3.1 Azimuthal and angular velocity profiles

In this part the properties of flow profiles for varying Taylor number will be investigated. An example of an instantaneous flow field is shown in figure 3(a), while the averaged field is shown in figure 3(c). In figure 3(b) the normalised radial velocity $\tilde{u} _{r}=u_{r}/u_{{\it\theta}}(r_{i})$ versus the dimensionless gap position $\tilde{r}=(r-r_{i})/(r_{o}-r_{i})$ is shown, from which it is apparent that, at this specific axial position (mid-height), a significant mean radial flow still exists for the lower Taylor numbers up to approximately $10^{9}$ . These are Taylor vortices, which at $a=0$ disappear for higher Taylor numbers (Lathrop et al. Reference Lathrop, Fineberg and Swinney1992). With increasing Taylor number, the azimuthal velocity profile in the bulk becomes flatter (figure 3 d–f), but it is still apparent that the strong curvature of a radius ratio of ${\it\eta}=0.5$ creates a significant asymmetry between the inner and outer boundary layers.

In figure 4, profiles up to the middle of the gap are shown, normalised to the wall-normal distance $y^{+}$ from both the inner and outer cylinders, and to the azimuthal velocity $u^{+}$ (figure 4 a,b) and angular velocity ${\it\omega}^{+}$ (figure 4 c,d). The wall-normal distance, azimuthal velocity and angular velocity for the inner cylinder are defined as

The same parameters for the outer cylinder are defined as with the viscous length scale ${\it\delta}_{{\it\nu},i}={\it\nu}/u_{{\it\tau},i}$ and the friction velocity $u_{{\it\tau},i}=\sqrt{{\it\tau}_{w,i}/{\it\rho}}$ , containing the wall shear stress ${\it\tau}_{w,i}=T/2{\rm\pi}r_{i}^{2}L$ , with the torque $T$ , where the subscript $i$ may be replaced by $o$ . As a consequence, ${\it\delta}_{{\it\nu},i}/{\it\delta}_{{\it\nu},o}={\it\eta}$ , $u_{{\it\tau},i}/u_{{\it\tau},o}=1/{\it\eta}$ and ${\it\tau}_{w,i}/{\it\tau}_{w,o}=1/{\it\eta}^{2}$ . The torque values are from Merbold et al. (Reference Merbold, Brauckmann and Egbers2013), where a (non-transparent) TC set-up with identical dimensions and boundary conditions was used. The local torque can be different from the average torque, following the large-scale roll structures. This causes the imperfect matching with the viscous sublayer $u^{+}=y^{+}$ and ${\it\omega}^{+}=y^{+}$ , especially visible in figure 4(b,d).

As the driving is increased, the profiles slowly approach the Prandtl–von Kármán log law, although at these Taylor numbers the log layer is not yet fully developed. For comparison, profiles at higher Taylor numbers and aspect ratio of ${\it\eta}=0.716$ from Huisman et al. (Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013) are also plotted. The data show good agreement with direct numerical simulations (DNS) at the same radius ratio ${\it\eta}=0.5$ from Chouippe et al. (Reference Chouippe, Climent, Legendre and Gabillet2014) at $Ta=1.8\times 10^{8}$ , Ostilla-Mónico et al. (Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014c ) at $Ta=1.0\times 10^{9}$ and $Ta=1.0\times 10^{10}$ , and Ostilla-Mónico et al. (Reference Ostilla-Mónico, Verzicco, Grossmann and Lohse2016) at $Ta=1.1\times 10^{11}$ .

