Turbulence strength in ultimate Taylor-Couette turbulence

We provide experimental measurements for the effective scaling of the Taylor-Reynolds number within the bulk $\text{Re}_{\lambda,\text{bulk}}$, based on local flow quantities as a function of the driving strength (expressed as the Taylor number Ta), in the ultimate regime of Taylor-Couette flow. The data are obtained through flow velocity field measurements using Particle Image Velocimetry (PIV). We estimate the value of the local dissipation rate $\epsilon(r)$ using the scaling of the second order velocity structure functions in the longitudinal and transverse direction within the inertial range---without invoking Taylor's hypothesis. We find an effective scaling of $\epsilon_{\text{bulk}} /(\nu^{3}d^{-4})\sim \text{Ta}^{1.40}$, (corresponding to $\text{Nu}_{\omega,\text{bulk}} \sim \text{Ta}^{0.40}$ for the dimensionless local angular velocity transfer), which is nearly the same as for the global energy dissipation rate obtained from both torque measurements ($\text{Nu}_{\omega} \sim \text{Ta}^{0.40}$) and Direct Numerical Simulations ($\text{Nu}_{\omega} \sim \text{Ta}^{0.38}$). The resulting Kolmogorov length scale is then found to scale as $\eta_{\text{bulk}}/d \sim \text{Ta}^{-0.35}$ and the turbulence intensity as $I_{\theta,\text{bulk}} \sim \text{Ta}^{-0.061}$. With both the local dissipation rate and the local fluctuations available we finally find that the Taylor-Reynolds number effectively scales as Re$_{\lambda,\text{bulk}}\sim \text{Ta}^{0.18}$ in the present parameter regime of $4.0 \times 10^8<\text{Ta}<9.0 \times 10^{10}$.


Introduction
Taylor-Couette (TC) flow, the flow between two coaxial co-or counter-rotating cylinders, is one of the idealized systems in which turbulent flows can be paradigmatically studied due to its simple geometry and its resulting accessibility through experiments, numerics, and theory. In its rich and vast parameter space, various different flow structures can be observed (Taylor 1923;Chandrasekhar 1981;Andereck et al. 1986; van Gils et al. 2011;Huisman et al. 2014;van der Veen et al. 2016a). For recent reviews, we refer the reader to Fardin et al. (2014) for the low Ta range and Grossmann et al. (2016) for large Ta.
The driving strength of the system is expressed through the Taylor number defined as where r i,o are the inner and outer radii, d = r o − r i the gap width, ω i,o the angular velocities of the inner and outer cylinders, ν the kinematic viscosity of the fluid, σ TC = (1 + ρ) 4 /(4ρ) 2 ≈ 1.06 a pseudo-Prandtl number employing the analogy with Rayleigh-Bénard (RB) flow (Eckhardt et al. 2007), and ρ = r i /r o the radius ratio. The response of the system is generally described by the two response parameters Nu ω and Re w . The first is the Nusselt number Nu ω = J ω /J ω,lam , with the angular velocity transfer J ω = r 3 (u r ω − ν∂ r ω) A,t , where A,t denotes averaging over a cylindrical surfaces of constant radius and over time. ω = u θ /r is the angular velocity and J ω,lam = 2ν(r i r o ) 2 (ω i − ω o )/(r 2 o − r 2 i ) is the angular velocity transfer from the inner to the outer cylinder for laminar flow. Nu ω describes the flux of angular velocity in the system, and is directly linked to the torque through the Navier-Stokes equations. The second response parameter of the flow is the so-called wind Reynolds number Re w = σ bulk (u r )d/ν, where σ bulk (u r ) is the standard deviation of the radial component of the velocity inside the bulk. Re w quantifies the strength of the secondary flows. In the ultimate regime of turbulence, where both the boundary layers (BL) and the bulk are turbulent (Ta 3 × 10 8 ), it was experimentally found that Nu ω ∼ Ta 0.40 , in the Taylor number regime of 10 9 to 10 13 , independent of the rotation ratio a = −ω o /ω i and radius ratio ρ ( van Gils et al. 2011;Huisman et al. 2014;Paoletti & Lathrop 2011;. This scaling has been identified, using the analogy with RB flow, with the ultimate scaling regime Nu ω ∼ Ta 1/2 L(Ta), where the log-corrections L(Ta) are due to the presence of the BLs. (Grossmann & Lohse 2011). The wind Reynolds number Re w was found experimentally to scale as Re w ∼ Ta 0.495 within the bulk flow ; very close to the 1/2 exponent that was theoretically predicted by Grossmann & Lohse (2011). Here, remarkably, the log-corrections cancel out.
