Optimal Taylor-Couette turbulence

Taylor-Couette (TC) flow (the flow between two coaxial, independently rotating cylinders) is, next to Rayleigh-Bénard (RB) flow (the flow in a box heated from 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); Pfister & Rehberg (1981); DiPrima & Swinney (1981); Smith & Townsend (1982); Mullin et al. (1982, 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 et al. (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 et al. (2010); Ravelet et al. (2010); van Hout & Katz (2011). Ji et al. (2006); Burin et al. (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 $2\times 10^{6}$. 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 $r,\phi$ and $z$. Next to the geometric ratio $\eta=r_{i}/r_{o}$ between the inner cylinder radius $r_{i}$ and the outer cylinder radius $r_{o}$, and the aspect ratio $\Gamma=L/d$ of the cell height $L$ and the gap width $d=r_{o}-r_{i}$, the dimensionless control parameters of the system can be expressed either in terms of the inner and outer cylinder Reynolds numbers $Re_{i}=r_{i}\omega_{i}d/\nu$ and $Re_{o}=r_{o}\omega_{o}d/\nu$, respectively, or in terms of the ratio of the angular velocities

and the Taylor number

Here, according to the theory by Eckhardt, Grossmann & Lohse (2007) (from now on called EGL), $\sigma=(((1+\eta)/2)/\sqrt{\eta})^{4}$ (thus $\sigma=1.057$ for the current $\eta=0.716$ of the used TC facility) can be formally interpreted as a ‘geometrical’ Prandtl number and $\nu$ is the kinematic viscosity of the fluid. With $r_{a}=(r_{i}+r_{o})/2$ and $r_{g}=\sqrt{r_{i}r_{o}}$, the arithmetic and the geometric mean radii, the Taylor number can be written as

The angular velocity of the inner cylinder $\omega_{i}$ is always defined as positive, whereas the angular velocity of the outer cylinder $\omega_{o}$ 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 $\tau$ 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’

where $\rho_{fluid}$ is the density of the fluid between the cylinders and

is the conserved angular velocity flux in the laminar case. The reason for the choice (4) is that

is the relevant conserved transport quantity, representing the flux of angular velocity from the inner to the outer cylinder. This definition of $J^{\omega}$ is an immediate consequence of the Navier-Stokes equations. (The authors would like to point out that (6) appeared first, in a different notation, in Busse (1972), where $J^{\omega}$ was called the ‘torque’. Equations (3.4) and (4.13) of Eckhardt et al. (2007) are analogous to (3.2) and (3.4) of Busse (1972).) Here $u_{r}$ ($u_{\phi}$) is the radial (azimuthal) velocity, $\omega=u_{\phi}/r$ the angular velocity, and $\left<\dots\right>_{A,t}$ characterizes 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 $\eta=0.716$ and for $Ta>10^{11}$ as well as the variable $a$ well off the stability borders $-\eta^{2}$ and $\infty$, are as follows: (i) in the $(Ta,a)$ representation, $Nu_{\omega}(Ta,a)$ within the experimental precision factorizes into $Nu_{\omega}(Ta,a)=f(a)F(Ta)$; (ii) $F(Ta)=Ta^{0.38}$ for all analysed $-0.4\leq a\leq 2.0$ in the turbulent regime; and (iii) $f(a)=Nu_{\omega}(Ta,a)/Ta^{0.38}$ has a pronounced maximum around $\mbox{$a_{opt}$}\approx 0.4$. Also Paoletti & Lathrop (2011), at slightly different $\eta=0.725$, found such a maximum in $f(a)$, namely at $\mbox{$a_{opt}$}\approx 0.35$. For this $a_{opt}$ the angular velocity transfer amplitude $f(\mbox{$a_{opt}$}(\eta))$ 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 $-0.4\omega_{i}\lesssim\omega_{o}<0$, the angular velocity transport flux is still further enhanced as compared to the case of fixed outer cylinder $\omega_{o}=0$. This stronger turbulence is attributed to the enhanced shear between the counter-rotating cylinders. Only for strong enough counter-rotation $\omega_{o}<-\mbox{$a_{opt}$}\omega_{i}$ (i.e. $a>\mbox{$a_{opt}$}$) 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 $Nu_{\omega}(Ta,a)/Ta^{\gamma}=f(a)$ as a function of $a$ and a theoretical interpretation of this maximum, including a speculation on how it depends on $\eta$. We also put our findings in the perspective of the earlier results on highly turbulent TC flow by Lathrop et al. (1992b, a) and Lewis & Swinney (1999) and on recent results on highly turbulent Rayleigh-Bénard 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 $\langle\omega(r)\rangle_{t}$ as 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 $a=\mbox{$a_{opt}$}$, has a vanishing angular velocity gradient $\partial\omega/\partial r$ in the bulk of the flow. We identify the location of the neutral line $r_{n}$, defined by $\langle\omega(r_{n})\rangle_{t}=0$ for fixed height $z$, finding that it remains still close to the outer cylinder $r_{o}$ for weak counter-rotation, $0<a<\mbox{$a_{opt}$}$, but starts moving inwards towards the inner cylinder $r_{i}$ for $a\gtrsim\mbox{$a_{opt}$}$. Finally we show that the turbulent flow organization totally changes for $a\gtrsim\mbox{$a_{opt}$}$, 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 section 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 section 3. Section 4 and 5 provide laser Doppler anemometry (LDA) measurements on the angular velocity radial profiles and on the turbulent flow structures inside the TC gap. The paper ends, in section 6, with a summary, further discussions of the neutral line inside the flow, and an outlook.

