2005 [b.dubrulle] stability and turbulent transport in taylor-couette flow from analysis of...

Stability and turbulent transport in Taylor–Couette flow from analysis of experimentaldataB. Dubrulle, O. Dauchot, F. Daviaud, P.-Y. Longaretti, D. Richard, and J.-P. Zahn Citation: Physics of Fluids 17, 095103 (2005); doi: 10.1063/1.2008999 View online: http://dx.doi.org/10.1063/1.2008999 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/17/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Suppression of turbulent resistivity in turbulent Couette flow Phys. Plasmas 22, 072304 (2015); 10.1063/1.4926582 Bi-stability in turbulent, rotating spherical Couette flow Phys. Fluids 23, 065104 (2011); 10.1063/1.3593465 Taylor‐Couette flow stability: effect of vertical density stratification and azimuthal magnetic fields AIP Conf. Proc. 733, 165 (2004); 10.1063/1.1832146 Linear theory of MHD Taylor‐Couette flow AIP Conf. Proc. 733, 71 (2004); 10.1063/1.1832138 Experimental studies on the effect of viscous heating on the hydrodynamic stability of viscoelasticTaylor–Couette flow J. Rheol. 47, 1467 (2003); 10.1122/1.1621423

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Stability and turbulent transport in Taylor–Couette flow from analysisof experimental data

B. Dubrulle, O. Dauchot, and F. DaviaudCNRS URA 2464 GIT/SPEC/DRECAM/DSM, CEA Saclay, F-91191 Gif-sur-Yvette, France

P.-Y. LongarettiLAOG UMR 5571 CNRS Université J. Fourier, F-38041 Grenoble, France

D. RichardNASA Ames Research Center, MS 245-3, Moffet Field, California 94035

J.-P. ZahnLUTh CNRS UMR 8102, Observatoire de Paris, F-92195 Meudon, France

�Received 31 August 2004; accepted 29 June 2005; published online 12 September 2005�

This paper provides discussion and prescription about stability and transport in the Taylor–Couetteexperiment, a rotating shear flow with shear perpendicular to the rotation axis. Such geometryfrequently occurs in geophysical or astrophysical context. The prescriptions we obtain are the resultof a detailed analysis of the experimental data obtained in several studies of the transition toturbulence and turbulent transport in Taylor–Couette flow. We first introduce a new set of controlparameters, based on dynamical rather than geometrical considerations, so that they may be relevantto any rotating shear flows in general and not only to Taylor–Couette flow. We then investigate thetransition thresholds in the supercritical and the subcritical regime in order to extract their generaldependencies on the control parameters. The inspection of the mean profiles provides us with somegeneral hints on the turbulent to laminar shear ratio. Then the examination of the torque data allowsus to propose a decomposition of the torque dependence on the control parameters in two terms, onecompletely determined by measurements in the case where the outer cylinder is at rest, the other onebeing a universal function deduced here from experimental fits. As a result, we obtain a generalexpression for the turbulent viscosity and compare it to existing prescriptions in the literature.Finally, throughout the paper we discuss the influence of additional effects such as stratification ormagnetic fields. © 2005 American Institute of Physics. �DOI: 10.1063/1.2008999�

I. INTRODUCTION

One of the basic principles of fluid mechanics is theso-called “Reynolds similarity principle:” no matter theircomposition, size, nature, different flows obeying the sameequations with the same control parameters exhibit the samedynamics. This principle has been widely used in engineer-ing to build, e.g., prototypes of bridges to be tested in windtunnels before construction. To obtain easy-to-use prototypeswith realistic control parameters, one then decreases the sizebut increases the velocity of the in-flowing wind so as tokeep constant the Reynolds number controlling the dynamicsof the flow. This principle may also be of great interest forcertain astrophysical flows, whose dynamics could well beapproached by simple laboratory flows. A good example isthe circumstellar disk. In Ref. 1, it has been shown that undersimple, but founded approximations, its equation of motionis similar to that of an incompressible rotating shear flow,with penetrable boundary conditions and cylindrical geom-etry. This kind of flow can be achieved in the Taylor–Couetteexperiment, where the fluid is sheared between two coaxialcylinders rotating at different speed, while penetrable bound-ary conditions can be imposed using porous material. Onmore general grounds, the Taylor–Couette device is also anexcellent prototype to study transport properties of most as-

trophysical or geophysical rotating shear flows with shearperpendicular to the rotation axis: depending on the rotationspeed of each cylinder, one can obtain various flow regimeswith increasing or decreasing angular velocity and/or angularmomentum.

The Taylor–Couette flow is a classical example of asimple system with complex and rich stability properties, andit can serve as a prototype of anisotropic, inhomogeneousturbulence. It has therefore motivated a great amountof laboratory experiments, and is even the topic of amajor biennial international conference. Tagg �seehttp://carbon.cudenver.edu/rtagg� has conducted a bibliogra-phy on Taylor–Couette flow, which gives a good idea of theprototype status of this flow.

Here, we make use of the many results obtained so farfor the Taylor–Couette experiment, regarding transition toturbulence and turbulence properties, to propose a practicalprescription for the turbulent viscosity as a function of theradial position and the control parameters. It reads

�t = F�Re,R�,��Slam

SSr2, �1�

where F is a function of the control parameters defined in

Sec. II B, and Slam / S is the ratio of the laminar to the mean

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shear, which encodes all the radial dependence as illustrated

in Sec. IV B. S and r are the typical shear and radius of theconsidered flow. For convenience, our notations are summa-rized in Table I. Most of the results we use here have beenpublished elsewhere, except recent experimental results ob-tained by Richard.2 Our work therefore completes and gen-eralizes the approach pioneered by Zeldovich,3 with subse-quent contributions by Refs. 4–6, in which usually only oneaspect of the experiments has been considered. An applica-tion of these findings to circumstellar disks using theReynolds similarity principle can be found in Hersant et al.1

thereby providing a physical explanation of several observ-able indicators of turbulent transport.

II. TAYLOR–COUETTE FLOW

A. Stationary flow

The Taylor–Couette flow is obtained in the gap d be-tween two coaxial cylinders of radii ri,o, rotating at indepen-dent velocities �i,o around an axis which is usually vertical.For generality purpose and in order to allow further compari-son with astrophysical flows, the velocity field at the innercylinder boundary may have a nonzero radial component.

The hydrodynamic equation of motions for an incom-pressible flow are given by

�tu + u · � u = −1

�� p + ��u ,

�2�� · u = 0,

where � and � are, respectively, the fluid density and kine-matic viscosity, u is the velocity, and p is the pressure. Equa-tion �2� admits a simple basic stationary solution, with axialand translation symmetry along the cylinders’ rotation axis�the velocity only depends on r�. It is given by a flow withzero axial velocity, and radial and azimuthal velocities givenby

ur =K

r,

�3�

u� = Ar1+� +B

r,

where A and B are constants and �=K /�. This basic laminarstate depends on three constants A, B and K, which can berelated to the rotation velocities at the inner and outerboundaries:7

A =ro

−�

1 − ��+2 ��o − �2�i� ,

�4�

B =ri

2

1 − ��+2 ��i − �o�� ,

where �=ri /ro and �=K /�=ur�ri�ri /� is the radial Reynoldsnumber, based on the radial velocity through the wall of theinner cylinder.

TABLE I. Notations.

Superscript and subscript conventions

X Any flow variable �e.g., component of velocity�X* Dimensionless part of X

X Mean part of X

X Typical value of X

X� Relates to inviscid flows

XPC Relates to Plane Couette flows

Xsg Small-gap approximation of X

Xwg Wide-gap approximation of X

Xlam Laminar part of X

Xturb Turbulent part of X

Xsub Relates to subcritical flows

Xsup Relates to supercritical flows

X+ Relates to cyclonic flows

X− Relates to anti-cyclonic flows

X= �Xr ,X� ,Xz� Inertial frame cylindricala components of X

X= �Xr ,X ,Xz� Rotating frame cylindricala components of X

Hydrodynamical quantities

u Inertial frame velocity vector

w Rotating frame velocity vector

p, Fluid pressure, generalized pressure

�=u� /r Angular velocity

�rf Angular velocity of the rotating frame

S Velocity shear

Slam; S rd�lam /dr; rd� /dr

r=�riro Typical radius

S=Slam�r� Typical shear value

L=r2� Specific angular momentum

T Torque

N Brunt–Vaissala frequency �stratified case�

Magnetic quantities

B Magnetic field

�0 Magnetic permeability

VA Alfvén velocity

�m Magnetic diffusivity

Geometric and physical quantities

� Kinematic viscosity

�t Turbulent viscosity

� Mass density

ri,o Inner, outer cylinder radii

�i,o Inner, outer cylinder angular velocity

d=ro−ri Gap

�=ri /ro Radius ratio �dimensionless measure of the gap�H Cylinders height

Dimensionless quantitiesStandard dimensionless form

�L�=d Unit of length

�T�=d2 /� Unit of time

Ri,o=ri,o�i,o /� Reynolds number of rotating cylinders �Taylor–Couetteflow�

Dynamical dimensionless form

�L�=d Unit of length

�T�= S−1 Unit of time

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The radial circulation is quantified by the value of �; it ispositive for outward motions. For impermeable cylinders,�=K=0 and one has the “classical” Taylor–Couette flow.For permeable cylinders, one obtains a Taylor–Couette flowwith radial circulation. The strength of the radial circulationcan be controlled by using more or less porous cylinders,8

and acting on the pressure gradient. Figure 1 provides anexample of the influence of the radial circulation on the azi-muthal profile.