In this study we characterize the local response of the flow with an alternate response parameter based on the standard deviation of the azimuthal velocity σ(u θ ) and the microscales of the turbulence, i.e. the Taylor-Reynolds number which is defined as Re λ = u ′ λ/ν, where u ′ is the rms of the velocity fluctuations and λ is the Taylor micro-scale.
Re λ is often used in the literature to quantify the level of turbulence in a given flow, ideally for homogeneous and isotropic turbulence (HIT), where it should be calculated from the full 3D velocity field. In experiments however, the entire flow field is generally not accessible. Assuming isotropy (which is most of the time not strictly fulfilled), the dissipation rate ǫ (in Cartesian coordinates) can be reduced to ǫ = 15ν (∂u/∂x) 2 t , where u is the component of the velocity in the streamline direction x. In this way, the Taylor micro-scale is then redefined as λ = u 2 / (∂u/∂x) 2 . Examples where this procedure has been followed in spite of the lack for perfect isotropy include turbulent RB flow (Zhou et al. 2008), the flow between counter-rotating disks (Voth et al. 2002), von Kármán flow (Zimmermann et al. 2010), or channel flow (Martínez Mercado et al. 2012). In all cases the isotropic form of Re λ is still chosen as a robust way to quantify the strength of the turbulence. It is in this spirit that we aim to calculate Re λ in turbulent Taylor-Couette flow, albeit in a region sufficiently far away from the BLs (bulk). Such a calculation allows for a quantitative comparison between the turbulence generated in TC flow and the one produced by other canonical flows, i.e. pipe, channel, RB, von Kármán flow, etc. Following this route, we define the bulk Taylor-Reynolds number for TC flow as where σ θ,t (u θ (r, θ, t)) is the standard deviation of the azimuthal velocity in the azimuthal direction and over time. σ bulk (u θ ) is then the average of the azimuthal velocity fluctuations profile over the bulk and ǫ bulk the bulk-averaged dissipation rate. Note that the subscript r bulk means that we average in the radial direction but only for 0.35 < (r − r i )/d < 0.65, i.e. the middle 30% of the gap (see also §3.1).
Multiple prior estimates of Re λ in TC flow can be found in the literature: Huisman et al. (2013) calculated it using a combination of the local velocity fluctuations and the global energy dissipation rate ǫ global , where the latter is obtained from torque measurements denoted by τ through ǫ = τ ω i /m, where m is the total mass. Lewis & Swinney (1999), however, estimated Re λ at midgap (r = (r − r i )/d = 0.5) with the local velocity fluctuations and a local dissipation rate estimated indirectly through the velocity spectrum E(k) in wave number space k, i.e. ǫ = 15ν k 2 E(k)dk. In this calculation, Taylor's frozen flow hypothesis was used to get the θ-dependence for the azimuthal velocity u θ , i.e. u(θ + dθ, t) = u(θ, t − rdθ/U ), where U is the mean azimuthal velocity. To the best of our knowledge, however, a truly bulk-averaged calculation of Re λ,bulk (based on local quantities) has hitherto never been reported in the literature. Of particular interest is how this quantity scales with Ta in the ultimate regime, and how this scaling is connected to that of Nu ω and Re w .
As TC flow is a closed flow system, the global energy dissipation rate ǫ global is connected to both the driving strength Ta and Nu ω by (Eckhardt et al. 2007) In the ultimate regime this implies an effective scaling of the global energy dissipation rateǫ global ∼ Ta 1.40 . A calculation of Re λ in the bulk does not require the global energy dissipation rateǫ global , but the bulk-averaged energy dissipation rate, ǫ bulk in combination with the bulk averaged velocity fluctuations σ bulk (u θ ), see equation (1.2). In general, velocimetry techniques like Particle Image Velocimetry (PIV) can provide σ bulk (u θ ) directly, thus the challenge of the calculation is to correctly estimate ǫ bulk . While the global energy dissipation rate ǫ global (equation (1.4)) can be obtained from torque measurements, an estimate of ǫ bulk requires the knowledge of the local dissipation rate ǫ(r, θ, t) as it is shown in equation (1.3). For fixed height along the cylinders, the dissipation rate profile ǫ(r) = ǫ(r, θ, t) θ,t is connected to the global energy dissipation rate through ǫ global = (π(r 2 o − r 2 i )) −1 ro ri ǫ(r)2πrdr. We note that due to the non-trivial interplay between bulk and turbulent BLs in the ultimate regime, it is not known a priori that ǫ bulk and ǫ global will scale in the same way: local measurements are needed to confirm this assumption.