When comparing the azimuthal velocity $u^{+}$ to the angular velocity ${\it\omega}^{+}$ , i.e. figure 4(a) to 4(c) and figure 4(b) to 4(d), it can be seen that the angular velocity profiles curve upwards more compared to the azimuthal ones. This is consistent with the theoretical argument of Grossmann et al. (Reference Grossmann, Lohse and Sun2014) that the angular velocity profile ${\it\omega}^{+}$ is closer to a log law than the azimuthal velocity profile $u^{+}$ . The effect is much more pronounced for the smaller ${\it\eta}=0.5$ here as compared to the ${\it\eta}=0.716$ from Huisman et al. (Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013). Note that, for example, the lower-Taylor-number $u^{+}$ profiles at ${\it\eta}=0.716$ are above the profiles measured here, but that they cross when represented as ${\it\omega}^{+}$ . This clearly shows the influence of the large curvature for small-radius-ratio TC set-ups.

Figure 5. The scaling of the wind Reynolds number $Re_{w}$ and turbulence intensity $I$ with Taylor number for inner cylinder rotation $a=0$ . (a,b) The standard deviation of the radial velocity (a) and azimuthal velocity (b) over 10 000 frames (400 s) and the azimuthal direction, normalised by the inner cylinder velocity, as a function of the normalised radial position. This corresponds to the turbulence intensity $I$ in the case of the azimuthal velocity. See the legend in figures 3 or 4 for the values of the Taylor numbers. The three areas A ( $\tilde{r}\in [1/8,3/8]$ ), B ( $\tilde{r}\in [3/8,5/8]$ ) and C ( $\tilde{r}\in [5/8,7/8]$ ) indicate regions over which ${\it\sigma}$ is averaged for (c–e). The black dots represent the position of the start of the outer layers at $y^{+}=50$ . (c) Plot showing $Re_{w}$ versus $Ta$ averaged over $\tilde{r}$ values corresponding to regions A, B, C and all three combined ( $\tilde{r}\in [1/8,7/8]$ ). In addition, data from Huisman et al. (Reference Huisman, van Gils, Grossmann, Sun and Lohse2012) at ${\it\eta}=0.716$ with their fit are shown, which are fully in the ultimate turbulent regime. These data use the equivalent to area B for averaging. (d) Turbulence intensity $I$ versus $Ta$ averaged over the same regions as before. The fits are $I=0.27Ta^{-0.066}$ for area A, $I=0.33Ta^{-0.076}$ for area B, $I=0.41Ta^{-0.098}$ for area C and $I=0.32Ta^{-0.079}$ for the three areas combined. (e) Data in (c) compensated by $Ta^{3/7}$ . The fits are $Re_{w}=0.162Ta^{0.434}$ for area A, $Re_{w}=0.193Ta^{0.424}$ for area B, $Re_{w}=0.240Ta^{0.402}$ for area C and $Re_{w}=0.192Ta^{0.421}$ for the three areas combined. The standard errors of the fitted exponents are 0.007, 0.007, 0.005 and 0.005, respectively.

3.2 Wind Reynolds number and turbulence intensity

In the analogy between TC and Rayleigh–Bénard flow, the unifying theory of Grossmann & Lohse (Reference Grossmann and Lohse2000) predicts that the wind Reynolds number in the classical turbulent regime scales as

In contrast, in the ultimate turbulent regime, the scaling was predicted to be $Re_{w}\propto Ta^{1/2}$ (Grossmann & Lohse Reference Grossmann and Lohse2011), which was confirmed experimentally by Huisman et al. (Reference Huisman, van Gils, Grossmann, Sun and Lohse2012) for TC flow with ${\it\eta}=0.716$ for $Ta$ up to $6.2\times 10^{12}$ .

The highest Taylor number we achieve in the present measurements is $Ta=6.2\times 10^{9}$ , which for ${\it\eta}=0.5$ with the higher transitional Taylor number of $Ta=10^{10}$ (Merbold et al. Reference Merbold, Brauckmann and Egbers2013; Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014c ) implies that we are in the classical turbulent regime. Therefore we expect the theoretically predicted $3/7$ scaling to hold. This classical scaling was measured in Rayleigh–Bénard convection (He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012), but to our knowledge has not yet been confirmed for TC flow.

