Optimal Taylor-Couette turbulence

Abstract

Strongly turbulent Taylor-Couette flow with independently rotating inner and outer cylinders with a radius ratio of \eta = 0.716 is experimentally studied. From global torque measurements, we analyse the dimensionless angular velocity flux Nu_\omega(Ta, a) as a function of the Taylor number Ta and the angular velocity ratio a = -\omega_o/\omega_i in the large-Taylor-number regime 10^{11} \lesssim Ta \lesssim 10^{13}. We analyse the data with the common power-law ansatz for the dimensionless angular velocity transport flux Nu_\omega(Ta, a) = f(a)Ta^\gamma, with an amplitude f(a) and an exponent \gamma. The data are consistent with one effective exponent \gamma = 0.39\pm0.03 for all a. The
amplitude of the angular velocity flux f(a) = Nu_\omega(Ta, a)/Ta^0.39 is measured to be maximal at slight counter-rotation, namely at an angular velocity ratio of a_opt = 0.33\pm0.04. This value is theoretically interpreted as the result of a competition between the destabilizing inner cylinder rotation and the stabilizing but shear-enhancing outer cylinder counter-rotation. With the help of laser Doppler anemometry, we provide angular velocity profiles and identify the radial position r_n of the neutral line.
While for moderate counter-rotation -0.40 \omega_i \lesssim \omega_o < 0, the neutral line still remains close to the outer cylinder and the probability distribution function (p.d.f.) of the bulk angular velocity is observed to be monomodal. For stronger counter-rotation the neutral line is pushed inwards
towards the inner cylinder; in this regime the p.d.f. of the bulk angular velocity becomes bimodal, reflecting intermittent bursts of turbulent structures beyond the neutral line into the outer flow domain, which otherwise is stabilized by the counter-rotating outer cylinder. Finally, a hypothesis is offered allowing a unifying view for all these various results.

Full text

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Journal of Fluid Mechanics
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Optimal Taylor–Couette turbulence
Dennis P. M. van Gils, Sander G. Huisman, Siegfried Grossmann, Chao Sun and Detlef Lohse
Journal of Fluid Mechanics / Volume 706 / September 2012, pp 118 ­ 149
DOI: 10.1017/jfm.2012.236, Published online: 
Link to this article: http://journals.cambridge.org/abstract_S0022112012002364
How to cite this article:
Dennis P. M. van Gils, Sander G. Huisman, Siegfried Grossmann, Chao Sun and Detlef Lohse 
(2012). Optimal Taylor–Couette turbulence. Journal of Fluid Mechanics,706, pp 118­149 
doi:10.1017/jfm.2012.236
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J. Fluid Mech. (2012), vol. 706, pp. 118–149.
doi:10.1017/jfm.2012.236
c ? Cambridge University Press 2012
118
Optimal Taylor–Couette turbulence
Dennis P. M. van Gils1, Sander G. Huisman1, Siegfried Grossmann2,
Chao Sun1and Detlef Lohse1†
1Physics of Fluids Group, Faculty of Science and Technology, Impact and MESA+ Institutes, and
Burgers Center for Fluid Dynamics, University of Twente, 7500AE Enschede, The Netherlands
2Department of Physics, Renthof 6, University of Marburg, D-35032 Marburg, Germany
(Received 27 November 2011; revised 31 March 2012; accepted 15 May 2012;
first published online 3 July 2012)
Strongly turbulent Taylor–Couette flow with independently rotating inner and outer
cylinders with a radius ratio of η = 0.716 is experimentally studied. From global
torque measurements, we analyse the dimensionless angular velocity flux Nuω(Ta,a)
as a function of the Taylor number Ta and the angular velocity ratio a = −ωo/ωi in
the large-Taylor-number regime 1011? Ta ? 1013and well off the inviscid stability
borders (Rayleigh lines) a = −η2for co-rotation and a = ∞ for counter-rotation. We
analyse the data with the common power-law ansatz for the dimensionless angular
velocity transport flux Nuω(Ta,a) = f(a)Taγ, with an amplitude f(a) and an exponent
γ. The data are consistent with one effective exponent γ = 0.39 ± 0.03 for all a,
but we discuss a possible a dependence in the co- and weakly counter-rotating
regimes. The amplitude of the angular velocity flux f(a) ≡ Nuω(Ta,a)/Ta0.39is
measured to be maximal at slight counter-rotation, namely at an angular velocity
ratio of aopt= 0.33 ± 0.04, i.e. along the line ωo= −0.33ωi. This value is theoretically
interpreted as the result of a competition between the destabilizing inner cylinder
rotation and the stabilizing but shear-enhancing outer cylinder counter-rotation. With
the help of laser Doppler anemometry, we provide angular velocity profiles and in
particular identify the radial position rn of the neutral line, defined by ?ω(rn)?t= 0
for fixed height z. For these large Ta values, the ratio a ≈ 0.40, which is close to
aopt= 0.33, is distinguished by a zero angular velocity gradient ∂ω/∂r = 0 in the bulk.
While for moderate counter-rotation −0.40ωi? ωo< 0, the neutral line still remains
close to the outer cylinder and the probability distribution function of the bulk angular
velocity is observed to be monomodal. For stronger counter-rotation the neutral line is
pushed inwards towards the inner cylinder; in this regime the probability distribution
function of the bulk angular velocity becomes bimodal, reflecting intermittent bursts
of turbulent structures beyond the neutral line into the outer flow domain, which
otherwise is stabilized by the counter-rotating outer cylinder. Finally, a hypothesis is
offered allowing a unifying view and consistent interpretation for all these various
results.
Key words: rotating turbulence, Taylor–Couette flow, turbulent transition
1. Introduction
Taylor–Couette (TC) flow (the flow between two coaxial, independently rotating
cylinders) is, next to Rayleigh–B´ enard (RB) flow (the flow in a box heated from
†Email address for correspondence: d.lohse@utwente.nl.