The energy dissipation rate ǫ is key for Kolmogorov's scaling prediction of the velocity structure functions (SFs) in HIT, namely D LL (s) = C 2 (ǫs) 2/3 for the second order longitudinal structure function and D N N (s) = C 2 (4/3)(ǫs) 2/3 for the second order transverse structure function within the inertial range, neglecting intermittency corrections (Pope 2000;Frisch 1995). The Kolmogorov constant was measured to be C 2 ≈ 2.0 and is believed to be universal (Sreenivasan 1995). The exponents for the scaling of the p-th order SFs (ζ ⋆ p ) have been measured and found to differ from Kolmogorov's original prediction p/3: the difference between them are attributed to the intermittency of the flow (Benzi et al. 1993;She & Leveque 1994;Lewis & Swinney 1999;Huisman et al. 2013). However, second order SFs along with the classical Kolmogorov scaling ζ 2 = 2/3 have been successfully used to estimate ǫ in fully developed turbulence (Voth et al. 2002;Blum et al. 2010;Zimmermann et al. 2010;Chien et al. 2013). One can then expect only a moderate underestimation of ǫ since the intermittency correction to the exponent of the second order SFs is small ζ ⋆ 2 − 2/3 ≈ 0.03, where ζ ⋆ 2 is the measured exponent of the second order SFs in TC flow using extended self-similarity (ESS) (Lewis & Swinney 1999;Huisman et al. 2013).
In this paper we make use of local flow measurements using planar Particle Image Velocimetry (PIV) to find σ bulk (u θ ) and using the scaling of the second order (p = 2) SFs we estimate ǫ bulk . The advantage of PIV over other flow measuring technique such as Laser-Doppler or Hot-wire anemometry is the possibility to access the whole velocity field at the same time in the r − θ plane, i.e. u = u r (r, θ, t)ê r + u θ (r, θ, t)ê θ , from which we can obtain directly the θ-dependence of the velocities. Unlike in the calculation of Lewis & Swinney (1999) and Huisman et al. (2013), in this work, we do not need to invoke Taylor's hypothesis in the calculation of Re λ,bulk . We only explore the case of inner cylinder rotation (a = 0), where there is virtually no stable structures (Taylor rolls) left when the driving strength is sufficiently large (Ta 10 8 ) (Huisman et al. 2014). In this way, the calculation is independent of the axial height z and thus there is no need for an axial average (van Gils et al. 2012).

Experimental apparatus
The PIV experiments were performed in the Taylor-Couette apparatus as described in Huisman et al. (2015). This facility provides an optimal environment for PIV experiments in TC flow, due to its transparent outer cylinder and top plate. The radii of the setup are r i = 75 mm and r o = 105 mm, and thus ρ = r i /r o = 0.714, which is very close to ρ = 0.724 and ρ = 0.716 from Lewis & Swinney (1999) and Huisman et al. (2013), respectively. The height ℓ equals 549 mm, resulting in an aspect ratio Γ = ℓ/d = 18.3. The excellent temperature control of the setup allows us to perform all the experiments at a constant temperature of 26.0 • C with a standard deviation of 15 mK. The measurements are done at midheight z = ℓ/2 in the r − θ plane. The flow is seeded with fluorescent polyamide particles with diameters up to 20 µm and with an average particle density of ≈ 0.01 particles/px. The laser sheet we use for illumination is provided by a pulsed laser (Quantel Evergreen 145 laser, 532 nm) and has a thickness of ≈ 2.0 mm. The measurements are recorded using a high-resolution camera at a framerate of f = 1 Hz. The camera we use is an Imager sCMOS (2560 px × 2160 px) 16 bit with a Carl Zeiss Milvus 2.0/100. The camera is operated in double frame mode which leads to an interframe-time ∆t ≪ 1/f . In figure 1a a schematic of the experimental setup is shown. In order to obtain a large amount of statistics, we capture 1500 fields for each of the 12 different Taylor numbers explored. The velocity fields are calculated using a "multi-pass" method with a starting window size of 64 px × 64 px to a final size of 24 px × 24 px with 50% overlap. This allows us to obtain a resolution of dx = 0.01d. When using the local Kolmogorov length scale in the flow (see §3.3), we find that dx/η bulk ranges from ≈ 1.6 (Ta = 4.0 × 10 8 ) to ≈ 10 (Ta = 9.0 × 10 10 ).