The wind Reynolds number $Re_{w}$ is extracted from the velocity field $u_{r}({\it\theta},r,z,t)$ by calculating the standard deviation ${\it\sigma}(u_{r}(r,{\it\theta},t))$ over time and the azimuthal direction, which is then averaged over a certain range in the gap (see figure 5). When looking at the shape of the $\tilde{r}$ dependence of ${\it\sigma}(u_{r})$ in figure 5(a), it is apparent that an asymmetry is present. The fluctuations exhibit a maximum around a quarter of the gap width away from the inner cylinder. It is likely that the strong curvature of the ${\it\eta}=0.5$ set-up causes this asymmetry. Calculating the wind Reynolds number thus comes with an arbitrariness in the area to choose for averaging the profile of ${\it\sigma}(r)$ , although excluding the boundary layers is reasonable because the interest lies in the convective bulk transport, as opposed to the diffusive transport in the boundary layers. The dependence of $Re_{w}(Ta)$ is shown in figure 5(c) for different regions of averaging. Region B lies within the outer layer of both the inner and outer cylinders for all Taylor numbers.

From figure 5(e) it can be seen that a clear power-law scaling is present, with an exponent that ranges from 0.402 to 0.434, depending on the region of averaging. The scaling in regions A, B and A + B + C is consistent with the theoretically predicted classical regime scaling of $3/7\approx 0.429$ (Grossmann & Lohse Reference Grossmann and Lohse2000) within the experimental uncertainties.

A similar analysis is performed on the azimuthal velocity instead of the radial velocity, as shown in figure 5(b,d). Again, the difference of the shearing velocity ( $u_{i}-u_{o}$ ), which in the case of $a=0$ simplifies to $u_{i}$ , is used to normalise the standard deviation. This results in a turbulence intensity $I={\it\sigma}(u_{{\it\theta}})/u_{i}$ , analogous to the turbulence intensity defined in Rayleigh–Bénard convection. The difference in rotation velocities of TC flow is analogous to the temperature difference in Rayleigh–Bénard convection (Grossmann & Lohse Reference Grossmann and Lohse2000). We find that the turbulence intensity has maxima in both the boundary layers, similar to the findings of Huisman et al. (Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013), and a relatively flat region in the bulk. The turbulence intensity $I(r)$ is averaged over the same regions as in figure 5(a) and shows negative scaling: $I\propto Ta^{{\it\gamma}}$ with $-0.1<{\it\gamma}<-0.07$ , depending on the region of averaging. This result is in rough agreement with the findings in Rayleigh–Bénard flow (Shang, Tong & Xia Reference Shang, Tong and Xia2008; Lakkaraju et al. Reference Lakkaraju, Stevens, Verzicco, Grossmann, Prosperetti, Sun and Lohse2012), where the authors found a scaling exponent of $-0.2<{\it\gamma}_{RB}<-0.1$ in the classical regime, depending on the Rayleigh number and the location of the temperature measurements. It is expected that ${\it\gamma}\rightarrow 0$ for $Re\rightarrow \infty$ , possibly explaining the higher exponent measured here because of a more turbulent system. Additionally, TC flow, with its curved boundaries, might have a fundamentally different scaling exponent.

Figure 6. Dependence of flow profiles on axial position for rotation ratios $a=0$ (a,b,e,f) and $a=0.2$ (c,d,g,h) for $Ta=4.2\times 10^{9}$ . The normalised radial velocity is $\tilde{u} _{r}=u_{r}(r)/(u_{i}-u_{o})$ and the azimuthal velocity is $\tilde{u} _{{\it\theta}}=(u_{{\it\theta}}(r)-u_{o})/(u_{i}-u_{o})$ . The data are represented in a colour map (a–d) and as profiles in (e–h).