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 $\eta=0.716$ and $\Gamma=11.68$. 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 $\pm 0.5$K) in the range $19^{\circ}$C - $26^{\circ}$C. The accuracy in setting and maintaining a constant $a$ is $\pm 0.001$ based on direct angular velocity measurements of the T³C facility. To reduce edge effects, similarly as in Lathrop et al. (1992b, a), the torque is measured at the middle part (length ratio $L_{mid}/L=0.578$) of the inner cylinder. Lathrop et al. (1992b)’s 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,

To control edge effects and ensure that they are indeed negligible in the strongly turbulent case under consideration here ($10^{11}<Ta<10^{13}$ and $-0.40\leq a\leq 2.0$, so well off the instability borders), we have measured time series of the angular velocity $\omega({\mbox{\boldmath$r$}},t)=u_{\phi}({\mbox{\boldmath$r$}},t)/r$ for various heights $0.32<z/L<1$ and radial positions $r_{i}<r<r_{o}$ with laser Doppler anemometry (LDA). We employ a backscatter LDA configuration set-up with a measurement volume of 0.07 mm x 0.07 mm x 0.3 mm. The seeding particles (PSP-5, Dantec Dynamics) have a mean radius of $r_{seed}=2.5~\mu$m and a density of $\rho_{seed}=1.03$ g cm^-3.

We estimate the minimum velocity difference $\Delta v=v_{seed}-v_{fluid}$ between a particle $v_{seed}$ and its surrounding fluid $v_{fluid}$ needed for the drag force $F_{drag}=6\pi\mu r_{seed}\Delta v$ to outweigh the centrifugal force $F_{cent}(r)=4\pi{r_{seed}}^{3}\left(\rho_{seed}-\rho_{fluid}\right)v^{2}/(3r)$. We put in $v=5$ m s^-1 as a typical azimuthal velocity inside the TC gap at mid-gap radial position $r=0.24$ m, with $\rho_{fluid}=1.00$ g cm^-3 as the density and $\mu=9.8\times 10^{-4}$ kg m^-1 s^-1 as the dynamic viscosity of water at 21^∘C. This results in $\Delta v=2{r_{seed}}^{2}\left(\rho_{seed}-\rho_{fluid}\right)v^{2}/(9\mu r)\approx 5\times 10^{-6}$ m s^-1, which is several orders of magnitude smaller than the typical velocity fluctuation inside the TC-gap of order $10^{-1}$ m 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 et al. (2012b). Figure 2(a) shows the height dependence of the time-averaged angular velocity at mid-gap, $\tilde{r}=(r-r_{i})/(r_{o}-r_{i})=1/2$, for $a=0$ and $Ta=1.5\times 10^{12}$, corresponding to $Re_{i}=1.0\times 10^{6}$ and $Re_{o}=0$. 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 $\omega$ 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 $\omega(r,t)=0$, as the upper plate is at rest for $a=0$ or $\omega_{o}=0$, 95% of the edge effects on $\omega$ occur in such a thin fluid 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 et al. (1984, 1986); Andereck et al. (1986); Tagg (1994)), is not visible at all in the time-averaged angular velocity profile $\langle\omega(z,\tilde{r}=1/2,t)\rangle_{t}$.