Identifying the bulk region
The profiles of the velocity fluctuations for both components of the velocity as a function of Ta are shown in figure 2a. The distance from the inner cylinder is represented by the normalized radiusr = (r − r i )/d. When normalized with the velocity of the inner cylinder r i ω i , both profiles collapse for all Ta numbers in most of the gap width around the value of 0.03. Only very close the inner and outer cylinder, the fluctuations increase (decrease) for the azimuthal (radial) component. In our calculation of Re λ,bulk (equation (1.1)), we use σ bulk (u θ ) as our velocity scale as u θ is the primary flow direction. Here, we are essentially assuming that the radial and axial velocity fluctuations, on average, have the same order of magnitude, i.e. σ bulk (u θ ) ≈ σ bulk (u r ) (the result is zindependent). In order to give an impression of how valid this assumption is, in figure 2b we show the ratio of the velocity fluctuations throughout the gap. We notice that within the bulk region, the ratio is between 1.0 and 1.6 for all analyzed Ta numbers; consistent with what one would expect for reasonably isotropic flows. Surprisingly, the ratio within the bulk increasingly deviates from unity as the driving is increased. The same observation is also observed in turbulent TC-flow (Ta ∈ [5.8 × 10 7 , 6.2 × 10 9 ]) for a wider gap η = 0.5, where also the ratio within the bulk increasingly deviates from unity with increasing Ta. In that case however, it seems to reach a value of ≈ 1.8 for the largest Ta (van der Veen et al. 2016b). Since the same observation is found in two different studies (with two different experimental setups), we believe this is a feature of TC-flow; however, a more rigorous theoretical explanation has yet to be provided. Another interesting feature of the profiles in figure 2b is that they become flatter as the turbulence level is increased, reflecting an increase in spatial homogeneity. Note that these results do not suggest readily that the flow is in a HIT state. What this merely shows is that there is a special region (bulk) where the flow becomes more homogeneous as compared to regions close to the solid boundaries and it is reasonably isotropic. This justifies that our calculation is based on an isotropic form of Re λ as was also used in other studies (Lewis & Swinney 1999;Voth et al. 2002;Zhou et al. 2008;Zimmermann et al. 2010;Martínez Mercado et al. 2012).
Next, we define the bulk region as r bulk ≡ r−r i ∈ [0.35d, 0.65d], wherein the magnitude of the velocity fluctuations for both u r and u θ are roughly constant. This definition of the bulk was previously used by Huisman et al. (2012) who measured the scaling of Re w in the ultimate regime. The same definition is also consistent with other studies (Smith & Townsend 1982;Lewis & Swinney 1999), where the bulk region is identified as the r domain wherein the normalized specific angular momentum remains constant (L θ = r u θ θ,t /(r 2 i ω i ) ≈ 0.5) for all Ta. In figure 2c we showL θ (r) and we find a good collapse of the profiles within our definition of the bulk. Here, it is seen that the value of L θ is indeed around 0.5 within the bulk.