3.3 Roll structures

The phenomena of roll structures, turbulent plumes and logarithmic velocity profiles in TC flow are intimately connected (Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ,Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse c ). We will first show an example of roll structures that are observed in TC flow. Flow profiles at several heights for both $a=0$ and $a=0.2$ are measured, allowing us to visualise roll structures as shown in figure 6. At this reasonably high Taylor number of $Ta=4.2\times 10^{9}$ , no clear roll structures exist for inner cylinder rotation only. The radial velocity shows some very faint indications of rolls, with a maximum value of $\tilde{u} _{r}=0.01$ in the centre of the gap. In contrast, in the case of counter-rotation ( $a=0.2$ ), very strong roll structures can be seen. The fact that there are strong rolls at optimal counter-rotation and no clear structures at inner cylinder rotation only is corroborated by recent work for higher radius ratio (Tokgoz et al. Reference Tokgoz, Elsinga, Delfos and Westerweel2011; Huisman et al. Reference Huisman, van der Veen, Sun and Lohse2014; Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ).

In the radial flow between two rolls at approximately $h/L=0.50$ to $h/L=0.52$ , there is very strong outwards radial flow (see figure 6 c). Specifically, the height $0.52L$ corresponds to the bottom of a roll, while $0.50L$ corresponds to the top of the roll below. Exactly in between the rolls at $0.51L$ , the radial flow surprisingly exhibits a local minimum. One might a priori expect symmetry around mid-height ( $0.50L$ ) instead of at $0.51L$ as observed. However, symmetry breaking of large-scale turbulent flow structures in symmetrical set-ups has been observed before, not only in TC flow (Huisman et al. Reference Huisman, van der Veen, Sun and Lohse2014; van der Veen et al. Reference van der Veen, Huisman, Dung, Tang, Sun and Lohse2016), but also in von Kármán flow (López-Caballero & Burguete Reference López-Caballero and Burguete2013). In figure 6(d,h) it can be seen that, in the case of positive (outward) radial flow, the inner cylinder velocity is advected outwards, shifting the profiles upwards at the inner half of the gap. For negative (inward) radial flow, the profiles shift downwards in the outer half of the gap. In summary, the roll structures are represented by a large secondary flow of $u_{r}$ , which advects velocity from the cylinders and changes the azimuthal velocity profiles. In the following two subsections we will elaborate on the characteristics of turbulent plumes and their effect on the velocity profiles.

3.4 Turbulent plumes

In turbulent TC flow, structures detach from the boundary layers, called herring-bone streaks (Dong Reference Dong2007) or velocity plumes (Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ), in correspondence to the thermal plumes in the analogous turbulent Rayleigh–Bénard convection. These plumes in TC flow are large-scale spatial and temporal fluctuations in the velocity fields that detach from either the inner or outer cylinder and advect velocity from the respective cylinder. Because of the sufficiently high frame rate of 25 f.p.s. and large amount of data recorded in our experiments, these plumes can be resolved in time, and the typical velocity can be extracted. We are interested in understanding how these plumes are affected by roll structures and whether some scaling relation exists for the typical plume velocity as a function of the driving parameter  $Ta$ .

Figure 7. Overview of turbulent burst velocity depending on axial position, for both pure inner cylinder rotation $a=0$ (a–f) and counter-rotation of $a=0.2$ (g–l), at $Ta=4.2\times 10^{9}$ . The radial velocity dependence is shown in (a,c,d,g,i,j) and the azimuthal velocity in (b,e,f,h,k,l). (a,b,g,h) Example of the radial and azimuthal velocity as a function of the normalised gap distance and time for the axial position of $0.50L$ . Only 10 s of the total 200 s per experiment are shown for clarity. Diagonal lines with a positive (negative) slope correspond to plumes coming from the inner (outer) cylinder. The extracted plume velocity (see text for method) is shown as a black line. (c,e,i,k) As above, for every height normalised by the total height of the cylinder $L$ . (d,f,j,l) The inverse of the resulting normalised burst velocity. For $a=0$ the inner cylinder velocity is $1.1~\text{m}~\text{s}^{-1}$ and for $a=0.2$ the inner and outer cylinder velocities are $0.91~\text{m}~\text{s}^{-1}$ and $-0.36~\text{m}~\text{s}^{-1}$ , respectively. The error bars show the width of the test function at 0.95 times the peak value.