To double check that this $z$ independence holds not only at mid-gap $\tilde{r}=1/2$ but also for the whole radial $\omega$ profiles, we measured time series of $\omega(z,\tilde{r},t)$ 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 $0.1<\tilde{r}<0.2$, 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 section 3; and (ii) the local profile and fluctuation measurements done close to mid-height $z/L=0.44$, which will be shown and analysed in sections 4 and 5, are representative for any height in the middle part of the cylinder.

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 ($Re_{i}$, $Re_{o}$) 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 $Nu_{\omega}(Ta,a)$ is shown in figure 6. One immediately observes a pronounced maximum in $Nu_{\omega}(Ta,a)$ 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 $Nu_{\omega}$ by the approximate effective scaling $\sim Ta^{0.39}$. In this way we identify a universal effective scaling $Nu_{\omega}(Ta,a)\propto Ta^{0.39}$ by 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 $\gamma(a)$ scatter with $a$, but at most very slightly depend on $a$; see figure 7(b, c). For different linear fits below and above $\mbox{$a_{opt}$}=0.33$ (actually below and above $a_{bis}=0.368$ or $\psi=0$, as will be introduced later on), we obtain $\gamma=0.378+0.028a\pm 0.01$ for $a<\mbox{$a_{opt}$}$, the exponent slightly decreasing towards less counter-rotation, and a constant exponent $\gamma=0.394\pm 0.006$ for increasing counter-rotation beyond the optimum. The trend in the exponents for $a<\mbox{$a_{opt}$}$ is small and compatible with a constant $\gamma=0.39$ and a merely statistical scatter of $\pm 0.03$. It is in this approximation that $Nu_{\omega}(Ta,a)$ factorizes.

We now wonder whether the distinguishing property of the flow at $a_{opt}$ (maximal angular velocity transport) is also reflected in other flow characteristics. We therefore performed LDA measurements of the angular velocity profiles in the bulk, close to mid-height, $z/L=0.44$, at various $-0.3\leq a\leq 2.0$: see table 3 for a list of all measurements, figure 12 for the mean profiles $\left<\omega(r,t)\right>_{t}$ at fixed height $z$, and figure 13 for the rescaled profiles $(\left<\omega(r,t)\right>_{t}-\omega_{o})/(\omega_{i}-\omega_{o})$, also at fixed height $z$. With our present LDA technique, we can only resolve the velocity in the radial range $0.04\leq\tilde{r}\leq 0.98$; there is no proper resolution in the inner and outer boundary layers. Because the flow close to the inner boundary region requires substantially more time to be probed with LDA, due to disturbing reflections of the measurement volume on the reflecting inner cylinder wall necessitating the use of more stringent Doppler burst criteria, we limit ourselves to the range $0.2\leq\tilde{r}\leq 0.98$.

From figure 12 it is seen that for nearly all co- and counter-rotating cases $-0.3\leq a\leq 2.0$ the slope of $\left<\omega(\tilde{r},t)\right>_{t}$ is negative. Only around $a=0.40$ do we find a zero mean angular velocity gradient in the bulk. This case is very close to $\mbox{$a_{opt}$}=0.33\pm 0.04$ and $a_{bis}=0.368$. The normalized angular velocity gradient as a function of $a$ is shown in figure 14(a). Indeed, it has a pronounced maximum and zero mean angular velocity gradient very close to $a=a_{bis}\approx\mbox{$a_{opt}$}$, the position of optimal angular velocity transfer. van Hout & Katz (2011), who performed PIV measurements of TC flow in air around $Ta\sim 10^{10}$, report a similar trend of a diminishing angular velocity gradient in the center of the TC gap for their investigated $a$ coming from $10.79$ down towards $0.70$, i.e. well in the counter-rotating regime. We speculate that their trend would continue and result in a diminishing slope of angular velocity when decreasing $a$ further to their (unreported) case of $a_{opt}$. Interestingly, the result of a zero angular velocity gradient across the gap is quite similar to what can be found in Taylor vortex flow, for which the axially averaged circumferential momentum or velocity is nearly uniform across the gap in the bulk (see e.g. Marcus (1984); Werne (1994)). We note that in strongly turbulent RB flow the temperature also has a (practically) zero mean gradient in the bulk – see, for example, the recent review by Ahlers et al. (2009).