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Optimal Taylor–Couette turbulence
119
below and cooled from above), the most prominent ‘Drosophila’ on which to test
hydrodynamic concepts for flows in closed containers. For outer cylinder rotation
and fixed inner cylinder, the flow is linearly stable. In contrast, for inner cylinder
rotation and fixed outer cylinder, the flow is linearly unstable thanks to the driving
centrifugal forces (see e.g. Taylor 1923; Coles 1965; DiPrima & Swinney 1981; Pfister
& Rehberg 1981; Mullin, Pfister & Lorenzen 1982; Smith & Townsend 1982; Mullin,
Cliffe & Pfister 1987; Pfister et al. 1988; Buchel et al. 1996; Esser & Grossmann
1996). The case of two independently rotating cylinders has been well analysed for
low Reynolds numbers (see e.g. Andereck, Liu & Swinney 1986). For large Reynolds
numbers, where the bulk flow is turbulent, studies have been scarce – see, for example,
the historical work by Wendt (1933) or the experiments by Andereck et al. (1986),
Richard (2001), Dubrulle et al. (2005), Borrero-Echeverry, Schatz & Tagg (2010),
Ravelet, Delfos & Westerweel (2010) and van Hout & Katz (2011). Ji et al. (2006)
and Burin, Schartman & Ji (2010) examined the local angular velocity flux with laser
Doppler anemometry in independently rotating cylinders at high Reynolds numbers
(to be defined below) up to 2 × 106. Recently, in two independent experiments, van
Gils et al. (2011b) and Paoletti & Lathrop (2011) supplied precise data for the global
torque scaling in the turbulent regime of the flow between independently rotating
cylinders.
We use cylindrical coordinates r,φ,z. Next to the geometric ratio η = ri/robetween
the inner cylinder radius ri and the outer cylinder radius ro, and the aspect ratio
Γ = L/d of the cell height L and the gap width d = ro− ri, the dimensionless control
parameters of the system can be expressed either in terms of the inner and outer
cylinder Reynolds numbers Rei= riωid/ν and Reo= roωod/ν, respectively, or in terms
of the ratio of the angular velocities
a = −ωo/ωi
(1.1)
and the Taylor number
Ta =1
4σ (ro− ri)2(ri+ ro)2(ωi− ωo)2ν−2.
(1.2)
Here, according to the theory by Eckhardt, Grossmann & Lohse (2007, from now on
called EGL), σ = [((1 + η)/2)/√η]4(thus σ = 1.057 for the current η = 0.716 of the
used TC facility) can be formally interpreted as a ‘geometrical’ Prandtl number and
ν is the kinematic viscosity of the fluid. With ra= (ri+ ro)/2 and rg=√riro, the
arithmetic and the geometric mean radii, the Taylor number can be written as
Ta = r6
ar−4
gd2ν−2(ωi− ωo)2.
(1.3)
The angular velocity of the inner cylinder ωi is always defined as positive, whereas
the angular velocity of the outer cylinder ωo can be either positive (co-rotation) or
negative (counter-rotation). Positive a thus refers to the counter-rotating case on which
our main focus will lie.
The response of the system is the degree of turbulence of the flow between the
cylinders (e.g. expressed in a wind Reynolds number of the flow, measuring the
amplitude of the r and z components of the velocity field) and the torque τ that is
necessary to keep the inner cylinder rotating at constant angular velocity. Following
the suggestion of EGL, the torque can be non-dimensionalized in terms of the laminar
torque to define the (dimensionless) ‘Nusselt number’
Nuω=
τ
2πLρfluidJω
lam
,
(1.4)

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D. P. M. van Gils, S. G. Huisman, S. Grossmann, C. Sun and D. Lohse
where ρfluidis the density of the fluid between the cylinders and
Jω
lam= 2νr2
ir2
o
ωi− ωo
r2
o− r2
i
(1.5)
is the conserved angular velocity flux in the laminar case. The reason for the choice
(1.4) is that
Jω= Jω
lamNuω= r3(?urω?A,t−ν∂r?ω?A,t)
(1.6)
is the relevant conserved transport quantity, representing the flux of angular velocity
from the inner to the outer cylinder. This definition of Jωis an immediate consequence
of the Navier–Stokes equations. (The authors would like to point out that (1.6)
appeared first, in a different notation, in Busse (1972), where Jωwas called the
‘torque’. Equations (3.4) and (4.13) of Eckhardt et al. (2007) are analogous to
equations (3.2) and (3.4) of Busse (1972).) Here ur (uφ) is the radial (azimuthal)
velocity, ω = uφ/r the angular velocity, and ?···?A,tcharacterizes averaging over time
and a cylindrical surface with constant radius r. With this choice of control and
response parameters, EGL could work out a close analogy between turbulent TC and
turbulent RB flow, building on Grossmann & Lohse (2000) and extending the earlier
work of Bradshaw (1969) and Dubrulle & Hersant (2002). This was further elaborated
by van Gils et al. (2011b).
The main findings of van Gils et al. (2011b), who operated the TC set-up, known
as the Twente turbulent Taylor–Couette system or T3C, at fixed η = 0.716 and for
Ta > 1011as well as the variable a well off the stability borders −η2and ∞,
are as follows: (i) in the (Ta,a) representation, Nuω(Ta,a) within the experimental
precision factorizes into Nuω(Ta,a) = f(a)F(Ta); (ii) F(Ta) = Ta0.38for all analysed
−0.4 ? a ? 2.0 in the turbulent regime; and (iii) f(a) = Nuω(Ta,a)/Ta0.38has a
pronounced maximum around aopt≈ 0.4. Also Paoletti & Lathrop (2011), at slightly
different η = 0.725, found such a maximum in f(a), namely at aopt≈ 0.35. For this
aopt the angular velocity transfer amplitude f(aopt(η)) for the transport from the inner
to the outer cylinder is maximal. From these findings one has to conclude that, for
not too strong counter-rotation −0.4ωi? ωo< 0, the angular velocity transport flux is
still further enhanced as compared to the case of fixed outer cylinder ωo= 0. This
stronger turbulence is attributed to the enhanced shear between the counter-rotating
cylinders. Only for strong enough counter-rotation ωo< −aoptωi (i.e. a > aopt) does
the stabilization through the counter-rotating outer cylinder take over and the transport
amplitude decrease with further increasing a.
The aims of this paper are to provide further and more precise data on the
maximum in the conserved turbulent angular velocity flux Nuω(Ta,a)/Taγ= f(a)
as a function of a and a theoretical interpretation of this maximum, including a
speculation on how it depends on η. We also put our findings in the perspective
of the earlier results on highly turbulent TC flow by Lathrop, Fineberg & Swinney
(1992a,b) and Lewis & Swinney (1999) and on recent results on highly turbulent
Rayleigh–B´ enard flow by He et al. (2012). We think that all these experiments
achieve the so-called ‘ultimate regime’ in which the boundary layers are already
turbulent. Next we provide laser Doppler anemometry (LDA) measurements of the
angular velocity profiles ?ω(r)?tas functions of height, and show that the flow close
to the maximum in f(a), for these asymptotic Ta and deep in the instability range
at a = aopt, has a vanishing angular velocity gradient ∂ω/∂r in the bulk of the
flow. We identify the location of the neutral line rn, defined by ?ω(rn)?t= 0 for
fixed height z, finding that it remains still close to the outer cylinder ro for weak