binning guarantees a constant spatial resolution ds = rdθ along the direction of s, when the radial variable r is fixed (see the sketch in figure 1b). The choice of ds is limited by the resolution of the PIV experiments dx and it is chosen such as to not filter out any intermittent fluctuations in the flow. The energy dissipation rate profiles for both directions are calculated as follows. For fixed r and Ta, ǫ LL is chosen as the maximum of s −1 (δ LL (r, s)/C 2 ) −2/3 such that s lies inside the inertial range. In the same manner, ǫ N N is taken as the maximum of s −1 (δ N N (r, s)/(4C 2 /3)) −2/3 with the same restriction for s. This operation is repeated for every r and Ta, leading to the dissipation rate profiles shown in figure 3. In this figure, the ǫ-profiles are made dimensionless asǫ(r) = ǫ(r)/(d −4 ν 3 ). Near the solid boundaries, this figure shows that the dissipation rates (LL and NN) differ from each other: ǫ LL increases while ǫ N N decreases, which is consistent with the measurement of the velocity fluctuations (figures 2ab). However, as one moves into the bulk region, the discrepancy between them decreases until eventually both dissipation rates intersect. The crossing remains within the bulk region, independent of Ta, and does not seem to occur at any particular radial position. Only in the case of HIT, the dissipation rates obtained from both SFs are exactly the same. However, as indicated in figures 2ab, the flow tends to be more homogeneous within the bulk. We expect then that, regardless of the structure function (longitudinal or transverse) used, the energy dissipation rate obtained from either direction should, on average, be nearly the same within the bulk. In this study we will show that this is indeed the case, which means that ǫ bulk can be obtained either from the dissipation rate in the LL direction ǫ LL or from that in the NN direction ǫ N N . A similar approach is followed in Ni et al. (2011), where both SFs are calculated in RB flow within the sub-Kolmogorov regime where the flow is found to be nearly homogeneous and isotropic at the center of the cell.
In figure 3, we have included the dimensionless dissipation rateǫ u = (d 4 /ν 3 ) (ν/2)(∂u i /∂x j + ∂u j /∂x i ) 2 V,t obtained from Direct Numerical Simulations (DNS) for ρ = 0.714, Γ = 2 and Ta = 2.15 × 10 9 from Zhu et al. (2017). Here, the V,t denotes the average over the entire volume and time respectively. This includes the boundary layers, that we explicitly avoid in our r bulk definition. When comparing the profile obtained from numerics and from our data for Ta = 3.6 × 10 10 we notice that both agree rather well, thus mutually validating each other.
By averaging the ǫ-profiles in the bulk (figure 3), we finally find the bulk-averaged dissipation ratesǫ LL,bulk = ǫ LL (r) r bulk andǫ N N,bulk = ǫ N N (r) r bulk . In order to validate the calculation, in figure 4 we show the bulk-averaged longitudinal D LL and transverse D N N SFs for every Ta. Here, we compensate the SFs as s −1 (D LL (s)/C 2 ) 2/3 and s −1 (D N N (s)/(4/3)C 2 ) 2/3 such that their units match that of the dissipation rate. The horizontal axis is normalized with the corresponding bulk-averaged Kolmogorov length scale (see §3.3). According to Kolmogorov's scaling, within the inertial regime (s ∈ [15η, L 11 ]), where L 11 is the integral length scale obtained from the azimuthal velocity, each compensated curve (fixed r and Ta) should be proportional to the dissipation rate in the bulk. Here we see that our estimates for the bulk-averaged dissipation rates are located within the plateau regions, demonstrating the self-consistency of the calculation. In the same figure, the separation of length scales in the flow can also be seen. Note in particular how such separation between η and L 11 increases with Ta. The integral length scale L 11 (Ta) in figure 4 is calculated using the integral of the autocorrelation of the azimuthal velocity in the azimuthal direction and averaged over the bulk region.

The dissipation rate in the bulk
In figure 5a we show the scaling of bothǫ LL,bulk andǫ N N,bulk . We find that the dissipation rate extracted from both directions scale effectively asǫ bulk ∼ Ta 1.40 , with a nearly identical prefactor. This shows that the local energy dissipation rate scales in the same way as the global energy dissipation rateǫ ∼ Ta 1.40 . Correspondingly, this implies that the local Nusselt number scales as Nu ω,bulk ∼ Ta 0.40 . In the same figure  (figure 5a), we includeǫ of (Ostilla-Mónico et al. 2014), obtained from both DNS, and Huisman et al. (2014) torque measurements from the Twente Turbulent Taylor-Couette (T 3 C) experiment. The compensated plot (figure 5b) reveals that both the local and global energy dissipation rate scale indeed as Ta 1.40 with the ratio ǫ bulk /ǫ global ≈ 0.1. In the regime of ultimate TC turbulence, it was suggested that both turbulent BLs extend throughout the gap until they meet around d/2 (Grossmann & Lohse 2011). The turbulent BLs give rise to the logarithmic correction L(Ta) in the scaling of the Nusselt number, which changes the scaling from Nu ω ∼ Ta 1/2 to effectively Nu ω ∼ Ta 1/2 L(Ta) ∼ Ta 0.40 (van Gils et al. 2011;Huisman et al. 2012). With equation (1.4) one obtains the effective scaling of the global energy dissipation rateǫ global ∼ Ta 3/2 L(Ta) ∼ Ta 1.40 . It  In both figures, the black dashed line is 15η while the colored short vertical lines are located at L11/η for each Ta: The inertial range is approximately bounded by these two lines. The colored stars show the maxima of each curve which correspond to ǫ(r) r bulk .
