reynolds numbers in the range
74% 43 20 1190 1,800−56,000 6.3×104 . Re . 1.2×106. To decrease Re down to
0% 1.1 0.8 1000 570,000−1,200,000
laminar regimes, i.e., to a few tens, we need a fluid with
akinematic viscosityathousandtimes greaterthanthat
TABLEI Dynamicviscosityµ(10−3Pa.s)atvarioustemper-
atures, density ρ (kg.m−3) at 20oC and achievable Reynolds ovfiswcoastietry.iWs0e.9th5u×s1u0s−e399m%2.-sp−u1reatgl2y0coeCro(lHwohdicghmkainn,em19a4t7ic)
number range for various mass concentrations C of glycerol
andshouldbeabletostudytherange50.Re.900. To
in water.
coverawiderangeofReynoldsnumbersandmatchthese
two extreme ranges, we then use different mixes of glyc-
erol and water, at temperatures between 15 and 35oC.
canbefoundinfigure1(a). Weusebladeddiskstoensure The physical properties of these mixtures are given in
inertialstirring. The impellersconsistof185mmdiame- tableI,whereC isthemasspercentageofglycerolinthe
terstainless-steeldiskseachfittedwith16curvedblades, mixture. Solutions samples are controlled in a Couette
ofcurvatureradius50mmandofheighth=20mm(fig- viscometer.
ure1b). Thedistancebetweentheinnerfacesofthedisks Thetemperatureoftheworkingfluidismeasuredwith
is H = 180 mm, which defines a flow volume of aspect a platinum thermoresistance (Pt100) mounted on the
ratioH/Rc =1.8. Withthecurvedblades,thedirections cylinder wall ({r = 1 ; z = 0}). To control this tem-
ofrotationarenolongerequivalentandwecaneitherro- perature, thermoregulated water circulates in two heat
tate the impellers anticlockwise —with the convex face exchangers placed behind the impellers. Plexiglas disks
of the blades forward, direction (+) or clockwise (with canbemountedbetweentheimpellersandheatexchang-
the concave face of the blades forward, direction (−). ers to reduce the drag of the impellers-back-side flows.
Theimpellersaredrivenbytwoindependentbrushless They are at typically 50 mm from the impellers back
1.8kW motors,withspeedservoloopcontrol. Themax- sides. However,these disks reduce the thermal coupling:
imal torque they can reach is 11.5 N.m. The motor ro- theyareusedinturbulentwaterflowsandtakenawayat
tation frequencies {f1 ; f2} can be varied independently low Reynolds number.
in the range 1 ≤ f ≤ 15 Hz. Below 1 Hz, the speed
regulation is not efficient enough, and the dimensional
quantities are measured with insufficient accuracy. We
will consider for exact counter-rotating regimes f1 = f2 B. Experimental tools, dimensionless measured quantities
the imposed speed of the impellers f. and experimental errors
Theexperimentalsetupisthusaxisymmetricandsym-
metric towards rotations of π around any radial axis Several techniques have been used in parallel: flow
passing through the centre O (R -symmetry), and we visualisations with light sheets and air bubbles, torque
π
will consider here only R -symmetric mean solutions, measurements and velocity measurements.
π
though mean flows breaking this symmetry do exist for Flow visualisations are made in vertical planes illumi-
these impellers, at least at very high-Reynolds numbers natedbyapproximately2mmthicklightsheets. Welook
(Ravelet et al., 2004). A detailed study of the Reynolds attwodifferentpositions withrespecttothe flow: either
number dependence of the “global turbulent bifurca- the central meridian plane where the visualised compo-
tion”isoutofthescopeofthepresentarticleandwillbe nents are the radial and axial ones or in a plane almost
presented elsewhere. Also, since we drive the impellers tangenttothecylinderwallwherethe azimuthalcompo-
independently, there is always a tiny difference between nent dominates. Tiny air bubbles (less than 1 mm) are
f1 andf2 andtheRπ-symmetryofthe systemcannotbe used as tracing particles.
considered as exact. In the following, we will keep us- Torques are measured as an image of the current con-
ingthissymmetry—veryusefultodescribetheobserved sumption in the motors given by the servo drives and
patterns– but we will keep in mind that our system is have been calibrated by calorimetry. Brushless motors
only an approximation of a Rπ-symmetric system. The are known to generate electromagnetic noise, due to the
consequences on the dynamics will be analysed in the Pulse-Width-Modulation supply. We use armoured ca-
discussion (§VI.A). bles and three-phase sinusoidal output filters (Schaffner
In the following, all lengths will be expressed in units FN5010-8-99), and the motors are enclosed in Faraday
ofR . Wealsousecylindricalcoordinates{r; z}andthe cages, which enhances the quality of the measurements.
c
originisontheaxisofthecylinder,andequidistantfrom The minimal torques we measured are above 0.3 N.m,
the twoimpellers totakeadvantageofthe R -symmetry and we estimate the error in the measurements to be
π
(seefigure1a). Thetimeunitisdefinedwiththeimpeller ±0.1 N.m. The torques T will be presented in the di-
4
mensionless form: A. Basic state at very low Reynolds number
K =T (ρR5(2πf)2)−1 At very low-Reynolds number, the basic laminar flow
p c
respects the symmetries of the problem. It is stationary,
axisymmetric and R -symmetric. This state is stable at
Velocity fields are measured by Laser Doppler Ve- π
Re = 90, where we present a flow visualisation in fig-
locimetry (LDV). We use a single component DAN-
ure 2(a-b). In figure 2(a), the light sheet passes through
TEC device, with a He–Ne Flowlite Laser (wave length
the axis of the cylinder. The visualisedvelocities are the
632.8 nm) and a BSA57N20 Enhanced Burst Spectrum
radial and axial components. The poloidal part of the
Analyser. The geometry of the experiment allows us to
flow consists of two toric recirculation cells, with axial
measure in one point either the axial component V (t)
z pumping directed to the impellers.
or the azimuthal component V (t). Though the time-
θ
averagedvelocity field V is not a solution of the Navier- The flow is also made of two counter-rotating cells,
separatedby an azimuthal flat shear-layer,which can be
Stokes equations, it is a solenoidal vector field, and it is
seen in figure 2(b) where the light sheet is quasi-tangent
axisymmetric. We thus use the incompressibility condi-
tion ∇·V = 0 to compute the remaining radial compo- to the cylinder wall. Both the azimuthal and axial com-
ponent vanishes in the plane z = 0 which is consistent
nent V .
r
with the axisymmetry and the R -symmetry. This flat
The measurements of the time-averaged velocity field π
shear-layer is sketched in figure 2(e). A LDV velocity
areperformedona{r×z}=11×17grid,usingaweight-
field is presented in section V (figure 10c-d).
ing of velocities by the particles transit time, to get rid
ofvelocitybiasesasexplainedbyBuchhave et al.(1979).
This acquisition mode does not have a constant acqui-
B. First instability – stationary bifurcation
sition rate, so we use a different method for the acqui-
sition of well-sampled signals to perform temporal anal-
The first instability for this flow has been determined
ysis at single points. In this so-called dead-time mode,
by visualisation and occurs at Re=175±5 for both di-
we ensure an average data rate of approximately 5 kHz,
rectionsofrotation. Thebifurcationissupercritical,non-
and the Burst Spectrum Analyser takes one sample ev-
hysteretic, and leads to a stationary regime, with an az-
ery single millisecond such that the final data rate is
imuthal modulation of m=2 wave number. We present
1 kHz. For practical reasons, this method is well-suited
a visualisation of this secondary state in figure 2(c), at
for points close to the cylindrical wall, so we choose the
Re = 185. The axisymmetry is broken: one can see the
point {r = 0.9 ; z = 0} for the measurements in fig-
m=2modulationoftheshear-layer,alsosketchedinfig-
ures 3, 4 and 5. The signals are re-sampled at 300Hz by
ure 2(e). One can also note that R -symmetry is partly
a“SampleAndHold” algorithm(Buchhave et al.,1979). π
broken: thebifurcatedflowisR -symmetricwithrespect
Let us now consider the experimental error on the π
to two orthogonal radial axis only. This first instabil-
Reynolds number value. The speed servo-loop control
ity is very similar to the Kelvin–Helmholtz instability.
ensures a precision of 0.5% on f, and an absolute pre-
(Nore et al., 2003) made a proper theoretical extension
cision of ±0.002 on the relative difference of the im-
of the Kelvin–Helmholtz instability in a cylinder. Their
pellers speeds(f1−f2)/(f1+f2). The mainerroronthe
modelis basedonthe useoflocalshear-layerthicknesses
Reynolds number is thus a systematic error that comes
and Reynolds numbers to take into account the radial
from the estimation of the viscosity. As far as the vari-
variations in the cylindrical case.
ation of the viscosity with temperature is about 4% for
1oC and the variation with concentration is about 5% Weobservethism=2shear-layertorotateveryslowly
in a given direction with a period 7500f−1. This cor-
for 1% of mass concentration, we estimate the absolute
responds to a very low frequency, always smaller than
error on Re to ±10% (the temperature is known within
o the maximum measured dissymmetry of the speed servo
1 C). However, the experimental reproducibility of the
loop control between both independent motors (§II.B).
Reynolds number is much higher than ±10%. In the
This is probably the limit of the symmetry of our sys-
range 100 . Re . 500 we are able to impose Re within
tem, i.e., the pattern is at rest in the slowly rotating
±5.
frame where both frequencies are strictly equal (see dis-
cussion in §VI.A). For convenience, we will describe the
dynamics in this frame.
III. FROM ORDER TO TURBULENCE: DESCRIPTION The laminar m = 2 stationary shear-layer pattern
OF THE REGIMES is observed up to typically Re ≃ 300 where time-
dependence arises.
This section describes the evolution of the flow from To investigate the time-dependent regimes, we now
thelaminarregimetothefullydevelopedturbulence,i.e., also perform precise velocity measurements at a given
for 30 . Re . 1.2× 106. This wide-range study has pointintheshear-layer. Wemeasuretheazimuthalcom-
been carriedfor the negativesense of rotation(−) ofthe ponent v at {r = 0.9 ; z = 0}, using the dead-time
θ
propellers. acquisition mode (see §II).
5
Re < 175 m = 0
175 < Re < 330 m = 2, fixed
Re > 330 m = 2, rotating
(a) 0 π/2 π 3π/2 2π (e)
(b) (c) (d)
FIG. 2 Visualisation and schematics of the basic laminar flow for impellers rotating in direction (−). The lightning is made
with a vertical light sheet. Pictures are integrated over 1/25 s with a video camera, and small air bubbles are used as tracers.
PictureheightisH−2h=1.4R . LaminaraxisymmetricflowatRe=90,meridianview(a). Viewsinaplanenearthecylinder
c
wall at Re = 90 (b), Re = 185 (c) and Re = 345 (d). The development of the first m = 2 instabilities —steady undulation
(c) and rotating vortices (d)— is clearly visible on the shape of the shear-layer. We give sketches of the shear-layer for these
Reynoldsnumbersin (e).
Below,wedescribeandillustratethe observeddynam- spectrumwithjustatinypeak—1/1000oftheamplitude
icsandthebuilding-upofthechaoticandturbulentspec- measured at Re=330— at f =f and we have no data
a
tra. ThenextsectionIViscomplementary: wequantita- in-between to check an evolution.
tively characterizethe transitionsas much as we can, we
On the spectra, we observe the first harmonic, but
discuss the mechanisms and we finally propose a global
never shows the expected blade frequency 8f nor a mul-
supercritical view of the transition to turbulence.
tiple. So,itisnotclearifitcorrespondstothebasicfluid
instabilitymodeorjusttoasmallprecessingmodedueto
themisaligningoftheimpelleraxisoreventomechanical
C. From drifting patterns to chaos
vibrationstransmittedto the fluidthroughthe bearings.
Since the travelling-wave mode of the next paragraph is
We present time series of the velocity and power spec-
much stronger and richer in dynamics we will consider
traldensities atfive Reynolds numbers infigure 3: Re= that the signal at f =f is a “minor” phenomenon, i.e.,
a
330±5, Re=380±5, Re=399±5, Re=408±5 and
a perturbation of the steady m=2 mode.
Re=440±5.
Infigure3(a)themeanvelocityisnotzero,butaround
v =+0.17during the 600time units of acquisition,i.e.,
θ
during 600 disks rotations. This value of the velocity
1. Oscillation at impeller frequency
has no special meaning and depends on the phase be-
tween the fixed measurement point and the slowly drift-
The point at Re = 330±5 is the first point where a
ing shear-layer(§ III.B). The measurementpoint indeed
cleartemporaldynamicsisobserved: asharppeakinthe
stays on the same side of the shear-layer for this time-
spectrum (figure 3b) is present at the impeller rotation
series but, on much longer time scales, we measure the
frequencyf =f —emphasizedintheInsetoffigure3(a).
a m=2 shear-layerrotation typical period.
This oscillation exists for higher Re with the same small
amplitude: it is too fast to be explicitly visible on the Further observation on signal and spectrum of fig-
long time series of figure 3, but it is responsible for the ure 3a-b reveals some energy at low frequency around
large width of the signal line. f ≃f/30,correspondingtoslowlyrelaxingmodulations:
a
In comparison, a similar measurement performed at theslownessofthisrelaxationistheclearsignatureofthe
Re ≃ 260 reveals a flat signal with a very low flat- proximity of a critical point.
6
3
0.6 10
Re=330 (a) 102 Re=330 (b)
0.4 1
10
0.2 0
10
D −1
Vθ 0 S 10
0.2 P −2
10
−0.2 −3
10
−0.4 0.1 −4
220 225 230 10
−5
−0.6 10
0 100 200 300 400 500 600 −3 −2 −1 0 1 2
10 10 10 10 10 10
Time (f−1)
f /f
3 a
0.6 10
Re=380 (c) 102 (d)
0.4 1
10
0.2 0
10
D −1
Vθ 0 S 10
P −2
10
−0.2 −3
10
−0.4 −4
10
−5 Re=380
−0.6 10
0 100 200 300 400 500 600 700 800 −3 −2 −1 0 1 2
10 10 10 10 10 10
Time (f−1)
f /f
a
0.6 0.6
Re=399 (e) Re=407 (f)
0.4 0.4
0.2 0.2
Vθ 0 Vθ 0
−0.2 −0.2
−0.4 −0.4
−0.6 −0.6
0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800
Time (f−1) Time (f−1)
3
0.6 10
Re=440 (g) 102 (h)
0.4 1
10
0.2 0
10
D −1
Vθ 0 S 10
P −2
10
−0.2 −3
10
−0.4 −4
10
−5 Re=440
−0.6 10
0 200 400 600 800 1000 −3 −2 −1 0 1 2
10 10 10 10 10 10
Time (f−1)
f /f
a
FIG.3 Temporal signalsv (t)measuredbyLDVat {r=0.9 ; z=0}andpowerspectraldensities(PSD),at: (a-b)Re=330,
θ
(c-d) Re = 380, (e) Re = 399 , (f) Re = 408 and (g-h) Re = 440. f is the analysis frequency whereas f is the impellers
a
rotation frequency. Inset in (a): zoom over the fast oscillation at frequency f. In (e), a small part of the signal is presented
with time magnification (×4) and arbitrary shift to highlight the modulation at 6.2f−1. Power spectra are computed by the
Welch periodogram method twice: with a very long window to catch the slow temporal dynamics and with a shorter window
to reducefast scales noise.
2. Drifting/Travelling Waves figure 3(c-d). The mean velocity is now zero: the shear-
layer rotates slowly such that the measurement point
is alternatively in the cell rotating with the upper im-
For330<Re<389thevelocitysignalisperiodicwith
peller (v > 0) and in the cell rotating with the lower
θ
a low frequency f . This is illustrated at Re = 380 in
D
7
impeller (v < 0). Visualisations confirm that this cor- and a region of fast fluctuations above the injection fre-
θ
responds to a travellingwave(TW) ora drifting pattern quency also seems to arise. Although we did not carried
and also show that the m = 2 shear-layer is now com- detailedPoincar´eanalysisorequivalentandcannotchar-
posed of two vortices (figure 2d) and thus deserves the acterizeclearlyascenario,wefindthistransitionandthis
name “mixing-layer”. Along the equatorial line, one no- regime typical enough to call it “chaos” (see also IV.B).
tice that the parity is broken or the vortices are tilted
(Coullet and Iooss, 1990; Knobloch, 1996). The veloc-
ity varies between −0.3 . vθ . 0.3. The drift is still D. From chaos to turbulence: building a continuous
slowbutoneorderofmagnitudefasterthanthe driftde- spectrum
scribed above for the “steady” m = 2 pattern: one can
see two periods during 600 time units, i.e., fD = f/300 Increasing further the Reynolds number, one obtains
which is very difficult to resolve by spectral analysis ow- the situation depicted in figure 4. The time-spectrum is
ingtotheshortnessofthesignal(cf. captionoffigure3). now continuous but still evolving. We describe the two
At Re = 380 (figure 3c-d), the peak at the rotation fre- parts, below and above the impellers frequency f.
quency is still present, but starts to spreadand becomes
broadband. The power spectral density at frequencies
higher than 3f decreases extremely rapidly to the noise
1. Slow time scales
level. Let us note that R -symmetry remains only with
π
respect to a pair of orthogonal radial axis which rotates
The slow dynamics which has already been described
with the propagating wave.
atRe=440(figure 3g,h)could be thoughtas depending
only on the largest spatial scales of the flow. It is well
built above Re ≃ 103 (figure 3b). The mean velocities
3. Modulated Travelling Waves corresponding to each side of the mixing layer are of the
order of ±0.6 at Re=1.0×103 and above (figures 4a, c
For 389 < Re < 408 the signal reveals quasiperiodic- and g). The power spectral density below the injection
ity, i.e., modulated travelling waves (MTW), shown in frequency seems to behave with a f−1 power-law over
figure 3e-f at Re=399 and 408. The MTW are regular, twodecades(seediscussion§VI.B)foralltheseReynolds
i.e., strictly quasiperiodic below Re = 400 and irregular numbers(figure4). The spectraldensitysaturatesbelow
above. The modulation frequency —see magnified (×4) 10−2f.
piece of signalin figure 3e— is f =f/(6.2±0.2)what-
M
ever Re, even above Re = 400. It is much faster than
the drift frequency (fD ∼ f/200) and seems to be re- 2. Fast time scales
lated to oscillations of the mixing-layer vortex cores on
the movies. However, the fast time scales, usually interpreted as
a trace of small spatial scales fluctuations, evolve be-
tween Re = 1.0 × 103 and Re = 6.5 × 103. At the
4. Chaotic regime former Reynolds number, there are few fast fluctuations
decaying much faster than f−5/3 (figure 4(b)) and the
The upper limit of the regular dynamics is precisely intermittent changeoversare easy to identify in the tem-
and reproducibly Re = 400 and there is no hysteresis. poral signal in figure 4(a). If Reynolds number is in-
Fromthe visualisations,we observethatthe m=2 sym- creased, the fast (small scales) fluctuations get bigger
metry is now broken. The mixing-layer vortices, which and bigger and behave as a power law over a growing
are still globally rotating around the cell in the previous frequency range (figure 4e, f, h). The measured slope
direction, also behave more and more erratically with is of order of −1.55 over 1.5 decade, i.e., 10% lower
increasing Re: their individual dynamics includes excur- than the classical f−5/3. This value of the exponent
sions in the opposite direction as well as towards one or could be ascribed to the so-called “bottleneck” effect
the other impeller. The velocity signal also looses its (Falkovich,1994)andiscompatiblewiththevaluesgiven
regularity (see figure 3e-g). by Lohse and Mu¨ller-Groeling (1995) (−1.56±0.01) for
When this disordered regime is well established, e.g. a Taylor-micro-scale Reynolds number Rλ ≃ 100, which
for Re= 440 (figure 3g), it can be described as series of is an estimation for our flow based on the results of
almostrandomandfastjumpsfromonesidetotheother Zocchi et al. (1994).
side of the v = 0 axis. The peaks reached by the veloc-
ity are now in the range −0.4 . v . 0.4. The spectral
θ
analysisofthesignalatRe=440(figure3h),doesnotre- IV. QUANTITATIVE CHARACTERIZATIONOF THE
vealanywell-definedfrequencypeakanymore. However, TRANSITIONS
a continuum of highly-energetic fluctuations at low fre-
quency, below f =f and down to f =f/100,emerges. Thevariousdynamicstatesencounteredhavebeende-
a a
A smallbump at the rotationfrequency f is still visible, scribed and illustrated in the previous section. Now,
8
3
1.2 10
(a) 2
0.8 10 (b)
1
10
0.4 0
D 10
Vθ 0 S −1
P 10
−0.4 −2
10
−3
−0.8 10
−4 Re=1000
−1.2 10
0 100 200 300 400 500 600 700 800 −3 −2 −1 0 1 2
10 10 10 10 10 10
Time (f−1)
f /f
3 a
1.2 10
(c) 2
0.8 10 (d)
1
10
0.4 0
D 10
Vθ 0 S −1
P 10
−0.4 −2
10
−3
−0.8 10
−4 Re=1700
−1.2 10
0 200 400 600 800 −3 −2 −1 0 1 2
10 10 10 10 10 10
Time (f−1)
f /f
3 3 a
10 10
2 2
10 (e) 10 (f)
1 1
10 10
0 0
D 10 D 10
S −1 S −1
P 10 P 10
−2 −2
10 10
−3 −3
10 10
−4 Re=2700 −4 Re=3800
10 10
−3 −2 −1 0 1 2 −3 −2 −1 0 1 2
10 10 10 10 10 10 10 10 10 10 10 10
f /f f /f
a 3 a
1.2 10
(g) 2
0.8 10 (h)
1
10
0.4 0
D 10
Vθ 0 S −1
P 10
−0.4 −2
10
−3
−0.8 10
−4 Re=6500
−1.2 10
0 200 400 600 800 1000 1200 −3 −2 −1 0 1 2
10 10 10 10 10 10
Time (f−1)
f /f
a
FIG. 4 (a-b): Temporal signal v (t) measured by LDV at {r = 0.9 ; z = 0} and power spectral density at Re = 1.0×103.
θ
(c-d): TemporalsignalandpowerspectraldensityatRe=1.7×103. (e-f): Powerspectraldensitiesat2.7×103 and3.8×103.
(g-h): Temporalsignal andpowerspectraldensityat 6.5×103. Solidlinesinthepowerspectraplotsarepower-laweye-guides
of slope −1 and −5/3. Spectra are computed as explained in thecaption of figure3.
we wish to analyse some characteristic measurements — A. From order to turbulence: a global supercriticality
theamplitude ofthevelocityfluctuationsandtheirmain
frequencies—, extract thresholds and critical behaviours It is known that fully turbulent von K´arm´an flow can
and then address the question of the nature of the re- generate velocity fluctuations of typically 50% of the
ported transitions. driving impellers velocity. So, we compute the variance
v2 of the LDV-time-series versus the Reynolds num-
θ rms
9
10
0.3
/ fD (a)
ms 0.2 0 f 5
2V rθ 100
0.1
0
Re Re
c t
0 0.08
0 1000 3000 5000 7000
Re
0.06 (b)
s
FIG. 5 Variance of v (t) measured at {r = 0.9 ; z = 0} m
(vRs.eR−eR. eSo)l1i/d2,linfiett:ednoθbne-ltiwneeeanr fiRteof=th3e50foramndvθ2Rrems==25a00×. 2 rVθ 0.04
The regrecssion coefficient is R2 = 0.990, and the fit gives 0.02
Re =328±8with95%confidenceinterval. Theintersection
betcweenthisfitandtheasymptoticvaluevθ2rms ≃0.27gives 1050 200 250 300 350 400 450 500 550
Re =3.3×103. Re
t
FIG. 6 (a) Low-frequency f of the quasi-periodic regime of
D
velocity v (t) measured at {r = 0.9 ; z = 0} (circles) and
θ
drift frequency of the m = 2 shear-layer pattern from flow
ber. This quantity is homogeneous to a kinetic energy
visualisations (squareswith highhorizontal error barsdueto
and may be referred to as the azimuthal kinetic energy
poorertemperaturecontrol). Thesolidlineisalinearfitoff
D
fluctuations in the mixing layer. With this method, we
between thetwo thresholds Re =330 and Re =400,
TW chaos
consider altogetherthe broadbandfrequency responseof indicated by vertical dotted lines. (b) Zoom of figure 5. The
the signal. The resultsarereportedinfigure5forallthe dashedlineisalinearfitofthelowestdatabetweenRe=350
measurementsperformedbetween260.Re.6500. Ex- and Re = 450. Close to the threshold, it crosses the dash-
ceptattime-dependencethreshold,thisquantitybehaves dottedlinewhichcorrespondstothevelocityduetothedrift
very smoothly: it can be fitted between Re = 350 and and estimates thelevel of imperfection.
Re= 2500 with a law in the square root of the distance
to a threshold Re ≃330 (figure 5):
c
v2 ∝(Re−Re )1/2. 1. Onset of time-dependence
θ rms c
Thedriftfrequencyf ofthe travellingwavesbehaves
D
Since we will show below that Re is precisely the linearly with Re above a threshold Re very close to
c TW
threshold for time-dependence, we can make here the 330. Bothmeasurementmethodsagreeevenifthevisual-
hypothesis that v2 is a global order parameter for isationsdeserveslargeerrorbarsinReatleastduetothe
θ rms
the transition to turbulence, i.e., for the transition from shortnessofourrecordsandto apoorerthermalcontrol.
steadyflowtoturbulentflowtakenasawhole. Withthis So, the fit is made on velocity data only. We observe
point of view the transition is globally supercritical. some level of imperfection in the quasi-periodic bifurca-
tion, due to the pre-existing slow drift below Re : we
TW
always observe the mixing-layer to start rotating in the
sense of the initial drift.
B. Transitions from order to chaos We show in figure 6(b) a zoom of figure 5, i.e., the
amplitude ofthe kinetic energyfluctuations. We observe
We now turn to the very first steps of the transition thatbothquadraticamplitudefitandlinearfrequencyfit
totime dependence. We monitorthe mainfrequenciesof converge to exactly the same threshold ReTW = Rec =
themixinglayerdynamicsintheTW-andMTW-regimes 328. We can conclude that the low-frequency mode at
(see §III.C). In these regimes, even if only few periods fa = fD bifurcates at Re = 330 ± 5 through a zero-
are monitored along single time-series, we carefully es- frequency bifurcation for fD.
timate the period by measuring the time delay between The question is thus how the amplitude precisely be-
crossings of the v = 0 axis. These value are reported in haves at onset. There is obviously a lack of data in the
figure 6(a) with circles. In an equivalent way, the peri- narrow range 300 . Re . 350 (figure 6(b)). It is due
odicity of the travelling of the mixing-layer vortices on to the high temperature dependence of the viscosity in
the visualisations give complementary data, represented thisregime(Reynoldsvariedquitefastevenwiththermal
by squares on the same figure. control)andto some data loss at the time of experimen-
10
talruns. Despite this lack,wepresenttheseobservations 100
because of the consistency of the different types of data
—visualisations,LDV,torques—overthewideReynolds
number range. The horizontal line v2 = 0.02 in fig-
θ rms (b)
ure 6(b) corresponds to an amplitude of typically 0.15 Re Re
c t
for v , which is produced just by the initial shear-layer
θ
drift (see the maximum speed in figure 3a). This value
p
isingoodagreementwithalinearextrapolationoverthe K
lower rangeof figure 6(b) and thus againwith an imper-
fectbifurcationduetothedrift. Ifwereducethedriftby
better motor frequencies matching, the onset value of v
θ (s)
will depend on the longitude of the probe location and
−1
the parabolaoffigure6(b) couldperhapsbe observedon 10
Viscous Transition Inertial
the m=2 shear-layernodes.
2 3 4 5 6
10 10 10 10 10
Re
2. Transition to chaos
FIG. 7 Dimensionless torque K vs. Re in a log-log scale
p
for the negative sense of rotation (−) of the impellers. The
The transition to chaos is very sharply observed for
main data (◦) corresponds to the symmetric (s)-flow regime
Re > Re = 400. There is no hysteresis. Just above
chaos described in this part of the article. For completeness, the
the chaotic threshold in the MTW regime (figure 3f), high-torque branch (⋆) for Re & 104 corresponds to the (b)-
thesignalsometimesexhibitsafewalmost-quasi-periodic
flow regime (Ravelet et al., 2004), i.e., to the “turbulent bi-
oscillations still allowing us to measure a characteristic furcation”(see§IV.D). Sincebothmotorsdonotdeliverthe
frequency. The measured values have been also plotted same torque in this R -symmetry broken (b)-flow, the aver-
π
on figure 6(a) and are clearly above the linear fit. This age of both values is plotted. Relative error on Re is ±10% ;
could reveal a vanishing time scale, i.e., a precursor for absolute error of ±0.1 N.m on the torque. Rec and Ret are
the very sharp positive/negativejumps of v reported in the transition values computed from the fits of figure 5. The
the chaotic and turbulent regimes. θ single points, displayed at Re=5×105, correspond to mea-
surements in water, where K is extracted from a fit of the
We do not clearly observe any evidence of mode lock- p
dimensional torque in a+b×f2 for 2×105 .Re.9×105
ingbetweenthepresentfrequencieswhichareinthepro-
(Ravelet et al., 2005).
gression f/200 → f/6.2 → f and there is no trace of
sub-harmonic cascade on any of each. This could be
linked to a three-frequency scenario a` la Ruelle–Takens
(Manneville, 1990).
the minimum of the K (Re) curve (figure 7). One can
p
propose that below this Reynolds number, the power in-
jected at the impeller rotation frequency mainly excites
C. Transition to full turbulence
low frequencies belonging to the “chaotic” spectrum,
whereasaboveRe≃1000italsodrivesthe highfrequen-
1. Torque data
ciesthroughtheKolmogorov-Richardsonenergycascade.
Complementarytothelocalvelocitydata,information
can be collected on spatially integrated energetic data,
i.e., on torque measurements K (Re) (figure 7). The 3. Inertial turbulence
p
low-Reynolds viscous part will be described below (§ V)
aswellasthehigh-torquebifurcatedbranch(§ IV.D). In The(Re−Rec)1/2 behaviourcanbefittedthroughthe
the high-Reynolds number regimes, the torque reaches quasi-periodic and chaotic regimes, up to Re ∼ 3000.
an absolute minimum for Re ≃ 1000 and becomes inde- Here,the azimuthalkinetic energyfluctuations levelsat-
pendent of Re above 3300. urates at v2 ≃0.27, i.e., fluctuations of velocities at
θ rms
this point of the mixing-layer are of the order of 50% of
theimpellertipspeed. Thissaturationisalsorevealedby
2. From chaos to turbulence theProbabilityDensityFunctions(PDF)ofvθ presented
in figure 8. These PDF are computed for 16 Reynolds
Is there a way to quantitatively characterize the tran- numbers in the range 2.5 × 103 . Re . 6.5 × 103.
sition or the crossoverbetween chaosand turbulence? It One can notice the bimodal character of the PDF: the
seems to be no evidence of any special sign to discrimi- two bumps, which are symmetric, correspondto the two
nate betweenthe two regimes. Anempiricalcriterionwe counter-rotating cells. Furthermore, all these PDF col-
couldproposewouldbethecompletenessofthe(f /f)−1 lapse and are therefore almost independent of Re in this
a
low-frequency part of the spectrum, clearly achieved for range. This is also consistent with the spectral data of
Re = 1000 (figure 4b). This region also corresponds to figure 4(b-d) where the (f /f)−1 slowest time-scales re-
a