A transition of the flow structure at $a=a_{bis}\approx\mbox{$a_{opt}$}$ can also be confirmed in figure 13(a), in which we have rescaled the mean angular velocity at fixed height as

We observe that up to $a=a_{bis}$ the curves for $\langle\tilde{\omega}(\tilde{r})\rangle_{t}$ for all $a$ go through the mid-gap point $(\tilde{r}=1/2,\langle\tilde{\omega}(1/2)\rangle_{t}\approx 0.35)$, implying the mid-gap value $\langle\omega(1/2)\rangle_{t}\approx 0.35\omega_{i}+0.65\omega_{o}$ for the time-averaged angular velocity. However, for $a>a_{bis}$, i.e. stronger counter-rotation, the angular velocity at mid-gap becomes larger, as seen in figure 13(b).

Figure 14(b) shows the relative contributions of the molecular and the turbulent transport to the total angular velocity flux $J^{\omega}$ (6), i.e. for both the diffusive and the advective term. The latter always dominates by far with values beyond 99%, but at $a\approx a_{bis}$ the advective term contributes 100% to the angular velocity flux and the diffusive term nothing, corresponding to the zero mean angular velocity gradient in the bulk at that $a$. This special situation perfectly resembles RB turbulence for which, due to the absence of a mean temperature gradient in the bulk, the whole heat transport is conveyed by the convective term. In the (here unresolved) kinetic boundary layers the contributions just reverse: The convective term strongly decreases if $\tilde{r}$ approaches the cylinder walls at 0 or 1 since $u_{r}\rightarrow 0$ ($u_{z}\rightarrow 0$ in RB), while the diffusive term (heat flux in RB) takes over at the same rate, as the total flux $J^{\omega}$ is an $\tilde{r}$-independent constant.

From the measurements presented in figure 12, we can extract the neutral line $\tilde{r}_{n}$, defined by $\left<\omega(\tilde{r}_{n},t)\right>_{t}=0$ at fixed height $z/L=0.44$ for the turbulent case. The neutral line is, of course, part of a neutral surface throughout the TC volume. We expect that, for large $a\gg\mbox{$a_{opt}$}$, the flow will be so much stabilized that an axial dependence of the location of the neutral line shows up, in spite of the large $Ta$ numbers, in contrast to the cases $a\approx\mbox{$a_{opt}$}$ where we expect the location of the neutral line to be axially independent for large enough $Ta$. The results for $\tilde{r}_{n}$ are shown in figure 15. Note that, while for $a\leq 0$ (co-rotation) there obviously is no neutral line at which $\langle\omega\rangle_{t}=0$, for $a>0$ a neutral line exists at some position $\tilde{r}_{n}>0$. As long as it is still within the outer kinematic BL, we cannot resolve it. This turns out to be the case for $0<a\lesssim{a_{bis}}$. But for ${a_{bis}}\lesssim a$ the neutral line can be observed within the bulk and is well resolved with our measurements. So, again, we see two regimes: for $0<a\lesssim a_{bis}$ in the laminar case the stabilizing outer cylinder rotation shifts the neutral line inwards, but, due to the now free boundary between the stable outer $r$ range and the unstable range between the neutral line and the inner cylinder, the flow structures extend beyond $\tilde{r}_{n}$. Thus also in the turbulent flow case the unstable range flow extends to the close vicinity of the outer cylinder. The increased shear and the strong turbulence activity originating from the inner cylinder rotation are too strong and prevent the neutral line being shifted off the outer kinematic BL. As described in Esser & Grossmann (1996), this is the very mechanism that shifts the minimum of the viscous instability curve to the left of the $Re_{i}=0$ axis. Therefore, the observed behaviour of the neutral line position as a function of $a$ is another confirmation of the above idea that $a_{opt}$ coincides with the angle bisector $a_{bis}$ of the instability range in parameter space. The small-$Re_{o}$ and the large-$Re_{o}$ behaviours perfectly merge. Only for $a>\mbox{$a_{opt}$}$ is the stabilizing effect from the outer cylinder rotation strong enough, and the width of the stabilized range broad enough, so that a neutral line $\tilde{r}_{n}$ can be detected in the bulk of the TC flow. This behaviour is similar to what is reported by van Hout & Katz (2011), see their figure 6.