is remarkable how our local measurements of the local energy dissipation rate reveal the very same scaling due to L(Ta) as the global energy dissipation rate. In contrast, in RB flow it is shown that when the driving is in the order of 10 8 < Ra < 10 11 , i.e. far below the transition into the ultimate regime (BLs are still laminar),ǫ bulk ∼ Ra 1.5 (Shang et al. 2008;Ni et al. 2011). Note, however, that in that regime the global energy dissipation rateǫ global is still determined by the BL contributions,ǫ BL ≫ǫ bulk andǫ BL ≈ǫ global . Our measurements are thus consistent with the prediction of Grossmann & Lohse (2011), where even at such large Ta numbers, a rather intricate interaction between turbulent BLs and bulk flow prevails through the entire gap.
In order to further show the quality of the scaling, we show in figure 6 the same ǫ-profiles shown in figure 3 but now compensated with Ta −1.40 . For both the LL and NN direction, the dissipation rates for different Ta collapse throughout most of the gap, far away from the inner and outer cylinder. Within the bulk however, they are nearly constant and very close to the prefactors (≈ 5 × 10 −4 ) found from the scaling in figure 5a. When looking at the compensated data from DNS, we notice that the prefactor is in that case twice as large as ours (≈ 10 −3 ). The reason is that the nature of both calculations is different: While the data from DNS is obtained from averaging the 3D velocity gradients over the entire volume, we rely on the scaling of the second order SFs (without intermittency corrections) to approximate the local energy dissipation rate in the bulk at the maximum peak in the compensated curves (see §3.2).
In order to further characterize the turbulent scales in the flow, we calculate the Kolmogorov length scale in the bulk. Since there are two dissipation rates available, we define their corresponding Kolmogorov length scales as η LL,bulk = (ν 3 /ǫ LL,bulk ) 1/4 and η N N,bulk = (ν 3 /ǫ NN,bulk ) 1/4 . Becauseǫ bulk ∼ Ta 1.40 , the scaling ofη bulk = η bulk /d ∼ Ta −0.35 , which can be seen in figure 7a. Obviously, here we find a similar prefactor in both directions LL and NN too. The inset of the figure shows the corresponding compensated plot. For comparison, we include in the same figure the scaling from Lewis & Swinney (1999). When comparing it with our data we notice some differences in magnitude. While we average in the bulk and make use of PIV to obtain the spatial dependence of the velocities directly, the data from Lewis & Swinney (1999) was measured at a single point (r = 0.5) using Hot-wire anemometry and Taylor's frozen flow hypothesis. When fitting data to a power law, confidence bounds for every coefficient in the regression can be obtained, given a certain confidence level. In this paper, we use the standard 95% confidence for every fit, from which the uncertainty in the power law exponents (figures 5,6) were chosen as the middle point between the lower and upper bound of its corresponding confidence bound. This procedure is done for all the exponents reported throughout this paper.

The turbulent intensity in the bulk
The final step in the calculation of Re λ,bulk is to look at the azimuthal velocity fluctuations. Thus we average σ θ,t (u θ (r, θ, t)) (see equation (1.2)) from figure 2a in the bulk and find a good description by the effective scaling law (d/ν)σ bulk (u θ ) ≈ 11.3 × 10 −2 Ta 0.44±0.01 . In figure 7b, we show the turbulence intensity I θ,bulk = σ θ,t (u θ )/ u θ θ,t r bulk as a function of Ta. In this way, we are able to compare our data to the turbulence intensity scaling from Lewis & Swinney (1999). We find that the effective scaling I θ,bulk ∼ Ta −0.061±0.003 reproduces our data well. In the inset of the same figure we show the compensated plot throughout the Ta range. Similarly as with the Kolmogorov length scale described in section §3.3, we include in the same figure the scaling of Lewis & Swinney (1999). In this case, the exponent in our scaling is nearly identical to the one found by Lewis & Swinney (1999) with a slightly larger prefactor. We remind the reader once again that our average is done over the bulk region while the data of Lewis & Swinney (1999) is obtained at a single point at midgap.

The scaling of the Taylor-Reynolds number Re λ,bulk
Finally, with both the local dissipation rate and the local velocity fluctuations in the bulk, we calculate the corresponding Taylor-Reynolds number as a function of Ta, using both ǫ LL,bulk , ǫ N N,bulk and σ bulk (u θ ). The results can be seen in figure 8a where an effective scaling of Re λ,bulk ∼ Ta 0.18±0.01 is found for both directions. The compensated plot in figure 8b reveals the good quality of the scaling throughout the range of Ta. In order to highlight the difference between the different calculations, we also include the  Lewis & Swinney (1999) (black dashed line). The inset shows the compensated plots for the local quantities where the effective scaling ofη bulk ∼ Ta −0.35 is found to reproduce both directions. (b) Bulk-averaged azimuthal turbulent intensity. The data reveals an effective scaling of I θ,bulk ∼ Ta −0.061 . The (dashed black line) represents the local scaling I θ = 0.1Ta −0.062 atr = 0.5 as it was obtained from Lewis & Swinney (1999). The inset in (b) shows the corresponding compensated plot. estimate of Huisman et al. (2013) for Ta = 1.49 × 10 12 (Re λ = 106). We emphasize that our calculation is based entirely on local quantities (fluctuations and dissipation rate) whilst the estimate of Huisman et al. (2013) is done using a single point in space,r = 0.5, in combination with the global energy dissipation rate (equation (1.4)). Our scaling predicts that the local Taylor-Reynolds number at that Ta is approximately Re λ,bulk ≈ 217, roughly twice the value estimated by Huisman et al. (2013) for the same Ta.

Summary and conclusions
To summarize, we have measured local velocity fields using PIV in the ultimate regime of turbulence. We showed that both structure functions (longitudinal and transverse) yield similar energy dissipation rate profiles that intersect within the bulk, similarly as what is observed in Rayleigh-Bénard convection. When averaging these profiles within the bulk, this leads to an effective scaling ofǫ bulk ∼ Ta 1.40±0.04 , which is the same scaling as obtained for the global quantityǫ measured from the torque scaling Huisman et al. 2014). This result reveals the dominant influence of the turbulent BLs over the entire gap. Future work will show whether this also holds for higherorder velocity structure functions, as it does hold in other turbulent wall-bounded flows (de Silva et al. 2015).
Next, we showed that the Kolmogorov length scale scales asη bulk ∼ Ta 0.35±0.01 and the azimuthal turbulent intensity scales as I θ,bulk ∼ Ta −0.061±0.003 . In order to evaluate the turbulence level in the flow, we showed that with both local quantities at hand (dissipation rate and turbulent fluctuations), the bulk Taylor-Reynolds number scales as Re λ,bulk ∼ Ta 0.18±0.01 . Our calculation can be generalized by inserting our result for the ratio between the local and global energy dissipation rateǫ bulk /ǫ global = α ≈ 0.1 back into equation (1.1) and using equation (1.4) to relate ǫ global and Nu ω . The latter yields Re λ,bulk (Ta) = 1/α √ 15σ TC d 2 ν 2 (σ bulk (u θ )) 2 √ TaNu ω . (4.1) Thus, given the local variance of the velocity fluctuations and the global Nusselt number, the response parameter Re λ,bulk (Ta) can be calculated in the bulk flow (r ∈ [0.35, 0.65]) for the case of pure inner cylinder rotation (a = 0). In order to extend the calculation to the case a ≈ a opt ≈ 0.36, i.e. close to the rotation ratio for optimal Nu ω , where pronounced Taylor rolls exist (Huisman et al. 2014;, an extra averaging process in axial direction for both the velocity fluctuations and the dissipation rates would be needed.