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Optimal Taylor–Couette turbulence
121
FIGURE 1. Sketch of the T3C Taylor–Couette cell employed for the measurements presented
here (from van Gils et al. 2011a). The total height of the cell is L = 92.7 cm. The torque
measurements are made with the middle part of the cell with length Lmid= 53.6 cm, in
order to minimize edge effects. The outer and inner cylinder radii are ro= 27.94 cm and
ri= 20.00 cm, leading to a radius ratio of η = 0.716, a gap width of d = ro− ri= 7.94 cm,
an aspect ratio of Γ = L/d = 11.68, and an internal fluid volume of 111 litres. The inner and
outer cylinder angular velocities are denoted by ωiand ωo, respectively. By definition, ωi> 0,
implying that ωo< 0 or a = −ωo/ωi> 0 represents counter-rotation, on which we focus in
this paper. The top and bottom plates are attached to the outer cylinder.
counter-rotation, 0 < a < aopt, but starts moving inwards towards the inner cylinder ri
for a ? aopt. Finally we show that the turbulent flow organization totally changes for
a ? aopt, where the stabilizing effect of the strong counter-rotation reduces the angular
velocity transport. In this strongly counter-rotating regime, the probability distribution
function of the angular velocity in the bulk becomes bimodal, reflecting intermittent
bursts of turbulent structures beyond the neutral line towards the outer flow region,
which otherwise, i.e. in between such bursts, is stabilized by the counter-rotating outer
cylinder.
The outline of the paper is as follows. The experimental set-up is introduced in §2
and we discuss, additionally, the height dependence of the flow profile and finite size
effects. The global torque results are reported and discussed in §3. Sections4 and 5
provide LDA measurements on the angular velocity radial profiles and on the turbulent
flow structures inside the TC gap. The paper ends, in §6, with a summary, further
discussions of the neutral line inside the flow, and an outlook.
2. Experimental set-up and discussion of end-effects
The core of our experimental set-up, the Taylor–Couette cell, is shown in figure 1.
In the caption we give the respective length scales and their ratios. In particular, the
fixed geometric dimensionless numbers are η = 0.716 and Γ = 11.68. The details
of the set-up are given in van Gils et al. (2011a). The working liquid is water
at a continuously controlled constant temperature (precision ±0.5 K) in the range

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122
19–26◦C. The accuracy in setting and maintaining a constant a is ±0.001 based
on direct angular velocity measurements of the T3C facility. To reduce edge effects,
similarly as in Lathrop et al. (1992a,b), the torque is measured at the middle part
(length ratio Lmid/L = 0.578) of the inner cylinder. Lathrop et al.’s (1992b) original
motivation for this choice was that the height of the remaining upper and lower
parts of the cylinder roughly equals the size of a pair of Taylor vortices. While the
respective first or last Taylor vortex indeed will be affected by the upper and lower
plates (which in our T3C cell are attached to the outer cylinder), the hope is that in
the strongly turbulent regime the turbulent bulk is not affected by such edge effects.
Note that for the laminar case (e.g. for pure outer cylinder rotation), this clearly
is not the case, as has been known since Taylor (1923) – see, for example, the
classical experiments by Coles & van Atta (1966), the numerical work by Hollerbach
& Fournier (2004), or the review by Tagg (1994). For such weakly rotating systems,
profile distortions from the plates propagate into the fluid and dominate the whole
laminar velocity field. The velocity profile will then be very different from the
classical height-independent laminar profile (see e.g. Landau & Lifshitz 1987) with
periodic boundary conditions in the vertical direction,
D. P. M. van Gils, S. G. Huisman, S. Grossmann, C. Sun and D. Lohse
uφ,lam= Ar + B/r,
A =ωo− η2ωi
1 − η2
,
B =(ωi− ωo)r2
1 − η2
i
.
(2.1)
To control edge effects and ensure that they are indeed negligible in the strongly
turbulent case under consideration here (1011< Ta < 1013and −0.40 ? a ? 2.0, so
well off the instability borders), we have measured time series of the angular velocity
ω(r,t) = uφ(r,t)/r for various heights 0.32 < z/L < 1 and radial positions ri< r < ro
with LDA. We employ a backscatter LDA configuration set-up with a measurement
volume of 0.07 mm × 0.07 mm × 0.3 mm. The seeding particles (PSP-5, Dantec
Dynamics) have a mean radius of rseed= 2.5 µm and a density of ρseed= 1.03 g cm−3.
We estimate the minimum velocity difference ?v = vseed− vfluid between a particle
vseed and its surrounding fluid vfluid needed for the drag force Fdrag= 6πµrseed?v
to outweigh the centrifugal force Fcent(r) = 4πrseed3(ρseed− ρfluid)v2/(3r). We put
in v = 5 m s−1as a typical azimuthal velocity inside the TC gap at mid-gap
radial position r = 0.24 m, with ρfluid = 1.00 g cm−3as the density and µ =
9.8 × 10−4kg m−1s−1as the dynamic viscosity of water at 21◦C. This results
in ?v = 2rseed2(ρseed− ρfluid)v2/(9µr) ≈ 5 × 10−6m s−1, which is several orders of
magnitude smaller than the typical velocity fluctuation inside the TC gap of order
10−1m s−1, and hence centrifugal forces on the seeding particles are negligible.
We account for the refraction due to the cylindrical interfaces – details are given by
Huisman, van Gils & Sun (2012b). Figure 2(a) shows the height dependence of the
time-averaged angular velocity at mid-gap, ˜ r = (r − ri)/(ro− ri) = 1/2, for a = 0 and
Ta = 1.5 × 1012, corresponding to Rei= 1.0 × 106and Reo= 0. The dashed-dotted line
at z/L = 0.79 corresponds to the gap between the middle part of the inner cylinder,
with which we measure the torque, and the upper part. Along the middle part, the
time-averaged angular velocity is z-independent within 1%, as is demonstrated in the
inset, showing the enlarged relevant section of the ω axis. From the upper edge of
the middle part of the inner cylinder towards the highest position that we can resolve,
0.5 mm below the top plate, the mean angular velocity decays by only 5%. This finite
difference might be due to the existence of Ekman layers near the top and bottom
plate (Greenspan 1990). Since at z/L = 1 we have ω(r,t) = 0, as the upper plate is
at rest for a = 0 or ωo= 0, 95% of the edge effects on ω occur in such a thin fluid

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Optimal Taylor–Couette turbulence
123
1%
(b)
(a) 1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0 0.1 0.20.3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.3300.3350.3400.345 0.350
0 0.005 0.010 0.0200.0150.025
Split IC
FIGURE 2. (Colour online) Time-averaged axial profiles of the azimuthal angular velocity
inside the T3C measured with LDA at mid-gap, i.e. ˜ r = (r − ri)/(ro− ri) = 0.5, for the case
Rei= 1.0 × 106and Reo= 0 (corresponding to a = 0 and Ta = 1.5 × 1012). The height z
from the bottom plate is normalized against the total height L of the inner volume of the tank.
(a) The time-averaged angular velocity ?ω(z, ˜ r = 1/2)?tnormalized by the angular velocity of
the inner cylinder wall ωi. (b) The standard deviation of the angular velocity σω(z) normalized
by the angular velocity of the inner wall. The split between the middle and the top inner
cylinder sections is indicated by the dashed-dotted line at z/L = 0.79. As can be appreciated
in this figure, the end-effects are negligible over the middle section where we measure the
global torques as reported in this work. The velocities near the top plate, z/L = 1, are not
sufficiently resolved to see the boundary layer.
layer near the top (bottom) plate that we cannot even resolve it with our present LDA
measurements.
For the angular velocity fluctuations shown in figure 2(b), we observe a 25% decay
in the upper 10% of the cylinder, but again in the measurement section of the inner
cylinder 0.2 < z/L < 0.8 there are no indications of any edge effects. The plots of
figure 2 together confirm that edge effects are unlikely to play a visible role for
our torque measurements in the middle part of the cylinder. Even the Taylor vortex
roll structure, which dominates TC flow at low Reynolds numbers (Dominguez-Lerma,
Ahlers & Cannell 1984; Andereck et al. 1986; Dominguez-Lerma, Cannell & Ahlers
1986; Tagg 1994), is not visible at all in the time-averaged angular velocity profile
?ω(z, ˜ r = 1/2,t)?t.
To double check that this z independence holds not only at mid-gap ˜ r = 1/2 but
also for the whole radial ω profiles, we measured time series of ω(z, ˜ r,t) at three
different heights z/L = 0.34, 0.50 and 0.66. The radial dependence of the mean value
and of the fluctuations are shown in figure 3. The profiles are basically identical for
the three heights, with the only exception of some small irregularity in the fluctuations
at z/L = 0.34 in the small region 0.1 < ˜ r < 0.2, whose origin is unclear to us. Note
that in both panels of figure 3 the radial inner and outer boundary layers are again not
resolved; in this paper we will focus on bulk properties and global scaling relations.
Based on the results of this section, we feel confident to claim that: (i) edge effects
are unimportant for the global torque measurements done with the middle part of the
inner cylinder reported in §3; and (ii) the local profile and fluctuation measurements

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D. P. M. van Gils, S. G. Huisman, S. Grossmann, C. Sun and D. Lohse
(b)
(a)
0.45
0.50
0.40
0.20
0.55
0.30
0.35
0.25
0.40.81.00 0.20.6
0.05
0.04
0.03
0.02
0.01
0.4 0.81.00 0.20.6
FIGURE 3. Radial profiles of the azimuthal angular velocity as presented in figure 2, scanned
at three different heights z/L = 0.66,0.50 and 0.34, plotted against the dimensionless gap
distance ˜ r = (r − ri)/(ro− ri), again for Reo= 0 and Rei= 1 × 106. (a) The time-averaged
angular velocity ?ω(˜ r)?tnormalized by the angular velocity of the inner wall ωi. All profiles
fall on top of each other, showing no axial dependence of the flow in the investigated axial
range. (b) Standard deviation of the angular velocity σω(˜ r) normalized by the angular velocity
of the inner wall. The velocity fluctuations show no significant axial dependence in the
investigated axial range. The boundary layers at the cylinder walls are not resolved.
done close to mid-height z/L = 0.44, which will be shown and analysed in §§4 and 5,
are representative for any height in the middle part of the cylinder.
3. Global torque measurements
In this section we will present our data from the global torque measurements
for independent inner and outer cylinder rotation, which complement and improve
the precision of our earlier measurements in van Gils et al. (2011b). The data as
functions of the respective pairs of control parameters (Ta,a) or (Rei,Reo) for which
we performed our measurements are given in tabular form in table 1 and in graphical
form in figures 4(a) and 5.
A three-dimensional overview of the found parameter dependences of the angular
velocity transport Nuω(Ta,a) is shown in figure 6. One immediately observes a
pronounced maximum in Nuω(Ta,a) with a considerable offset from the line a = 0.
A more detailed view is obtained in cross-sections through figure 6 and in particular
in compensated plots as shown in figure 7(a), where we divided Nuω by the
approximate effective scaling ∼Ta0.39. In this way we identify a universal effective
scaling Nuω(Ta,a) ∝ Ta0.39by averaging over the complete Ta range, ignoring Ta
dependence and thus calling the scaling effective. If each curve for each a is fitted
individually, the resulting Ta scaling exponents γ(a) scatter with a, but at most very
slightly depend on a; see figure 7(b,c). For different linear fits below and above
aopt= 0.33 (actually below and above abis= 0.368 or ψ = 0, as will be introduced
later on), we obtain γ = 0.378 + 0.028a ± 0.01 for a < aopt, the exponent slightly
decreasing towards less counter-rotation, and a constant exponent γ = 0.394 ± 0.006
for increasing counter-rotation beyond the optimum. The trend in the exponents for

Page 9

Optimal Taylor–Couette turbulence
125
a
ψ
Ta
Ta
ωi
ωi
ωo
ωo
Rei
Rei
Reo
Reo
G
G
Nuω
Nuω
γ(a)
f(a)
min
max
min
max
min
max
min
max
min
max
min
max
min
max
(deg.)
(1011)
(1011)
(rad s−1)
(rad s−1)
(rad s−1)
(rad s−1)
(105)
(105)
(105)
(105)
(109)
(109)
(10−3)
2.000
43.1
4.07
34.0
10.5
30.7
−61.5
−21.1
1.72
4.99
−13.95
−4.82
0.49
3.29
91
212
0.397
2.71
1.000
27.2
2.07
62.3
11.3
61.9
−61.9
−11.3
1.84
10.12
−14.15
−2.58
0.49
10.30
129
492
0.386
4.84
0.714
17.7
1.52
57.1
11.4
69.3
−49.5
−8.1
1.84
11.31
−11.29
−1.84
0.47
12.08
143
603
0.399
6.21
0.650
15.0
1.85
52.3
12.0
62.6
−40.7
−7.8
2.12
11.25
−10.21
−1.92
0.58
11.91
162
621
0.396
6.58
0.600
12.8
1.52
52.2
11.4
65.8
−39.5
−6.9
1.97
11.59
−9.71
−1.65
0.53
12.57
162
656
0.392
6.98
0.550
10.3
1.63
53.7
11.9
67.4
−37.1
−6.5
2.11
12.13
−9.33
−1.63
0.57
13.42
168
691
0.401
7.23
0.500
7.7
1.43
57.5
11.7
73.3
−36.7
−5.9
2.04
12.97
−9.06
−1.43
0.55
15.32
172
762
0.398
7.66
0.450
4.9
1.33
66.5
12.0
83.0
−37.4
−5.4
2.04
14.43
−9.08
−1.28
0.54
18.07
177
836
0.396
7.98
0.400
2.0
1.15
85.4
12.6
93.6
−37.5
−5.0
1.97
16.93
−9.47
−1.10
0.52
22.96
184
937
0.386
8.36
0.368
0.0
2.65
63.2
18.9
89.2
−32.8
−7.0
3.05
14.90
−7.67
−1.57
1.08
18.20
250
864
0.389
8.60
0.350
−1.2
1.16
64.6
12.3
90.1
−31.5
−4.3
2.04
15.28
−7.47
−1.00
0.52
18.67
181
876
0.391
8.61
0.300
−4.5
1.15
65.0
12.4
93.0
−27.9
−3.7
2.11
15.91
−6.67
−0.89
0.52
18.36
185
859
0.383
8.60
0.250
−8.0
1.07
63.4
13.0
97.5
−24.4
−3.2
2.12
16.35
−5.71
−0.74
0.48
17.35
177
822
0.381
8.37
0.200
−11.6
2.03
67.7
19.0
105.8
−21.2
−3.8
3.05
17.59
−4.91
−0.85
0.83
17.57
219
805
0.375
8.07
0.143
−15.9
2.27
69.4
22.1
112.5
−16.1
−3.2
3.38
18.70
−3.73
−0.68
0.83
17.21
208
779
0.386
7.69
0.100
−19.3
4.78
60.0
31.5
112.6
−11.3
−3.2
5.10
18.08
−2.52
−0.71
1.56
14.72
269
717
0.395
7.37
0.000
−27.2
1.34
61.7
18.2
124.0
0.0
0.0
2.97
20.15
0.00
0.00
0.49
13.73
158
660
0.375
6.81
−0.140
−38.3
1.90
37.4
25.5
112.4
3.6
15.7
4.11
18.25
0.80
3.57
0.55
7.07
151
436
0.364
5.76
−0.200
−42.8
2.08
29.6
28.6
107.7
5.7
21.5
4.62
17.44
1.29
4.87
0.49
5.10
128
354
0.392
5.00
−0.400
−56.4
4.87
22.2
58.3
124.3
23.3
49.8
9.44
20.16
5.28
11.27
0.47
1.68
80
135
0.358
2.21
TABLE 1. The measured global torque data for the individual cases of fixed a ≡ −ωo/ωi at increasing Ta, as presented in figure 5,
equivalent to straight lines in the (Reo, Rei) parameter space, as presented in figure 4(a). We list the minimum and maximum values of
the driving parameters (Ta, ωi, ωo, Reiand Reo) and we list the minimum and maximum values of the response parameters (dimensionless
torque G(Ta,a) and dimensionless angular velocity transport flux Nuω(Ta,a) = f(a)F(Ta)). The variable a can be transformed to the angle
ψ, i.e. the angle in (Reo, Rei) space between the straight line characterized by a and that characterized by abis, describing the angle bisector
of the instability range; see figure 4(a) and (3.5). The second to last column lists the effective scaling exponent γ(a) obtained from fitting
Nuω(Ta,a) to a least-squares linear fit in log–log space for each individual case of a, as presented in figure 7(b,c). The prefactor f(a), as
given in the last column, is determined by fixing γ(a) at its average encountered value of γ ≈ 0.39 and by averaging the compensated
NuωTa−0.39over Ta, as presented in figure 10.

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126
D. P. M. van Gils, S. G. Huisman, S. Grossmann, C. Sun and D. Lohse
(× 106)
(× 104)
(a)
2.0
1.5
1.0
0.5
0
–1.5 –1.0–0.500.51.0 1.5
(× 106)
–100 –50050 100
2.4
1.8
1.2
0.6
0
–1.0 –0.50 0.51.0
(× 104)
(b)(c)
Angle bisector E. & G.
Stab. bound. Rayleigh
Stab. bound. E. & G.
240
180
120
60
0
FIGURE 4. (a) Reynolds number phase space showing the explored regime of the T3C as
symbols with pale colours. The dotted green lines are the boundaries between the unstable
(upper left) and stable (lower right) flow region, shown here for the radius ratio η = 0.716 as
experimentally examined in this work. The green line in the right quadrant is the analytical
expression for the stability boundary as found by Esser & Grossmann (1996), which recovers
to the Rayleigh stability criterion Reo/Rei= η for Rei,Reo? 1, the viscous corrections
decreasing ∝ Re−2
Esser & Grossmann (Rei∝ Re3/5
is sufficient, if a is not too large, i.e. away from the stability curve. Similar to van Gils et al.
(2011b), we define the parameter a ≡ −ωo/ωi as the (negative) ratio between the angular
rotation rates of the outer and inner cylinders. We hypothesize maximum instability and hence
optimal turbulence on the bisector of the unstable region, indicated by the solid red line.
(b,c) Enlargements of the Re space at different scales showing the curvatures of the stability
boundaries and the corresponding bisector (red). Above Rei,Reo> 105the viscous deviation
from straight lines becomes negligible.
o. The green line in the left quadrant also follows the stability boundary by
o), but is taken here as Reo= 0. This inviscid approximation

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Optimal Taylor–Couette turbulence
127
Ta
a
0
1.0
0.5
–0.5
1.5
2.0
1013
1012
1011
1.000
0.714
0.650
0.600
0.550
0.500
0.450
0.400
0.368
0.350
0.300
0.250
0.200
0.143
0.100
0.000
–0.140
–0.200
–0.400
17.7
15.0
12.8
10.3
7.7
4.9
2.0
0
–1.2
–4.5
–8.0
–11.6
–15.9
–19.3
–27.2
–38.3
–42.8
–56.4
a
2.000
27.2
43.1
FIGURE 5. The probed (Ta,a) parameter space, equivalent to the (Rei,Reo) space shown
in figure 4. Each horizontal data line corresponds to a global torque measurement on the
middle section of the inner cylinder at different constant a (and hence ψ). The (blue) filled
circles correspond to local measurements on the angular velocity at fixed Ta and a as will be
discussed in §4.
Ta
1013
1012
1011
a
0
1.0
0.5
–0.5
1.5
2.0
800
600
400
200
1000
800
600
400
200
0
1000
0
FIGURE 6. Three-dimensional (interpolated and extrapolated) overview Nuω(Ta,a) of our
experimental results. The colour and the height correspond to the Nuωvalue.

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128
D. P. M. van Gils, S. G. Huisman, S. Grossmann, C. Sun and D. Lohse
(c)
(× 10–3)
Ta
a
2
1011
3
4
5
6
7
8
9
1013
1012
0.42
0.40
0.38
0.36
0.34
–0.5
0.42
0.40
0.38
0.36
0.34
0 0.5 1.02.0 2.5 1.5
20–20 40–40 60 –600
(b)(a)
FIGURE 7. (a) Plot of Nuω(Ta,a), compensated by Ta0.39, for various a as a function of Ta,
revealing effective universal scaling. The coloured symbols follow the same coding as given
in the legend of figure 5. The solid line has the predicted exponent from (3.1) (cf. Grossmann
& Lohse 2011) and can be shifted arbitrarily in the vertical direction. Here we have used
Rew= 0.0424Ta0.495as found by Huisman et al. (2012a) for the case a = 0 over the range
4 × 109< Ta < 6 × 1012, and the von K´ arm´ an constant ¯ κ = 0.4 and b = 0.4, resulting in a
predicted exponent of 0.395 (solid line in panels (b) and (c)). (b,c) The Nuω(Ta) exponent
for each of the individual line series, fitted by a least-squares linear fit in log–log space, is
plotted (b) versus a and (c) versus ψ. Assuming a independence, the average scaling exponent
is γ = 0.39 ± 0.03, which is very consistent with the effective exponent γ = 0.387 of the
first-order fit on log10(Nuω) versus log10(Ta) in the shown Ta regime.
a < aopt is small and compatible with a constant γ = 0.39 and a merely statistical
scatter of ±0.03. It is in this approximation that Nuω(Ta,a) factorizes.
3.1. Ultimate regime
Van Gils et al. (2011b) interpreted the effective scaling Nuω(Ta,a) ∼ Ta0.38, similar
to our currently obtained γ = 0.39 ± 0.03, as an indication of the so-called ‘ultimate
regime’ – distinguished by both turbulent bulk and turbulent boundary layers. Such
scaling was predicted by Grossmann & Lohse (2011) for very strongly driven RB
flow. As detailed in Grossmann & Lohse (2011), it emerges from a Nuω(Ta) ∼ Ta1/2
scaling with logarithmic corrections originating from the turbulent boundary layers.
Remarkably, the corresponding wind Reynolds number scaling in RB flow does not
have logarithmic corrections, i.e. Rew∼ Ta1/2. These RB scaling laws for the thermal
Nusselt number and the corresponding wind Reynolds number have been confirmed
experimentally by Ahlers, Funfschilling & Bodenschatz (2011) for Nu and by He et al.
(2012) for Rew. According to the EGL theory this should have its correspondence
in TC flow. That leads to the interpretation of γ = 0.39 as an indication for the
ultimate state in the presently considered TC flow. Furthermore, Huisman et al.
(2012a) indeed also found from particle image velocimetry (PIV) measurements in the
present strongly driven TC system the predicted (Grossmann & Lohse 2011) scaling of
the wind, Rew∝ Ta1/2.

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Optimal Taylor–Couette turbulence
129
0.25
0.30
0.35
0.40
0.45
0.50
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.15
0.20
1013
1011
1010
Ta
109
107
1013
1011
1010
Ta
109
107
Swinney & Lewis (1999)
1012
108
1012
108
(d)
(c)
(b)
(a)
FIGURE
d(log10Nuω)/d(log10Ta) for the case of inner cylinder rotation only (a = 0). The black
solid line is our experimental data and the dark grey (red) dashed line is data from
Lewis & Swinney (1999, figure 3), but now transformed into (Ta,Nuω) space. The local
exponent is calculated by using a sliding least-squares linear fit over different intervals:
(a) ?(log10Ta) = 0.2, (b) 0.4, (c) 1.0 and (d) 2.0. This method is similar to that used in
Lewis & Swinney (1999). Our data reveal a detailed local sensitivity of the scaling exponent
on small Ta intervals; see (a). When fitting over wider Ta intervals, an overall increasing
γ(Ta) with Ta becomes apparent; see (d).
8. (Colour online)Local
Ta-dependentscaling exponent
γ(Ta) =
We note that in our available Ta regime the effective scaling law Nuω∼ Ta0.39
is practically indistinguishable from the prediction of Grossmann & Lohse (2011),
namely,
Nuω∼ Ta1/2L(Rew(Ta)),
(3.1)
with the logarithmic corrections L(Rew(Ta)) detailed in equations (7) and (9) of
Grossmann & Lohse (2011). The result from (3.1) is shown as a solid line in
figure 7(a), showing the compensated plot Nuω/Ta0.39. Indeed, only detailed inspection
reveals that the theoretical line is not exactly horizontal.
Thus, strictly speaking, there is a Ta dependence of the scaling exponent γ(Ta),
as was clearly evidenced by Lathrop et al. (1992a) and Lewis & Swinney (1999)
in a much larger Ta range. In figure 8 we present our local γ(Ta) for the case of
a = 0.000 and we compare it to the data from Lewis & Swinney (1999). Similar
to Lewis & Swinney (1999) we calculate γ(Ta) = d(log10Nuω)/d(log10Ta) by using
a sliding least-squares linear fit over a certain ?(log10Ta) range, as indicated by
the top left corner of panels (a–d). The narrow averaging range used in figure 8(a)
results into a strongly fluctuating γ(Ta). The origin of these fluctuations may be

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130
D. P. M. van Gils, S. G. Huisman, S. Grossmann, C. Sun and D. Lohse
Ta
PIV exp.
PIV extrap. exp.
DNS
Torque exp.
Torque extrap. exp.
DNS
104
105
106
103
102
106
108
1010
1012
103
102
101
104
Ta
10–3
10–2
10–1
100
106
108
1010
1012
104
104
105
106
103
102
(b)(a)
FIGURE 9. (a) The shear Reynolds number Res, and (b) the coherence length ?coh, estimated
as 10 times the Kolmogorov length scale ηK, over the TC gap width d, both versus the driving
strength (Ta, lower abscissa; Rei, upper abscissa) for the case of pure inner cylinder rotation
(a = 0). (a) The solid black line results from the experimental data obtained by PIV (Huisman
et al. 2012a). The extrapolation of these data (dashed black line) towards smaller Ta nicely
agrees with the DNS data (grey/red crosses) of Ostilla et al. (2012). The (pale blue) shaded
area indicates the transitional regime from moderate turbulence at lower Ta (turbulent bulk
with laminar BLs) into ultimate turbulence at higher Ta (turbulent bulk with turbulent BLs).
(b) The solid black line is experimental data obtained by global torque measurements (this
work). The extrapolation of these data (dashed black line) towards smaller Ta agrees with the
DNS data (grey/red crosses) of Ostilla et al. (2012). The (pale yellow) shaded area indicates
the transitional regime where spatial coherence becomes small enough to allow for a turbulent
bulk beyond this regime.
different turbulent flow states (e.g. different number of Taylor vortices); future studies
should shed more light onto this. When averaging over a wider Ta range, our data
recover a monotonically increasing γ(Ta) trend, as can be seen in figure 8(d),
which is in line with Lathrop et al. (1992a) and Lewis & Swinney (1999). Clearly,
with their large-Ta measurements, these authors also already were in the ultimate
TC regime.
This gives rise to the following question: Where does the ultimate turbulence
regime set in for turbulent TC flow? To find out, we calculate the shear Reynolds
number Res= Usδ/ν, where δ is the thickness of the kinetic boundary layer, still
being of Prandtl type, and Us is the shear velocity across δ. The latter we estimate
as Us= Ui− Uw. Correspondingly, we estimate the kinetic Prandtl–Blasius type BL
thickness as δ = aPBd/√(Rei− Rew) (see e.g. Landau & Lifshitz 1987), with aPB set
to 2.3. This results in a shear Reynolds number of Res= aPB
wind Reynolds number we take our experimental result based on PIV measurements
(Huisman et al. 2012a), namely Rew= 0.0424Ta0.495(in the Ta regime from 3.8 × 109
to 6.2 × 1012, for a = 0). This implies that the relative contribution of the wind
Rew/Rei= Uw/Ui is only around 4.6% in this regime. Nonetheless, we take it
into consideration in figure 9(a), in which we plot Res versus Ta, retrieving the
effective scaling Res= 2.02Ta0.25. That figure also shows the result of Ostilla et al.
(2012) from direct numerical simulation (DNS), who found Rew= 0.0158Ta0.53in
the Ta regime from 4 × 104to 1 × 107. Again, also here the relative contribution
of the wind Rew/Rei is very small, namely around 3%. These numerical results give
√(Rei− Rew). For the

Page 15

Optimal Taylor–Couette turbulence
131
an effective scaling of Res= 2.05Ta0.25, very similar to our experimental findings,
even prefactor-wise.
The Prandtl–Blasius type BL becomes turbulent for a shear Reynolds number larger
than a critical shear Reynolds number or transition shear Reynolds number Res,T,
which is known to be in the range between 180 and 420 (see e.g. Landau & Lifshitz
1987). This range is shown as shaded in figure 9: all of our experimental data points
of this present paper (solid line) are beyond that onset. So, indeed, we are in the
ultimate regime. In contrast, the numerical data points by Ostilla et al. (2012) are in
the Prandtl–Blasius regime with laminar-type boundary layers.
The transition between these two regimes occurs in between. The range 180–420
for the transitional shear Reynolds number Res,T here (i.e. for the present η and
a = 0) corresponds to a range between 3 × 107and 109for the transitional Taylor
number TaT and to a range between 5 × 103and 2 × 104for the transitional (inner)
Reynolds number Rei,T. This corresponds to the transitional Reynolds number found
by Lewis & Swinney (1999, see their figure 3), in which the transition to the ultimate
regime is identified at a Reynolds number Rei,T= 1.3 × 104. Below that value Lewis
& Swinney (1999) find a very steep increase of the local slope dlogNuω/dlogReiwith
Rei; beyond the transition the increase is much less. (Here we have translated Lewis
& Swinney’s finding into the notation of this present paper.) We stress again that the
values given in this and the next subsection hold for a = 0. How the values of the
transitional Reynolds or Taylor number depend on a remains an important question for
future research.
Both in our experiment and in the experiments by Lewis & Swinney (1999) the
logarithmic corrections in (3.1) are visible and have the consequence that the ‘real’
ultimate scaling Nuω∼ Ta1/2is never achieved. As explained in Grossmann & Lohse
(2011) – and, differently and with a different result, much earlier in Kraichnan (1962)
– these logarithmic corrections are a consequence of the logarithmic velocity profile
in the turbulent boundary layers. Only by destroying these logarithmic profiles by
extreme wall roughness as done in TC experiments by van den Berg et al. (2003) or in
RB experiments by Roche et al. (2001) or by replacing the walls by periodic boundary
conditions (and a volume forcing) as done in numerical simulations by Lohse &
Toschi (2003), Calzavarini et al. (2005) and Schmidt et al. (2012) can one recover
the 1/2 scaling exponent, which is obtained in the strict upper bound of Doering &
Constantin (1994).
3.2. Comparison of Taylor–Couette turbulence with Rayleigh–Bénard turbulence
To get an idea of the extension of the non-ultimate turbulence regime in TC flow,
we also estimate the coherence length ?coh, below which the spatial coherence of
structures in the flow becomes small enough to allow for developed turbulence in
the flow. Typically, one estimates the coherence length as a multiple of the (mean)
Kolmogorov length scale ηK= ν3/4/?1/4, namely ?coh≈ 10ηK. The factor of 10 between
these two length scales is motivated by the transition between viscous subrange and
inertial subrange, which is known to happen at a scale around 10ηK (see e.g. Effinger
& Grossmann 1987). Here ? is the mean energy dissipation rate. That can be obtained
from the angular velocity flux Jω(see (4.7) of EGL), namely
? =2(ωi− ωo)Jω
r2
o− r2
i
=2(ωi− ωo)Jω
lamNuω
r2
o− r2
i
,
(3.2)