One would expect that for much weaker turbulence $Ta\ll 10^{11}$, the capacity of the turbulence around the inner cylinder to push the neutral line outwards would decrease, leading to a smaller $a_{opt}$ for these smaller Taylor numbers. Numerical simulations by H. Brauckmann and B. Eckhardt of the University of Marburg ($Ta$ up to $1\times 10^{9}$) and independently ongoing DNS by Ostilla et al. (2012) of the University of Twente (presently $Ta$ up to $1\times 10^{8}$) seem to confirm this view.

For much weaker turbulence, one would also expect a more pronounced height dependence of the neutral line, which will be pushed outwards where the Taylor rolls are going outwards and inwards where they are going inwards. Based on our height dependence studies of section 2, we expect that this height dependence will be much weaker or even fully washed out in the strongly turbulent regime $Ta>10^{11}$ in which we operate the TC apparatus. However, in figure 15 we observe that the neutral line in the turbulent case lies more inside than in the laminar case, and this result is difficult to rationalize apart from assuming some axial dependence of the neutral line location, i.e. more outwards locations of the neutral line at larger and smaller height. In future work we will study the axial dependence of the neutral line in the turbulent and counter-rotating case in more detail.

For completeness, we also compare our experimental angular velocity profiles with those employed in the upper-bound theory by Busse (1972). Busse (1972) derived an expression for the angular velocity profiles in the limit of infinite Reynolds number. Translating that expression to the notation used in the present work gives

As already shown by Lewis & Swinney (1999), for the case $a=0$ excellent agreement between the Busse profile and the experimental one is found. However, for $a$ farther away from zero, there is a greater discrepancy between the experimental data and the profiles suggested by the upper-bound theory, as shown in figure 16(a). When we rescale the angular velocity profiles to $\tilde{\omega}$, according to (13), the profiles as given by the upper-bound theory (Busse, 1972) fall on top of each other for all $a$,

In contrast to the collapsing upper-bound profiles, the experimental data in figure 16(b) show a different trend. Clearly, the profiles suggested by the upper bound theory are in general not a good description of the physically realized profiles, apart from the $a=0$ case. Given the complexity of the flow, this may not be surprising.

While in this section we have only focused on the time-mean values of the angular velocity, in the following section we will give more details on the probability density functions (p.d.f.) in the two different regimes below and above $a_{bis}$ and thus on the different dynamics of the flow in these two different regimes.

Time series of the angular velocity at $\tilde{r}=0.60$ below the optimum amplitude at $a=0.35<a_{bis}=0.368$ (co-rotation dominates) and above the optimum at $a=0.60>a_{bis}=0.368$ (counter-rotation dominates) are shown in figure 17(a, b), respectively. While in the former case we always have $\omega(t)>0$ and a Gaussian distribution (see figure 18a), in the latter case we find a bimodal distribution with one mode fluctuating around a positive angular velocity and one mode fluctuating around a negative angular velocity. This bimodal distribution of $\omega(t)$ is confirmed in various p.d.f.s shown in figure 18. We interpret this intermittent behaviour of the time series as an indication of turbulent bursts originating from the turbulent region in the vicinity of the inner cylinder and penetrating into the stabilized region near the outer cylinder. We find such bimodal behaviour for all $a>a_{bis}$ (see figure 18d–i), whereas for $a<a_{bis}$ we find a unimodal behaviour (see figure 18a–c). Apart from one case ($a=0.50$) we do not find any long-time periodicity of the bursts in $\omega(t)$. In future work we will perform a full spectral analysis of long time series of $\omega(t)$ for various $a$ and $\tilde{r}$.