The time dependence of the profiles of the azimuthal and radial velocity can be represented by the spatio-temporal fields $u_{r}(r,t)$ and $u_{{\it\theta}}(r,t)$ , with the value of the velocities represented by a colour map, as shown in figure 7(a), for example. In this way, structures or fluctuations of the profiles that propagate inwards or outwards become visible as diagonal lines. It has to be stressed that these diagonals do not represent a non-zero mean radial flow, but only the fluctuations on these profiles that travel outwards or inwards. In order to assign a velocity to these plumes, we employ an analysis that is based on applying the following affine shearing transformation to the functions $u_{r}(r,t)$ and $u_{{\it\theta}}(r,t)$ :

with $\tilde{r}=(r-r_{i})/d$ , gap width $d$ and plume velocity $v_{b}$ . For a certain amount of shearing, the plumes that were represented by diagonal lines become vertical lines. To find this optimal value of shearing, we postulate that, when averaging the sheared function over $\tilde{r}$ , the standard deviation of the resulting time signal has a maximum as a function of the shearing value. At this optimal shearing, the plumes are represented by narrow peaks in the time signal, creating a high standard deviation. Using a golden section search algorithm (Kiefer Reference Kiefer1953), the optimal shearing value is found, which corresponds to the characteristic plume velocity $v_{b}$ . Note that plumes can go both inwards and outwards and that this $v_{b}$ corresponds to the ‘strongest’ plume direction, not necessarily to the mean plume velocity.

The dependence of the plume velocity on the axial position and the rotation ratio is shown in figure 7. Again, for inner cylinder rotation only ( $a=0$ ), no height dependence can be seen. The fluctuations on the radial velocity are moving outwards with a velocity $v_{b}/(u_{i}-u_{o})=0.25\pm 0.02$ ( $u_{o}=0$ , $2{\it\sigma}$ error). When the azimuthal velocity is considered, the picture looks different. Crossing diagonal lines can be seen, meaning that plumes emit from both the inner and outer cylinders. A maximum was found for inward plumes with an average velocity of $v_{b}/(u_{i}-u_{o})=-0.20\pm 0.02$ . The velocity of these plumes is of the same order, but it remains to be explained why the main plume direction is different for the two velocity components.

For counter-rotation ( $a=0.2$ ), the bottom half of figure 7 clearly indicates that roll structures are present. There is a correspondence between the sign of the mean radial velocity and the direction of these turbulent plumes, coming from either the inner or outer cylinder – compare figure 6(c) with figure 7(i,k). For both velocity components, the direction of the plumes corresponds to the direction of the mean radial velocity. The mean radial velocity forces the plumes strongly in the inward or outward direction, also typically creating a higher plume velocity than for the $a=0$ case. Whereas for the $a=0$ case there was some ambiguity in the direction of the plumes for the azimuthal velocity, for $a=0.2$ it exactly follows the roll structure. When looking at the radial velocity, the inward plumes are less pronounced than the outward ones, likely due to a tendency for the $u_{r}$ plumes to flow outwards, as could be seen for the $a=0$ case.

Figure 8. Turbulent plumes in the radial velocity for varying $Ta$ and $a=0$ . (a,b) As in figure 7. (c) The plume velocity normalised by the inner cylinder velocity ( $u_{o}=0$ ). Errors are of the same order as in figure 7(d).

In addition to the dependence on the axial position and the rotation ratio, in figure 8 the dependence of the plume velocity on the driving parameter $Ta$ is shown. With increasing driving strength, the velocity of the plumes also increases. When the plume velocity is normalised by $(u_{i}-u_{o})$ , it is however nearly constant over two decades of  $Ta$ .