In practice, even for impermeable cylinders, the flow isnot purely azimuthal. Because of the finite vertical extent ofthe apparatus, a large-scale–Ekman-like–circulation is estab-lished through the effect of the top and bottom boundaries.This circulation depends on the ratio of radii and velocities,and on the top and bottom boundary conditions.2,9 Its signa-ture is easy to detect by profile monitoring, or by measuringthe difference between the torque applied on correspondingsections of the inner and outer cylinder �see Ref. 10 and Sec.V F 2�. Of course this circulation is both radial and axial, andvaries along the cylinders’ axis. Its intensity is not easy tocontrol, since it is not fixed externally, but results from anontrivial equilibrium within the flow. Still, at a given axialposition, one may estimate this intensity by a fit of the lami-nar profile using �3� and �4�. To simplify the exploration ofthe parameter space, we shall restrict ourselves to the case of��0, and study separately the influence of this parameter. In

the laboratory the effects of circulation can be minimized byworking with tall cylinders or by considering only a fractionof the flow, in height, where the radial velocity is expected tobe the weakest. Specific influence of � on stability and trans-port properties will be considered in Secs. III D and V F 2.

B. Control parameters

Dimensional considerations show that there are onlyfour independent nondimensional numbers to characterizethe system, which can be chosen in various ways.

1. Traditional choice

The traditional choice is to consider d=ro−ri as the unitlength, and d2 /� as the unit time. The dimensionless equa-tions of motion then take the form

TABLE I. �Continued.�

Re= Sd2 /� Reynolds number

R�=2�rf / S Rotation number

R�� Rotation number at marginal stability �Re=��

R�c Critical rotation number

RC=d / r Curvature number

=H /d Aspect ratio

�=u�ri�ri /� Radial circulation Reynolds number

Influence of body forces

Fr=� /N Froude number �stratification�

Pr Prandtl number �stratification�Pm=� /�m Magnetic Prandtl number �magnetic field�

Transition Reynolds numbers

Rc First supercritical linear transition

Rg Minimal Reynolds number for self-sustainedturbulence

RT Transition to “hard” turbulence �as traced by torques�Physical quantities

G=T / ��H�2� Dimensionless torque

Gi Dimensionless torque with resting outer cylinder

Go Dimensionless torque with resting inner cylinder

h=G /Gi Universal function

q=−� ln � /� ln r

Local rotation number

f=Rg�R�

c �� ,��Function defining the unstable manifold

Q= �VAd�2 /�m�

Non-dimensional magnetic number

aThe correspondence between Cartesian and cylindrical axes is �x↔−� ,−�, and �y↔r�.

FIG. 1. Influence of the radial circulation onto the azimuthal profile. Line:case �=0; �: �=−1; � �=1; �: �=−10; �: �=10. The upper panel iswith �i /�o=0.86; the lower panel is with �i /�o=0. The radius ratio ri /ro

has been arbitrarily fixed at 0.7.

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�tu* + u* · � u* = − � p* + �u*,

�5�� · u* = 0,

with the boundary conditions

u*�ri� = ��1 − ��/�,Ri,0� ,

�6�u*�ro� = ��1 − ��,Ro,0� ,

where

� =riur�ri�

�,

� =ri

ro,

�7�

Ri =ri�id

�,

Ro =ro�od

�.

In the following, we omit the asterisk superscript indicatingnondimensional quantities. The present choice of unitamounts to defining the control parameters by nondimen-sional boundary conditions. When comparing flows that donot share the same geometry, it is of interest to identify con-trol parameters characterizing instead the dynamical proper-ties of the flows.

2. Dynamics motivated choice

In the case of rotating shear flows, it is convenient towrite the equations in an arbitrary rotating frame with angu-lar velocity �rf, choose again d as unit length, but the in-

verse of a typical shear S as unit time and r as a typicalradius. Furthermore, it is useful to introduce the “advectionshear term” proposed in Refs. 6 and 11:

w · ��w � w · � �wrer� + �rw · � �w/r��e + w · � �wzez�

�8�

so that the contribution of the mean flow derivative to themodified advection term vanishes when the flow is notsheared, for azimuthal axisymmetric flow. As a result, onehas

�tw + w · ��w = − � − R�ez � w + RC�w2

r/rer

−wwr

r/re + Re−1�w ,

�9�� · w = 0,

with boundary conditions

w�ri,o� = u�ri,o� −R�

2RC

r

re, �10�

where

Re =Sd2

�,

�11�

R� =2�rf

S,

RC = d/r �12�

are the dynamical control parameters for a given radial cir-culation of Reynolds number �. Re is an azimuthal Reynoldsnumber, measuring the influence of shear. R� is a rotationnumber, measuring the influence of rotation. RC is the curva-ture number. Note that the generalized pressure now alsoincludes the centrifugal force term.

In this general formulation, �rf is arbitrary. It is conve-

nient to choose �rf as a typical rate of rotation � so that onecan easily compare the Taylor–Couette case to the case of aplane shear in a rotating frame. For instance one can choose�rf so that w��ri�=−w��ro� in order to restore the symmetrybetween the two walls boundary conditions. This choice of

� amounts to fixing r by ��r�=u��r� / r=�. For consistency,

it is then convenient to choose S=Slam�r�. In this context andwith �=0, it is easy to relate the above control parameters�Re ,R� ,RC� to the traditional choice �Ri ,Ro ,��:

r = �riro,

Re =Sd2

�=

2

1 + ��Ro − Ri ,

�13�

R� =2�

S= �1 − ��

Ri + Ro

�Ro − Ri,

RC =1 − �

�1/2 .

The above control parameters have been introduced sothat their definitions apply to rotating shear flows in generaland not only to the Taylor–Couette geometry. It is very easyin this formulation to relate the Taylor–Couette flow to theplane Couette flow with rotation, by simply considering thelimit RC→0. Also, in the astrophysical context, one oftenconsiders asymptotic angular velocity profiles of the form��r��r−q where q fully characterizes the flow. In that caseq=−� ln � /� ln r=−2/R�, which is a simple relation to situ-ate astrophysical profiles in the control parameters space ofthe Taylor–Couette flows. From the hydrodynamic view-point, an important characteristic of the flow profile is thesign of the shear compared to the sign of the angular veloc-ity, which defines cyclonic and anti-cyclonic flows. For theco-rotating laminar Taylor–Couette flow, the sign of the localratio ��r� /S�r� is constant across the whole flow and is thussimply given by the sign of the rotation number �R��0 forcyclonic flows and R��0 for anti-cyclonic flows�. Finally,let us recall that an analogy exists between Taylor–Couetteand Rayleigh–Bénard convection �see Ref. 12 for details and



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Ref. 13 for a review�, which calls for an even larger gener-alization of the control parameters definition.

Figure 2 displays the characteristic values taken by thenew parameters in the usual parameter space �Ri ,Ro� for co-rotating cylinders. It also helps to situate cyclonic and anti-cyclonic flows, as well as prototypes of astrophysical flows.

III. STABILITY PROPERTIES

A. Inviscid limit and data sources for viscid flows

Stability is a notion encompassing at least two differentscenarii: �i� stability against infinitesimal disturbances—linear stability and �ii� stability against finite amplitudedisturbances—nonlinear stability. When the basic flow is un-stable against finite amplitude disturbance, but linearlystable, it is called subcritical. In contrast, the supercriticalcase occurs when the first possible destabilization is linear�see Refs. 14 and 15 for further details�.

In the inviscid limit �Re→��, and for axisymmetric dis-turbances, the linear stability properties of the flow are gov-erned by the Rayleigh criterion. The fluid is stable if theRayleigh discriminant is everywhere positive:

�

r�rL�r� � 0, �14�

where L�r�=r2��r� is the specific angular momentum. Ap-plying this criterion to the laminar profile leads to

�R� + 1��R� + 1 − r2/r2� � 0. �15�

Since �r /r�2 varies between 1/� and �, this means that inthe inviscid limit, the flow is unstable against infinitesimalaxisymmetric disturbances when R�

�−�R��R��+, where

R��−=−1, respectively, R�

�+=1/�−1, are the marginal stabil-ity thresholds in the inviscid limit �superscript �� in thecyclonic case �R��0, superscript ��, respectively, anti-cyclonic case �R��0, superscript ��. These Rayleigh limits

are also displayed in Fig. 2, where they are asymptotic sta-bility lines. As a matter of fact, this information is somewhatlimited:

• nonaxisymmetric disturbances can be more destabilizingthan axisymmetric ones, so that the flow could be linearlyunstable in part of the domain delineated by Eq. �15�;

• viscous damping reduces the linearly unstable domain;• finally, finite amplitude disturbances may seriously reduce

the stable domain.

In the following, we explore these three possibilities.First we recall the theoretical result obtained by Esser andGrossmann29 regarding the nonaxisymmetric disturbances.Then, we review the existing results on the effect of viscosityin the supercritical case, yielding a critical Reynolds numberas a function of the other parameters Rc�R� ,��. Finally, weinvestigate the subcritical stability limit, when the flow islinearly stable and try to figure out what is the behavior ofthe minimal Reynolds number for self-sustained turbulenceRg�R� ,��.

These boundaries can be estimated via different tools,depending on the type of experiment and on the availablemeasurements. In numerical experiments, the simplest wayto estimate the stability boundary in the linear case isthrough a modal decomposition and a monitoring of the realpart of the largest eigenvalue. In laboratory experiments, atleast three different methods have been used: �i� torque mea-surements, �ii� flow visualization, and �iii� mean velocityprofile measurements. Torque measurements have been tra-ditionally used in the past.9,16 Their advantage is their accu-racy and their flexibility to detect other transitions at largerReynolds numbers. Their disadvantage is their difficulty ofimplementation in the case where both cylinders are rotating.Flow visualizations allow to discriminate between laminarand turbulent flows but they suffer from the lack of quanti-tative information on the flow. Measuring the mean velocityprofile is a third alternative, which allows to deduce the criti-cal Reynolds number from deviation of velocity profiles withrespect to laminar case, or changes of regime. This techniqueis more local in nature, and requires advanced techniques ofin-flow measurements. In the following, we use data fromseveral sources, described in Table II. Except for the data ofRichard, all of them have been published. Those by Richardare available in his thesis manuscript.2 We take the opportu-nity of this synthesis to fit them into a larger perspective.

FIG. 2. Parameters space and some Taylor–Couette flow properties for co-rotating cylinders ��=0.72�. Flows with positive gradient of angular mo-mentum L=r2� but negative �respectively, positive� gradient of angularvelocity are referred to as Keplerian �respectively, stellar�. The shaded areacorresponds to Rayleigh unstable flows �supercritical case�.

TABLE II. Experimental data and sources.

� R� Source

1 �0,0.1� Ref. 17

0.983 �0 Ref. 18

0.724 −0.276 Ref. 19

0.7 �0,0.6� Ref. 2

0.7 �−1.5,−1� Ref. 2

�0.79,0.97� 1/�−1 Ref. 16

0.68,0.85,0.93 �−0.7,0.5� Ref. 9



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B. Supercritical case

Numerous experimental setups are used to study the sta-bility boundary in the linear case, starting from the earlyexperiments of Couette,20 Taylor,21 and Donnelly andFultz.22 The viscosity damps the instability until Re�Rc�R� ,��, corresponding to the transition from the laminarflow to the so-called Taylor vortices flow. Figure 3 displaysthe numerical data by Snyder,23 providing the stabilitythreshold Rc as a function of R� for three gap sizes ��=0.935,0.8,0.2�; it illustrates the influence of the curvatureon the instability threshold. The experimental data of Prigentet al.,18 at �=0.983, are also reported.

As �→1 �rotating plane Couette limit�, the stabilitycurve becomes symmetric around R�=−1/2 and diverges atR�=0 or −1. This is in agreement with the linear stabilitycriterion for the rotating plane Couette flow,24–26 a generali-zation of the linear stability of the nonrotating plane Couetteflow for all Reynolds number.27 The symmetry of the stabil-ity line actually reflects the geometrical symmetry of therotating plane Couette. The linearized equations of motionsare invariant by the transformation exchanging streamwiseand normal to the walls coordinates and velocities �corre-sponding to exchanging r with and ur and u, in Taylor–Couette�. This transformation changes R� into −1−R�,hence the symmetry around R�=−1/2. When � departs from1, curvature enters into play and breaks the symmetry result-ing in less and less symmetrical curves, as can be observedfor �=0.2.

The above stability boundary can be recovered numeri-cally by classical stability analysis, using, e.g., normal modeanalysis with numerical solutions.28 A very good approxi-mate analytical formula in the whole parameter space, in-cluding nonaxisymmetric modes, has recently been derivedby Esser and Grossmann.29 It is

Rc2�R� + 1��R� + 1 −

1

�x2 = − 1708� �1 − ��2��x�� − 1�

4

,

�16�

with

x�� = 1 +1 − �

2��

dn

d ,

dn

d=

�

1 − �� 1

��R� + 1�− 1 , �17�

�� = �1 − �� 1 + ��3

2�1 + 3��− �−1

,

where ��x� is a function equal to x if x�1 and equal to 1 ifx�1. Continuous lines in Fig. 3 give a good insight on thevalidity of the above formula. This stability curve has twointeresting approximations: in the small gap approximation,�→1, it gives

Rcsg =� 1708

− R��R� + 1�. �18�

This is exactly the stability criterion for the rotating planeCouette flow �Lezius and Jonston26�. This criterion actuallydescribes also well the stability curve at any gap, for R�

�2�−1 and works very well until �=0.8. In the wide gaplimit �→0, this criterion predicts

�Rcwg =�1708

4

1

��R� + 1�, �19�

where ��x� is the Heavyside function equal to 1 for x�0and 0 if x�0. This is in fact a very poor approximation evenfor �=0.2.

FIG. 3. The linear stability boundary.Numerical data by Snyder �: �=0.2;�: �=0.8, � �=0.935. Experimentaldata by Prigent et al. �: �=0.983.Continuous line: Lezius and Johnstonplane Couette or small gap limit stabil-ity criteria. Dashed line: Esser andGrossmann prediction for �=0.8 and�=0.2.



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Note that the formula �16� defines two critical rotationnumber for which the critical Reynolds number diverges:R�

c−=−1 and R�c+ such that R�

c+=−1+1/�x2. This number hasbeen computed for various 0.7��1 and is shown in Fig.4. One sees that it is very well approximated by the formulaR�

c+= �1−�� /�. This remark is used in the next section.In this supercritical situation, the flow undergoes several

other bifurcations following the first linear instability andturns into more and more complex patterns, eventually lead-ing to turbulence. Interestingly, at much larger Reynoldsnumber, an additional transition has been reported.30 It isdetected through a change in the torque dependence on theReynolds number, which could be associated with a feature-less turbulence regime. Sometimes called “hard turbulence,”this regime is observed for Re�RT. For reasons that willbecome clearer, we defer its discussion after the study of thetorque.

C. Subcritical case

In the absence of a general theory for globally subcriticaltransition to turbulence, the nonlinear stability boundary hasonly been explored experimentally. Wendt9 and Taylor16 con-sidered the case with inner cylinder at rest, corresponding toR�=1/�−1�0, at various gap size, using torque measure-ments. A more recent experiment by Richard2 explores thedomain −1.5�R��−1 and R��0.5, at fixed gap size �=0.7, using flow visualizations. Finally, the measurementsconducted in a rotating plane Couette flow ��=1� byTillmark and Alfredsson17 for R��0 are also available. Thecorresponding results are reported in Fig. 5, giving Rg as afunction of R� for different values of �.

This naive representation of the data hides some difficul-ties, especially on the cyclonic side. First, the data are pre-sented for different values of �. In the case of Taylor ��,each point corresponds to a different �. A linear fit ofTillmark’s data shows that the different values of � do notobey a simple law. Second, the representation seems toimpart to R�=0 a role in the cyclonic regime similar to

R�=−1 in the anti-cyclonic case. The linear stability analysisindicates that this is true only for �=1. As discussed in theprevious section, the correct value of the marginal stability isapproximately equal to the inviscid limit for the cycloniccase R�

c+�R��+=1/�−1. Taylor’s data are actually given at

this precise value of R�, imposed by the experimental con-figuration with resting inner cylinder. This limitation some-how restricts our possibility to extract the dependence ofRg

�+,−� on R� and �, both in the cyclonic and anti-cycloniccase. Here is what we can say about it.

The dependence of the critical Reynolds number on thecontrol parameter defines a general manifold Rg

�+,−��R� ,��.All data regarding this manifold are obtained close to itsintersection with the other manifold R�=R�

c�+,−�. A possiblecharacterization of the critical Reynolds number can there-fore be done first through the parametrization of the locus ofthe intersection between these two manifolds, namelyRg

�+,−��R�c�+,−�� ,��= f �+,−��, then through parametrization

of the variation of Rg+ with R�, in the vicinity of the inter-

section.Let us first consider the cyclonic case. One can use Tay-

lor’s and Wendt’s data to estimate f+�� as proposed byRichard and Zahn.5 The fact that the data are read from theoriginal figure of Taylor and Wendt, however, induces a natu-ral error bar in the determination of the critical Reynoldsnumber, as illustrated in Fig. 6, where several estimates, ob-tained by different authors, are reported.

Because of this error, it is difficult to give a precise fit ofthe function f+. One sees that the quadratic regime in 1−�given by f+��=1400+550 000�1−��2 and proposed byRichard and Zahn provides a good upper estimate of thefunction. A linear trend in 1−�, with slope 136 000 gives agood lower estimate of the data for 1−��0.1, as shown inFig. 6. Clearly, a more precise determination of this functionusing modern data would be most welcome. Note that at �→1, the function tends to a constant f+�1�=1400 that is

FIG. 4. The critical rotation number R�c+ as a function of the gap size �. �:

Computed from the analytic formula of Esser and Grossmann. Plain line:�1−�� /�.

FIG. 5. The nonlinear stability boundary. Cyclonic flow: �: Taylor datawith inner cylinder at rest and 0.7��0.935; �: Richard data with �=0.7; �: Tillmark data for rotating plane Couette flow �=1; dotted line:linear fit of Tillmark’s data; plain line: linear fit of Richard’s data. Anti-cyclonic flow: �: Richard data with �=0.7; plain line: linear fit of Richard’sdata.



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nothing but RgPC=1400, the global stability threshold mea-

sured independently by Tillmark and Alfredsson17 andDauchot and Daviaud31 in the nonrotating plane Couetteflow. The second step is to propose a linear development inR�−R�

c+, close to the above estimate:

Rg+�R�,�� = f+�� + a+��R� − R�

c+�� . �20�

For �=1 one recovers the linear fit proposed by Tillmark andAlfredsson �plotted and extrapolated in Fig. 5� for the rotat-ing plane Couette flow:

Rg+�R�,� = 1� = 1400 + 26 000R�, �21�

that is, a+��=1�=26 000. For �=0.7, the linear fit of Rich-ard’s data �plotted and extrapolated in Fig. 5� leads to a+��=0.7�=59 000.

In the anti-cyclonic case, the situation is simpler becauseR�

c−=−1 does not depend on �. On the other hand, data areavailable for a unique value �=0.7, so that one cannot esti-mate f−��. The only fit that can be performed in this state ofexperimental knowledge is

Rg−�R�,0.7� = f−�0.7� + a−�0.7�R� − R�

c− . �22�

One finds f−�0.7�=1300 and a−�0.7�=21 000 and the fit isdisplayed in Fig. 5. In the anti-cyclonic regime, at least forthis value of �, one recovers a dependence on the rotationsimilar to that of the plane Couette flow. Also remarkable isthe fact that f− is so close to Rg

PC in the nonrotating case.Altogether the data collected to date suggest that, in the

linearly stable regime, the Reynolds number of transition tosubcritical turbulence is well represented by

Rg±�R�,�� = f±�� + a±��R� − R�

c,± , �23�

with 1�105�1−��1� f+��1400+5.5�105�1−��2,f−�0.7�=1300 and 21 000�a±�59 000. It is difficult to dis-tinguish the effect of experimental procedures from the ef-

fects of gap width dependence in the present parameterrange.

D. Influence of radial circulation

1. Supercritical case

The influence of radial circulation on the linear stabilityonset has been studied numerically by Min and Lueptow.8

They observed that an inward radial flow and strong outwardflow have a stabilizing effect, while a weak outward flow hasa destabilizing effect. We may use their data to get moreprecise estimates for the case �=−3/2 �q=3/2, Kepleriancase�. Figure 7 shows the ratio Rc��=−3/2� /Rc��=0�−1 asa function of � for �o /�i=0. One sees that the variation isquasilinear. A best fit gives

Rc�� = − 3/2�Rc�� = 0�

= 1 + 0.12�1 − ��,�o

�i= 0. �24�

On the same graph, we show Rc��=−3/2� /Rec��=0�−1 as afunction of �o /�i for �=0.85. A best fit gives

Re+�� = − 3/2�Re+�� = 0�

= 1 + 1.16��o

�i2

, � = 0.85. �25�

FIG. 6. Subcritical thresholds in the cyclonic regime, obtained with theinner cylinder at rest, that is, R�=R�

c+. �: Wendt’s data; �: Taylor’s data; �:Richard’s data; plain and dotted line: fit of f+��=Rg

+�R�c+�� ,�� see text for

details�. The size of the symbol denotes different estimates by Richard andZahn �small�, Zeldovich �medium�, and present authors �large� based onpublished figures of Taylor and Wendt.

FIG. 7. Rc��=−3/2� /Rc��=0�−1 as a function of �a� � for �0 /�i=0; �b��0 /�i=0 for �=0.85. �: Data from Min and Lueptow. The dotted lines arethe fit Eqs. �24� and �25�.



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2. Subcritical case

The influence of the radial circulation on the nonlinearstability has not been systematically studied. However, wecan get partial answers from the experiments of Wendt9 andRichard,2 where the influence of the top and bottom circula-tion on the onset of stability has been studied. Both Richardand Wendt investigated the stability boundary with differentboundary conditions. One was with the bottom �Wendt andRichard� and top �Richard� attached to the outer cylinder. Inthis case, the circulation is mainly in the anti-clockwise di-rection in the lower part of the gap, with radial velocitiesoutward at the bottom ��0�. Another boundary conditionwas with the ends attached to the inner cylinder �at rest�. Inthat case, the circulation is in the opposite direction, withinward radial velocities at the bottom ��0�. A third bound-ary condition was intermediate between the two, with theends split in two annuli, each of them being attached to itsneighboring cylinder. Only modest differences of the stabil-ity boundary have been noticed between these three settings:at this aspect ratio ��0.7, the radial circulation induced bythe top and bottom boundaries modifies the subcriticalthreshold Reynolds number by less than 10 percent �accu-racy of the measurements�.

E. Influence of aspect ratio

Most of the experimental setups described in this paperhave a very large aspect ratio =H /d�1. Keplerian disksare characterized by a small aspect ratio H /d�0.01–0.1. Itwould be interesting to conduct systematic studies of the roleof in the stability and transport properties. The influence of onto the instability threshold, in the case of an outer cyl-inder at rest has been computed by Chandrasekhar28 andSnyder.23 This is illustrated in Fig. 8. The critical Reynoldsnumber is increased, as is decreased. It follows an approxi-mate law:

Rc� � = Rc�1 + −2� + O� −3� . �26�

This behavior can be understood if one says that as be-comes smaller, the smallest relevant length scale in the prob-lem becomes H instead of d. The relevant Reynolds numberhas thus to be corrected by a factor �H /d�2, hence, the −2

law. However, another experimental study by Park et al.32

suggests that the physical relevant length scale is �Hd in-stead of H. A possible explanation of the difference isthrough the Ekman circulation, which is present in experi-ments but not in numerical simulations. This circulation maycouple vertical and radial velocities, leading to an effectivelength scale. The only way to settle this issue is throughsystematic laboratory and numerical experiments of smalleraspect ratio.

F. Structural stability

Before closing this section, it is interesting to considerthe influence of additional physical forces that may be rel-evant to astrophysical flows. We shall only give a summaryof the main experimental or theoretical results obtained, re-ferring to the publications for more details.

1. Magnetic field

The influence of a vertical magnetic field on the stabilityof a Taylor–Couette flow has been studied theoretically28,33

and experimentally by Donnelly and Ozima34 using mercury.Applications to astrophysics have been discussed by Balbusand Hawley.35 This has motivated a lot of numerical work onthis instability. For references, see, e.g., Ref. 36.

In the inviscid limit, the presence of a magnetic fieldmodifies the Rayleigh criterion �14�. For example, in the caseof a magnetic field given by B=��0��0,V�

A�r� ,VzA�, the suf-

ficient condition for stability is now38,39

r2�r�2 −

1

r2�r�rV�A�2 � 0. �27�

Therefore, anti-cyclonic flow, with R��0 are now poten-tially linearly unstable in the presence of a magnetic fieldwith no azimuthal and radial component.35

The linear instability in the presence of dissipation hasonly been studied numerically. A first finding is that bound-ary conditions �e.g., insulating or conducting walls� are rel-evant to determining the asymptotic behaviors.28 The pro-posed explanation is that the magnetic field makes the flowadjoin the walls for longer distances, so that the viscous dis-sipation remains comparable to the Joule dissipation at allfields. A second observation is the importance of the mag-netic Prandtl number Pm=� /�m ��m is the magnetic diffusiv-ity� on the instability.36,37 On general grounds, it seems thatat small Prandtl numbers, the magnetic field stabilizes theflow in the supercritical case, while at large Prandtl numbers,the magnetic field destabilizes the flow. In the subcriticalcase, the magnetic field can excite a linear instability foranti-cyclonic flow, at any Prandtl number. This is illustratedin Fig. 9.

Scalings of critical Reynolds number with magneticPrandtl numbers have been found: in the supercritical

FIG. 8. Influence of aspect ratio onto the critical Reynolds number forinstability, in the case where the outer cylinder is at rest. �: Numerical databy Chandrasekhar. The dotted line is a power law fit 0.9 −2.



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case −1�R��0, the critical Reynolds number scales likePm−1/2.37 In the subcritical case R�=−1, the criticalReynolds number scales like Pm−1.

2. Vertical stratification

A vertical stable stratification added onto the flow playsthe same role as a vertical magnetic field at low Pm. In theinviscid limit, its presence changes the Rayleigh criteria intor2�r�

2�0.40,41 This means that all anti-cyclonic flows arepotentially linearly unstable. The role of dissipation on theinstability has been studied numerically41,42 andexperimentally.43,44 It is found that stratification stabilizes theflow in the supercritical case regime, while it destabilizes itin the subcritical anti-cyclonic regime. The critical Reynoldsnumber is found to scale with the Froude number �ratio ofrotation frequency to Brunt–Vaissala frequency� like Fr−2,and to scale with the Prandtl number �ratio of viscosity toheat diffusivity� like Pr−1/2.

3. Radial stratification

A radial temperature gradient applied to the flowchanges also its stability. In the inviscid limit, theRayleigh criterion is modified by the radial temperaturegradient into45,46 �cf. also http://www.couette-taylor2001.northwestern.edu/ct/abstract/mutabazi.pdf�

�

r�r�r2��1 −

�T

T0 −

�rT

T0r�2 � 0, �28�

where 1/T0 is the coefficient of thermal expansion and �T isthe temperature difference between the cylinders. The lastterm in �28� induces an asymmetry between the case withpositive �T and negative �T. An experimental study by Sny-der and Karlsson47 helps to quantify the role of dissipativeprocesses. It is found that both positive and negative �Thave a stabilizing effect when �T is small, and a destabiliz-ing effect when �T is large. A more complete exploration of

the parameter space would be welcome, since astrophysicaldisks are likely to be subject to this kind of stratification.

4. Summary

These studies point out an interesting dissymmetry be-tween the case R��0 �cyclonic flows� and R��0 �anti-cyclonic flows�. In many instances, the regime of linear in-stability is extended by the large-scale force into the wholedomain R��0. As a result, in the anti-cyclonic regime oneoften has to deal with a competition between a linear desta-bilization mechanism induced by the large-scale effect andthe subcritical transition controlled by the self-sustainedmechanism of the turbulent state.

IV. MEAN FLOW PROFILES

A. Supercritical case

Turbulent mean profiles have been measured recently fordifferent Reynolds numbers by Lewis and Swinney19 in thecase with the outer cylinder at rest. They observe that the

mean angular momentum L=ru� is approximately constant

within the core of the flow: L�0.5ri2�i for Reynolds num-

bers between 1.4�104 and 6�105. At low Reynolds num-ber, this feature can be explained by noting that reducing theangular momentum is a way to damp the linear instability,and, thus, to saturate turbulence. At larger Reynolds number,however, one expects the turbulence to be sustained by theshear in the same way as it is when there is no linear insta-bility at all. Accordingly, this constancy of the angular mo-mentum is quite a puzzling fact. Some understanding of thisbehavior can be obtained by observing that the mean profilesobtained by Lewis and Swinney are actually in good agree-ment with a profile obtained by Busse upon maximizing tur-bulent transport in the limit of high Reynolds number:48,49

u��r� = − �

r2S

8r+ r�� +

2 − 3� + 2�2

4�1 + �2�S . �29�

This profile bears some analogy with the laminar profile,which reads

u�,lam�r� = − �r2S

2r+ r�� +

1

2S . �30�

In the Busse solution, the shear profile S��r�=�r2S /4r2

= �1/4�Slam�r�. This ratio is analog to the value observed atvery large Reynolds number in the nonrotating plane Couetteflow.50 It is therefore a clear signature of the shear instability,with no discernable influence of rotation, at least for thelimited value of the rotation number �of the order of −0.28�considered by Lewis and Swinney. So it is interesting to testthe Busse asymptotic profile using other data, with differentrotation number. This will be the purpose of the next section,where Richard’s data will be used.

We may, however, not conclude this section without not-ing an intriguing property of the Busse solution. ConsideringR�,turb=2��r� /S��r�, we get from �29�

FIG. 9. Influence of body forces on the linear stability boundary. �: Withoutforces, �=0.2, numerical data from Snyder �1968�. With vertical constantmagnetic field, at Pm=1 �� and Pm=10−5 ��, �=0.27; numerical datafrom Rüdiger et al. 2003; �: with vertical stratification, �=0.2; data fromWhithjack and Chen �1974�.



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R�,turb = 4R� + 3�1 − ��2

�1 + �2�. �31�

So the condition −1�R�,turb�1/�−1 �“linear stability ofthe turbulent profile”� is satisfied provided R� follows:

−2 − 3� + 2�2

2�1 + �2�� R� �

1 − �

4�

�1 + ��2 − 3��1 − ��2

�1 + �2�,

�32�

that is, in the small gap limit, −1/4�R��0. As we shall seein the sequel, this is precisely the range of value where thetorque is extremum.

B. Subcritical case

For the turbulent flow following the subcritical transi-tion, we use the data of Richard,2 collected for differentReynolds numbers and rotation numbers. Figure 10 displaystypical turbulent mean profiles in both the cyclonic and anti-cyclonic cases for comparison with the laminar and theBusse profiles.

One notices the profile tendency to evolve from thelaminar one to the Busse solution, even if they are still veryfar away from the solution with maximal transport. In orderto evaluate how fast the convergence occurs, Fig. 11 displaysthe ratio of the turbulent mean shear to the laminar shear,

both estimated at r, i.e., S�r� /Slam�r�, as a function of theratio of the Reynolds number to the threshold for shear sus-tained turbulence, i.e., Re /Rg. One may indeed observe atendency of shear reduction as the Reynolds number in-creases, with a more rapid reduction for rotation numbercloser to 0. However, none of the cases studied by Richardapproaches the value 0.25 predicted by Busse. It would beinteresting to conduct higher Reynolds number experimentsat large value of the rotation number to check whether rota-tion merely slows down the convergence toward the 0.25value, or changes it into a number depending on the rotationnumber.

Also one may notice that the decrease of S�r� /Slam�r�with Re /Rg is much faster for cyclonic flows than for anti-cyclonic ones. Figure 12 may provide some hints on theorigin of this dissymmetry. The first one is obtained by

studying the radial variation of the ratio Slam / S at a givenRe /Rg, for different rotation number. This quantity yields theradial variation of the turbulent viscosity and thus is a goodtracer of transport properties. One may observe an interestingtendency for cyclonic flow to display enhanced �respec-trively, reduced� transport at the inner �respectively, outer�core boundary, while anti-cyclonic flow rather displays re-duced transport at the center, and enhanced transport at bothboundaries.

The second one is provided by the function

q�r� =2��r�

S�r�=

d ln �

d ln r, �33�

which may be viewed either as a local mean angular velocityexponent, or a local mean rotation number. This local expo-nent is also plotted in Fig. 12, for different rotation number,

FIG. 10. Turbulent mean velocity profiles from Richard at Re /Rg=1.6: �a�:cyclonic case �R�=0.39�; �b�: anti-cyclonic case �R�=−0.6�. Dotted line:laminar profile. Continuous line: Busse solution Eq. �29�.

FIG. 11. Ratio of the turbulent mean shear to the laminar shear variationwith the Reynolds number �data from Richard�. Cyclonic case: �: R�

=0.39; �: R�� 0.64,0.52�. Anti-cyclonic case: �: R�� −2,−1�.



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at Re /Rg=1.6. One clearly observes a tendency toward con-stancy of this local exponent in the core of the flow and abimodal behavior: cyclonic flow scatters toward q=0.5 whileanti-cyclonic flow scatters toward q=−1.5. We have ob-served a persistence of this behavior at larger Reynolds num-ber �up to at least Re /Rg=20�.

V. TORQUE MEASUREMENTS AND TRANSPORTPROPERTIES

The turbulent transport can be estimated via the torque Tapplied by the fluid to the rotating cylinders. Traditionally,one works with the nondimensional torque G=T /�H�2.30

For laminar flows, one can compute this torque analyticallyusing the laminar velocity profile. It varies linearly with theReynolds number:

Glam =2�

�1 − ��2�Re. �34�

When turbulence sets in, the torque applied to the cylindersincreases with respect to the laminar case. A good indicator

of the turbulent transport can then be obtained by measuringG /Glam.

A. Supercritical case

As noticed by Richard and Zahn,5 most of the torquemeasurements available in the literature concern the casewith the outer cylinder at rest �see, e.g., Refs. 19 and 30 andreferences therein�. In that case, we note that R�= �−1�1. An example of the variation of G /Glam with Reynoldsnumber is given in Fig. 13, in an apparatus with �=0.724.One observes three types of behaviors: below a Reynoldsnumber Rc, i.e., in the laminar regime, G /Glam=1. Above Rc,one observes a first regime in which G /Glam varies approxi-mately like a power law, with exponent 1 /2. In this regime,Taylor vortices are often observed. This regime continuesuntil Re=RT, where the torque increases at a faster pace, andthe power law steepens into an exponent closer to 1. Thisregime has been observed up to the highest Reynolds numberachieved in the experiment �of the order of 106�.

The experiment with inner cylinder rotating only coversflows such that R�=�−1. To check whether this kind ofmeasurement is typical of torque behaviors in the globallysupercritical case, one must rely on experiments in which theouter cylinder is also in rotation. Unfortunately, the onlytorque measurements available in this case are quite old9 andnot as detailed as in the case with inner cylinder rotating.More specifically, they do not extend all the way down to thetransition region between laminar and turbulent. In severalinstances in which large Reynolds number are achieved,however, one may observe a steepening of the relative torquetoward the G /Glam�Re regime already observed in the casewith inner cylinder rotating. On other measurements per-formed at lower Reynolds numbers, the relative torque dis-plays a behavior more closely related to the intermediateregime, with G /Glam�Re1/2. Altogether, this is an indicationthat in the globally supercritical case, the torque followsthree regimes:

FIG. 12. Mean profile at Re /Rg=1.6 for �a� turbulent transport, traced by the

ratio Slam / S; �b� local rotation number. �: R�=−1.31; �: R�=−1.41; �:R�=0.39; �: R�=0.51. Data are from Richard.

FIG. 13. Relative torque G /Glam as a function of the Reynolds number. �:Supercritical case with outer cylinder at rest; R�=−0.276; �=0.724. Dataare from Lewis and Swinney. �: Subcritical case, with inner cylinder at rest.R�=0.47; �=0.68. Data are from Wendt.



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G � �Re, Re � Rc,

G � �supRe3/2, Rc � Re � RT,

�35�G � �supRe2, Re � RT,

� =2��

�1 − ��2 ,

where �sup and �sup are constants to be specified later.

B. Subcritical case

The only torque measurements in the subcritical casewere performed by Wendt9 and Taylor16 in experiments withthe resting inner cylinder, and rotating outer cylinder.Wendt’s experiments were performed at three different val-ues of �, Taylor’s at eleven values of �. Taylor measure-ments cover sufficiently small value of Reynolds number sothat one can see that above a critical Reynolds number Re=Rg, the torque bifurcates from the laminar value toward aregime in which the relative torque G /Glam behaves like Re.An example is given in Fig. 13. Measurements by Wendt atlarger Reynolds number display no evidence for an addi-tional bifurcation. So, in the subcritical case, the torque pre-sumably follows only two regimes:

G � �Re, Re � Rg,

�36�G � �subRe2, Re � Rg,

where �sub is a constant that we specify in the next subsec-tion.

C. Connecting torque and thresholds

The different regimes identified in the previous subsec-tion enables an interesting connection between the torquevalue and the critical Reynolds number between two re-gimes. This is important because torque is not easy to mea-sure, especially in the case where both cylinders are rotating.On the other hand, critical Reynolds numbers between tworegimes may be more easily detected, using, e.g., flow visu-alizations.

Mathematical continuity of the torque as a function ofthe Reynolds number at the transitions allows determinationof the prefactors �sup, �sup, and �sub. In the supercritical case,one obtains

�sup = �Rc−1/2,

�37�

�sup = �supRT−1/2 =

�

�RcRT

,

and in the subcritical case

�sub =�

Rg, �38�

where � is known through �35�.Plugging these expressions in �35� and �36� enables

the expression of the torque as a function of the critical

Reynolds number Rc, RT, and Rg which then encode all thedependencies on R� and �. Our argument is admittedly verycrude, so it is important to test its validity on available data.Figure 14 shows the comparison between the real nondimen-sional torque measured in experiments, and the torque com-puted using only the critical Reynolds numbers. At lowReynolds number, there is a fairly large discrepancy but atlarge Reynolds, the approximate formula provides a goodestimate.

D. Critical Reynolds number for shear instability

The existence of two turbulent regimes in the supercriti-cal case is to be contrasted with the existence of only oneturbulent regime in the subcritical case. One possible inter-pretation is that the last turbulent regime corresponds to the“ultimate” �universal� turbulent regime achieved by the sys-tem, in which energy dissipation proceeds independently ofinitial conditions. A good argument in favor of this interpre-tation is the remark by Lathrop et al.30 that when the torquevaries like Re2, it does not depend anymore on molecularviscosity, like in the Kolmogorov theory for developed tur-bulence at large Reynolds numbers. Because of that, it islikely that the nature of the “instability” �mechanism� lead-ing to this ultimate regime is the same in the supercriticalcase and in the subcritical case, corresponding to a regimewhere turbulence is sustained by a shear mechanism. Thecritical Reynolds number for such a transition can be ob-tained by comparing �37� and �38�. One sees that the analogof Rg in the supercritical case is Rg

sup=�RcRT. A physicalbasis for this expression could be given using the observationthat in the supercritical case, the transition occurs in a turbu-lent state, where transport properties are augmented with re-spect to a quiescent, laminar case, in which all transport isensured by viscous diffusion. This results in a delayed tran-

FIG. 14. Relative torque G /Glam as a function of the Reynolds number,compared with its determination using critical Reynolds numbers. �: Glo-bally supercritical case with outer cylinder at rest. R�=−0.276; �=0.724.�Lewis and Swinney�. �: Globally subcritical case, with inner cylinder atrest. R�=0.47; �=0.68 �Wendt 1933�. Short dashed line: �Re /Rc with Rc

=90; dot-dashed line: Re /�RcRT with �RcRT=957; long-dashed line: Re /Rg

where Rg=32 688. The critical Reynolds numbers have been computed us-ing the results of Sec. III.



131.112.125.216 On: Wed, 04 Nov 2015 06:02:00

sition to the ultimate state, since the viscosity is artificiallyenhanced by an amount �t /�, where �t is the turbulent vis-cosity. Using �t /�=G /Glam, we thus get from �35� and �37�an estimate of the relevant threshold as

Rgsup = RT

�

�t= RT

�

�sup�RT

= �RcRT. �39�

At this stage of the analysis, Rg−, Rg

sup and Rg+, respectively,

define a function of R� and � on the intervals R��R�c−,

R�c−�R��R�

c+ and R�c+�R�, where we recall here that R�

c−

=−1 and R�c+=1/�−1. Their concatenation yields the critical

Reynolds number for shear instability, in a hypothetic rotat-ing shear flow where no primary �linear� instability ispresent. This is illustrated in Fig. 15, where a continuity isobtained on the cyclonic side between Tillmark’s data ��=1� and Wendt’s data ��=0.935� and on the anti-cyclonicside between Richard’s data ��=0.7� and Wendt’s data ��=0.68�.

E. How to use torque with resting outer cylinder

Torque measurements described in the previous sectionsuggest that at large enough Reynolds number, an “ultimate”regime is reached with quadratic variation with Reynoldsnumber. This suggests that in this regime, the interesting pa-rameter collecting the dependence in all the other parameters�rotation number and curvature� is the ratio of the torque inany configuration, to the torque measured in a special case.Because the case with resting outer cylinder is the most stud-ied, it is of practical interest to choose this case as the refer-ence, so that the relevant ratio is G /Gi, where Gi is thetorque when only the inner cylinder is rotating. According tothe above subsections, G /Gi is a function of only R� and �given by h�R� ,��=Rg�R��Ro=0� ,�� /Rg�R� ,��, where Rg isthe generalized threshold defined in the previous section anddisplayed in Fig. 15. Figure 16 indeed shows the ratio G /Gi,for different values of �, as a function of the rotation num-

ber. The measurements for −0.8�R��0.5 are direct mea-surements from the Taylor and Wendt experiments. The mea-surements for R��−1 and R��0.5 are indirectmeasurements, drawn from the experiment by Richard, fromwhich only critical numbers for stability were deduced. Inthat case, the torques have been computed using the resultsof the previous section. All these results show that the non-dimensional torque behaves as

G�Re,R�,�� = Gi�Re,��h�R�,�� , �40�

where Gi is the torque when only the inner cylinder is rotat-ing, and h�R� ,�� is the function of Fig. 16. Note that in thepresent range of �, the function h appears to vary very little

FIG. 15. Rg as a function of R� and �.Anti-cyclonic side: �: Richard data��=0.7�; �: Wendt data ��=0.68�.Cyclonic side: �: Tillmark data ��=1� and fit as in Sec. III; �: Wendtdata ��=0.935�; �: Richard data ��=0.7�. The lines are guides for theeyes to underline the continuity acrossthe supercritical to subcritical domainsfor similar values of �.

FIG. 16. Relative torque h=G /Gi as a function of R� and �. � �respec-tively, ��: estimation from Richard data ��=0.7�, based on critical Rey-nolds numbers, computed using results of Sec. III in the anti-cyclonic �re-spectively, cyclonic� case; � �respectively, ��: Wendt data ��=0.68,0.85,0.935�, the square size increasing with � in the anti-cyclonic�respectively, cyclonic� case.



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with �: most of the geometrical dependence is lumped intoGi. It would be interesting to conduct measurements atsmaller values of � to see whether this feature remains valid.

This universal function is very interesting because it pro-vides good insight into the influence of the rotation on thetorque. For rotation number −0.2�R��0, the torques aremaximal and equal to the torque measured when only theinner cylinder is rotating. Outside this range, the torquesdecrease, with a sharp transition toward a constant of theorder of 0.1 on the cyclonic side R��0. On the other side,the transition is softer, with an approximate quadratic inversevariation until the smallest available rotation numberR�=−1.5. From a theoretical point of view, the asymmetrycould be linked with the different stability properties of theflow on either side of the curve: for −1�R��1/�−1, theflow is linearly unstable, while it becomes liable to finiteamplitude instabilities outside this range. The variation weobserve can also be linked with experimental studies by Jac-quin et al.,51 re-analyzed by Dubrulle and Valdettaro.52 Theyshow that rotation tends to inhibit energy dissipation andobserved simple power laws linking the energy dissipationwith and without rotation via the rotation number based onlocal shear and rotation.

Finally, the previous discussion shows that the knowl-edge of the torque in the case with resting outer cylinder as afunction of Re and � is essential data to compute the torquein any other configuration. A theoretical model of the torquein that configuration has been proposed by Dubrulle andHersant,12 in the case where the boundary conditions at thecylinder are smooth. It gives

Gi =�3 + ��1/4��Re�3/2

�1 − ��7/4�1 + ��1/2 , Rc � Re � RT, �41a�

Gi = 0.33�3 + ��1/2

�1 − ��3/2�1 + ��Re�2

�ln��M��Re�2��3/2 ,

Re � RT, �41b�

M�� = 0.0001�1 − ��3 + ��

�1 + ��2 . �41c�

The quality of the fit can be checked in Fig. 17. For Re�Rc, the flow is laminar and the transport is ensured only byviscous diffusion.

F. Toward extended Reynolds similarity

The link between torque and critical Reynolds numberhas a powerful potential for generalization of the torque mea-surements performed in the laboratory for astrophysical orgeophysical flows. Indeed, all the additional complicationsstudied so far �aspect ratio, circulation, magnetic field, strati-fication, wide gap limit� have been found to shift the criticalReynolds number for linear stability by a factor function ofthis effect, like Rc�effect�0�=Rc�effect=0�F. Depending onthe situation, F can be interpreted as either a change in theeffective viscosity �magnetic field�, or a change in the effec-tive length scale �aspect ratio, wide gap�. If, on the other

hand, the scaling of the torque with Reynolds number �i.e.,the shear� remains unaffected by such a process, the compu-tations done in Sec. V E are easy to generalize through aneffective Reynolds number Re /F. Specifically, everythingthat has been said for the torque, in the ideal Taylor–Couetteexperiment, will still be valid with additional complicationprovided one replaces the Reynolds number by an effectiveReynolds number, taking into account the stability modifica-tion induced by this effect. This principle is by no meantrivial and must be used with caution, even though it mayappear as nothing more that an extension of the Reynoldssimilarity principle. In fact, it has been validated so far onlyin the case with vertical magnetic field, where it has beenindeed checked by Donnelly and Ozima34 that the torquescaling is unchanged by the magnetic field. In Ref. 1, weused this procedure in circumstellar disks, because we no-ticed that it gave the most sensible results. It would, how-ever, be important to check experimentally this “extendedReynolds similarity” principle.

1. Influence of boundary conditions

Experimental investigation of the Taylor–Couette flowwith different setups has shown that boundary conditionshave an influence on the torque. More precisely, it has beenshown that, with the outer cylinder at rest, the inclusion ofone53 or two54 rough boundaries increases the torque withrespect to the case with two smooth boundary conditions, athigh Reynolds numbers. In convective flows, a similar in-crease of transport properties is observed when changingfrom no-slip to stress-free boundary conditions.55 In bothcases, the increase occurs so as to improve the agreementbetween the observed value, and a value based on classicalKolmogorov theory. A theoretical study of Dubrulle56 ex-plains this feature through the existence or absence of loga-rithmic corrections �see formula �41b�� to scaling based onmolecular viscosity and large-scale velocity gradient in thevicinity of the boundary. Obviously, in the presence of a

FIG. 17. Influence of boundary conditions on torque. Case with two roughboundaries at �=0.724, �, data from Van den Berg et al.; at �=0.625, �,data from Cadot et al. The continuous lines are the formula �43�. Case withtwo smooth boundaries at �=0.724, �. Data from Lewis and Swinney. Thedotted and the dashed-dot lines are the formulas �41�.



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rough boundary, or under stress-free boundary conditions,mean large-scale velocity gradients are erased near theboundary, and no logarithmic correction develops.

For two rough boundary conditions, Cadot et al.54

measure Gi�0.22–0.3Re2 for �=0.625, while van den Berget al.53 observe Gi�0.43Re2 for �=0.73. The analogy withthermal convection12 suggests that Gi depends on its laminarvalue and on � and Re like

Gi

Glam= �rough

��3 + ��1 − ��1 + ��

�Re. �42�

Using �34�, and the experimental law, we find �rough=0.017,so that

Gi = 0.107��3 + ��

�1 + ��1 − ��3/2 ��Re�2. �43�

The comparison between this formula and the experiments ismade in Fig. 17. For reference, we also added the torque inthe case of two smooth boundary conditions, as given by�41b�.

We do not have any theory for the case with asymmetricboundary conditions �one rough, one smooth�. Laboratoryexperiments show that the torque lies in between the curvefor two smooth boundaries and the curve for two roughboundaries. The exact location, however, depends on localconditions in a nontrivial way �for example, it is differentwhen the rough condition applies to the �rotating� inner cyl-inder or to the �resting� outer cylinder�. The present experi-mental evidence therefore only allows the torque measure-ments with two smooth �respectively, two rough� boundaryconditions to be considered as lower �respectively, upper�bounds for the torque, in the case of complicated boundaryconditions.

2. Influence of radial circulation

Wendt measured the torque applied on the inner cylinder�including the bottom plate when it was attached to it� fordifferent gaps, and he showed that the circulation has aninfluence on the transport of angular momentum, even in thelaminar case. Specifically, he observed that an outward cir-culation at the no-slip bottom boundary ��0� increases thetorque, for given cylinder velocities, while an inward circu-lation ��0� decreases this torque. The difference can bequite important. At large Reynolds numbers, the relative in-crease of the torque G��0� /G��0� can be computed asa function of �. This is shown in Fig. 18. One observes aquasilinear variation for small gap:

Go�� 0�Go�� 0�

= 12.45�1 − ��, �i = 0. �44�

When applying intermediate boundary conditions �corre-sponding to � close to zero�, the torque lies about half-waybetween the two cases so that

Go�� 0�Go�� = 0�

= 4.75�1 − ��, �i = 0. �45�

Wendt explained this property by the fact that the rotationprofile near the bottom plate is modified such that the angu-lar momentum, at given radius r, is enhanced �respectively,reduced� when this plate is attached to the outer �inner� cyl-inder. The outward �inward� circulation near the bottom platethen transports more �less� angular momentum than the in-ward �outward� return flow near his upper free surface.

Coles and Van Atta10 pointed out that part of the torqueis applied to the end plates, which causes an imbalance be-tween the torques exerted on the cylinders only. Coles andVan Atta measured this imbalance as a function of theReynolds number for the case with inner cylinder at rest, for�=0.89, and with boundary conditions favoring an outward

FIG. 18. Influence of radial circulation on torque. �a� Go��0� /Go��0�as a function of � for �0 /�i=0 in the experiment of Wendt, at large Rey-nolds numbers. The squares are the data. The line is the fit Eq. �44�. �b�Ratio of torque applied to outer cylinder vs torque applied to the innercylinder in the case of an outward circulation, �=0.89.



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circulation. Their results are displayed in Fig. 18: one ob-serves an imbalance of the order of 30 to 50 percent, with thetorque on the inner cylinder being larger.

3. Influence of a vertical magnetic field

The influence of a constant vertical magnetic field on thetorque has been studied by Donnelly and Ozima.34 The mea-surements have been performed in the linear instability re-gime, with the outer cylinder at rest. It is observed that anincreasing magnetic field reduces the torque, so as to con-serve the Re3/2 scaling observed at zero magnetic field �Sec.V A�. The torque reduction is thus a function only of a non-dimensional magnetic field, and of �. Examples are providedin Fig. 19, for gap sizes 0.901 and 0.995 and Reynolds num-ber Re�2100.

The torque reduction can be quantified by the dimen-sionless number Q= �VA�2d2 /�m�, where VA is the Alfvénvelocity. It seems to follow a simple law:

Gi

Gi�Q = 0�=

e��1 + c��Q

, �46�

where e�� and c�� are functions of the gap size. Physically,this torque reduction may be due to the elongation of thecellular vortices which occurs as the magnetic field isincreased.34 Mathematically, the reduction can be understoodusing the connection between torque and critical number. Inthis framework, Chandrasekhar28 observes that the additionof a magnetic field onto a flow heated from below imparts tothe liquid an effective kinematic viscosity �Vz

Ad�2 /�m. Onlythe component of the field parallel to the gravity vector iseffective. This makes the critical Reynolds number for sta-bility proportional to 1+Q /Q0, where Q0 is a constant. Usingthe expression linking the torque and the critical Reynoldsnumber in the linearly unstable regime �Eqs. �35� and �37��,this leads to the scaling �46�.

G. Turbulent viscosity

The turbulent viscosity in the direction perpendicular to

the shear �t can be estimated via the mean torque T appliedby the fluid to the rotating cylinders and the mean turbulentvelocity profile. Indeed, this torque induces a stress equal to

T

rA=

��2

2�r2G = ��tr�ru�

r, �47�

where A is the area of a cylindrical fluid element at radius r,� is the fluid density, and u� is the mean azimuthal viscosity.Since a similar formula applies in the laminar case, with �t

=�, one simply gets

�t

�=

Slam

S

G

Glam. �48�

Using the expression of Re, RC and Glam �34�, we thus get thesimple expression

�t =1

2�RC

4 Gi�Re,��Re2 h�R�,��

Slam

SSr2 = �t

*Sr2. �49�

Here, we have expressed the turbulent viscosity in units ofthe typical shear and radius of the flow as �t

* �in Richard andZahn5 it was called ��. This nondimensional parameter en-compasses all the interesting variations of the turbulent vis-cosity as a function of the radial position r and the controlparameters Re, RC �or �� and R�. The radial variation is

given through the ratio Slam / S as illustrated in Sec. IV B.This ratio is 1 near the boundaries and it increases in the coreof the flow, due to the turbulent shear reduction. However, itstypical value varies only weakly with the Reynolds number,ranging from 1 in the laminar case to 4 in the limit of infiniteReynolds number �see Sec. IV A�. In fact, most of the varia-tion with Re is through Gi�Re ,�� /Re2, which can be deter-mined through torque measurements �Secs. V A and V B�,with a theoretical expression provided in Sec. V E forsmooth boundaries, and Sec. V F for rough boundaries. Allthe variations with R� are through the function h which hasbeen empirically determined in Sec. V E and plotted in Fig.16. The dependence on the curvature is subtler since it ap-pears in all the above dependencies.

An example of variation of the dimensionless turbulentviscosity �t

* with Reynolds number for �=0.72 is providedin Fig. 20 for smooth and rough boundary conditions. Onesees that at high enough Reynolds number, this function be-comes independent of the Reynolds number for roughboundary conditions, while it decreases steadily in thesmooth boundary cases, due to logarithmic corrections. Thisweak Reynolds number variation is in contrast with standardturbulent viscosity prescription, based on dimensional con-sideration à la Kolmogorov.

Finally, let us compare our results with previous resultsfor the turbulent viscosity in rotating flows. Using a turbulentclosure model of turbulence, Dubrulle57 derived �t=2

�10−3R�−2Sr2. This formula reflects the correct behavior in

terms of R� �see Sec. V C� but fails to reproduce theReynolds dependence in the case of smooth boundary con-

FIG. 19. Influence of vertical magnetic field on torque in the case with innercylinder rotating. Ratio of torque with magnetic field to the torque withoutmagnetic field in same experimental conditions, as a function of Q, thenondimensional magnetic number. The symbols are the data. The lines arethe fit �46�. Data are from Donnelly and Ozima. �: �=0.901, Re=2105; theconstants for the fit are b=0.803 and c=0.001; �: �=0.995, Re=2050; theconstants used for the fit are b=0.920 and c=0.002.



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ditions. For rough boundary conditions, our formula predicts

a turbulent viscosity going like �t=8�10−3R�−2Sr2 for R�

�−0.5 and �t=2�10−3R�−2Sr2 for R��0.5�10−4, in the

wide gap limit. The formula of Dubrulle is therefore in be-tween these two predictions. Richard and Zahn5 used Tay-lor’s measurements16 to derive the value �t

*=1.5±0.5�10−5. These measurements are performed for Re�5�104, with the inner cylinder at rest. At �=0.7, one has

R�=0.4 and from Fig. 16, h�0.4,0.7�=0.05. For Slam / S weadopt a value equal to 2, as suggested by Sec. IV �interme-diate value between 1 and 4, since we are at intermediateReynolds numbers�. Finally, from Fig. 20, we getRC

4Gi�Re ,0.7� /Re2=5�10−4 so that we finally obtain �t*=8

�10−6, an estimate close to the one proposed by Richard andZahn.

VI. CONCLUSION AND HINTS FOR FUTURE WORK

The present work enables us to derive a prescription forthe turbulent viscosity, and hence for the turbulent transportin the Taylor–Couette flow. This prediction clearly indicatesthe dependencies on the Reynolds number and the rotationnumber. The curvature effect is much more difficult to iso-late, at least with the available data, since it appears in all theterms of the prescription. Especially on the cyclonic side,where the Rayleigh criterion depends on the curvature, it isimpossible without any phenomenological arguments toseparate the curvature effect from the rotation one within theset of data used here. Since we wanted to remain as close aspossible to the existing data, we decided not to resort to anyphenomenological arguments in the present paper �for suchan analysis see, e.g., Longaretti and Dauchot11�.

The introduction of new control parameters, which arebased on the dynamical properties of the flow rather than onits geometry, allows us to envision some application of ourresult to rotating shear flows in general, even if one shouldremain cautious with the details of the boundary conditions.These new control parameters have a rather general basis,

but they remain global quantities. It would be interesting tofurther extend this approach, by introducing local dynamicalcontrol parameters, so that one could conduct a local study ofthe stability and transport properties.

In order to validate the above prescription, it is clearlynecessary to confront it to more experimental data. In theTaylor–Couette configuration, a number of additional mea-surements would be welcome. On the anti-cyclonic side, inthe subcritical regime, only one value of the curvature hasbeen investigated, so that we have very little idea of its in-fluence.

In the supercritical regime, we introduce an importantnew concept Rg

sup and relate it to RT and Rc. We propose tointerpret it as the critical Reynolds number for transition to ashear sustained turbulence, in the absence of linear stabilitymechanism. It would be important to test and fit our formulafor other values of the gap size. This concept could also betested in other general rotating shear flows.

Also, we have proposed to relate the torque measure-ments �a difficult experimental task� to the threshold deter-mination. These conjectures should be checked against moredata. Finally, we have tried to provide some indications onthe influence of external effects such as stratification, ormagnetic fields. Clearly the lack of experimental data here issuch that very little could be done and a definitive effortshould be conducted in this direction.

It would also be interesting to apply our findings to othergeometries involving rotating shear flows, so as to checkwhether the new control parameters we introduced can beused to define universality classes and draw general resultsabout stability and transport in these flows. In this regard, wemay already notice an interesting potential difference be-tween geometries where the shear is in a direction orthogonalto the rotation axis �Taylor–Couette flow, rotating planeCouette flow, rotating shear layer� and geometries where theshear is along the axis of rotation �von Kármán flow�. Ifuniversality classes are to be found, we believe they willnecessarily split according to this difference. This conjecturewould be worth testing.

Still, it is, to our knowledge, the first time that usingmost of the existing experimental studies a practical pre-scription for the turbulent viscosity is proposed. It can cer-tainly be improved, but we believe that, even at the presentlevel, it can already bring much insight into the understand-ing of some astrophysical and geophysical flows. As an ex-ample, this prescription has been applied to astrophysicaldisks �where the shear is indeed perpendicular to the rotationaxis� in Ref. 1. It leads to an estimate of the transport prop-erties in good agreement with observations.

ACKNOWLEDGMENTS

We would like to thank L. Marié, F. Hersant and F.Busse for fruitful discussions. D.R. is currently supported bya National Research Council Associateship.

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FIG. 20. Influence of Reynolds number on turbulent viscosity. Case withtwo rough boundaries at �=0.724, �, data from Van den Berg et al. Thecontinuous line is drawn using formula �43�. Case with two smooth bound-aries at �=0.724, �, data from Lewis and Swinney. The dotted and thedashed-dot lines are drawn using formulas �41�.



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