Figure 9. Angular velocity profile dependence on mean radial flow for counter-rotation $a=0.2$ and $Ta=4.2\times 10^{9}$ at varying axial positions (data from figure 6 h). (a) Angular velocity profiles near the inner cylinder ( $\tilde{r}\in [0,1/2]$ ). The angular velocity ${\it\omega}^{+}$ and wall-normal distance $y^{+}$ are defined in the caption of figure 4, the colours and line styles of the profiles correspond to those in figure 6. The grey area is the fitting domain $y^{+}\in [50,800]$ , chosen to encompass the start of the outer region up to mid-gap for both inner and outer cylinders. (b) Angular velocity profiles near the outer cylinder ( $\tilde{r}\in [1/2,1]$ ). (c) Normalised radial velocity $\tilde{u} _{r}$ as in figure 6(g). (d) The von Kármán constant ${\it\kappa}_{{\it\omega}}$ found by fitting ${\it\omega}^{+}=1/{\it\kappa}_{{\it\omega}}\ln y^{+}+B$ for $y^{+}\in [50,800]$ near the inner (IC) and outer (OC) cylinder. The horizontal axis shows the normalised wall-normal velocity, equal to $\tilde{u} _{r}$ and $-\tilde{u} _{r}$ for the IC and OC, respectively, averaged over $\tilde{r}\in [1/4,3/4]$ . The error bars have a total length of four times the standard error of the fit. (e) The standard error of the fitting parameter ${\it\kappa}_{{\it\omega}}$ .

3.5 Logarithmic velocity profiles

It has been proposed that the logarithmic temperature profiles in Rayleigh–Bénard convection and logarithmic velocity profiles in TC flow are triggered by plume ejection (Ahlers et al. Reference Ahlers, Bodenschatz and He2014; Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ; van der Poel et al. Reference van der Poel, Ostilla-Mónico, Verzicco, Grossmann and Lohse2015). These authors also support the notion that parts of the boundary layer can be turbulent, while others are not, and that the transition to the ultimate regime entails plume emission from the full extent of the boundary layer.

We will now further investigate the effect of the plumes on the logarithmic nature of the velocity profiles. As we have seen before, the roll structure that appears for counter-rotation of $a=0.2$ provides strong mean radial flows, which cause plume emission away from either the inner or outer cylinder.

In figure 9(a) the angular velocity at the inner cylinder is shown. It can be seen that there is a significant variation in the shape of the profiles, and that profiles at the axial positions where a strong positive radial flow exists (top lines in figure 9 c) follow the shape of a logarithmic law more closely. This effect is even stronger at the outer cylinder in figure 9(b), where profiles at an axial position with negative radial flow, so a flow away from the outer cylinder, correspond best to a log law.

To quantify this phenomenon, the velocity profiles are fitted with the modified (Grossmann et al. Reference Grossmann, Lohse and Sun2014) Prandtl–von Kármán law of the wall: ${\it\omega}^{+}=1/{\it\kappa}_{{\it\omega}}\ln y^{+}+B$ , with both ${\it\kappa}_{{\it\omega}}$ and $B$ as fitting parameters. We quantify the deviation from a log profile by the standard error of the fitting parameter ${\it\sigma}({\it\kappa}_{{\it\omega}})$ . A small value of ${\it\sigma}({\it\kappa}_{{\it\omega}})$ corresponds to a profile that closely follows a log law. In figure 9(d) it can be seen that, for a mean wall-normal velocity away from the cylinder (positive $\langle \tilde{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{n}\rangle$ ), the value of ${\it\kappa}_{{\it\omega}}$ more closely approaches the known classical value of 0.40. Additionally, the error of the fit is much smaller for positive values of the outward wall-normal velocity as compared to negative values. This means that, for regions with strong radial flow away from a cylinder surface, the velocity profiles follow a log law more closely. In the previous section it was found that, at these regions, plumes are emitted from the cylinder surface. We therefore postulate that turbulent plumes are responsible for logarithmic boundary layers. This is in line with results from DNS (van der Poel et al. Reference van der Poel, Ostilla-Mónico, Verzicco, Grossmann and Lohse2015; Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ).