Ultimate Turbulent Taylor-Couette Flow

The flow structure of strongly turbulent Taylor-Couette flow with Reynolds numbers up to ${\mathrm{Re}}_i=2\times {10}^6$ of the inner cylinder is experimentally examined with high-speed particle image velocimetry (PIV). The wind Reynolds numbers ${\mathrm{Re}}_w$ of the turbulent Taylor-vortex flow is found to scale as ${\mathrm{Re}}_w\propto {\mathrm{Ta}}^{1/2}$, exactly as predicted by Grossmann and Lohse [Phys. Fluids 23, 045108 (2011).] for the ultimate turbulence regime, in which the boundary layers are turbulent. The dimensionless angular velocity flux has an effective scaling of ${\mathrm{Nu}}_{\omega }\propto {\mathrm{Ta}}^{0.38}$, also in correspondence with turbulence in the ultimate regime. The scaling of ${\mathrm{Nu}}_{\omega }$ is confirmed by local angular velocity flux measurements extracted from high-speed PIV measurements: though the flux shows huge fluctuations, its spatial and temporal average nicely agrees with the result from the global torque measurements.

Figure 1

${\mathrm{Re}}_w$ vs Ta. The data from repeated experiments at midheight are plotted as separate (blue) dots, showing the quality of the reproducibility and the statistical stationarity of the measurements. We have averaged azimuthally, over time, and in the bulk flow ($0.23\mathrm{m}\le r\le 0.25\mathrm{m}$). The straight line is the best fit ${\mathrm{Re}}_w=0.0424{\mathrm{Ta}}^{0.495\pm 0.010}$ and the (red) dashed line is the Kraichnan prediction [13] Eq. (2). The inset shows the compensated plot ${\mathrm{Re}}_w/{\mathrm{Ta}}^{1/2}$ vs Ta. The horizontal (green) line is the prediction (1) of Ref. 12.

Figure 2

(a) Snapshot of the instantaneous convective angular velocity flux, measured at $z=L/2$, for $\mathrm{Ta}=1.5\times {10}^{12}$. The ($r$, $\theta$) plane and time averaged flux is found to be equal to $⟨{\mathrm{Nu}}_{\omega }{⟩}_{\theta ,r,t}=325$. A corresponding movie is available as supplemental material [33]. (b) Local normalized convective $\omega$-flux as functions of $r^{\prime }=(r-r_i)/(r_o-r_i)$ for 6 heights varying between $0.5\le z/L\le 0.73$, for $\mathrm{Ta}=1.5\times {10}^{12}$. The black solid line is the average of the 12 experiments ($⟨{\mathrm{Nu}}_{\omega }{⟩}_{\theta ,r,z,t}$), which is very close to the expected value $326\pm 6$ from global torque measurements [5]. (c) Local normalized convective angular velocity flux vs radial position $r^{\prime }$ for various rotation rates, measured at $z=L/2$. From bottom to top we have repeated experiments for $\mathrm{Ta}=3.8\times {10}^9$, $1.5\times {10}^{10}$, $6.2\times {10}^{10}$, $3.8\times {10}^{11}$, $1.5\times {10}^{12}$, and $6.2\times {10}^{12}$. The dashed lines for the three highest Ta represent the ${\mathrm{Nu}}_{\omega }$ value derived from global torque measurements [5].

Figure 3

Local convective angular velocity flux as a function of Taylor number. The blue dots are results obtained from PIV measurements and show a scaling of ${\mathrm{Nu}}_{\omega }\propto {\mathrm{Ta}}^{0.45\pm 0.04}$. The green and orange dots are repeated measurements at a height of $z=L/2+d/2$ and $z=L/2+d$, respectively. The black data points are obtained from global torque measurements and show a scaling that is less steep: ${\mathrm{Nu}}_{\omega }\propto {\mathrm{Ta}}^{0.38}$. The dashed green line is obtained by matching two log-layers [3], and has a slope of 0.37 at $\mathrm{Ta}={10}^9$, and 0.41 at $\mathrm{Ta}={10}^{13}$. The red line is from the turbulent boundary layer theory of ref. [12]. It has a slope of 0.43 around $\mathrm{Ta}={10}^9$ and 0.44 around $\mathrm{Ta}={10}^{13}$. Dark red data points are obtained by means of global torque measurements [4].

Figure 4

Results of three experiment with varying rotation rate resulting in $\mathrm{Ta}=3.8\times {10}^{11}$, $1.5\times {10}^{12}$, and $6.2\times {10}^{12}$, colored in red, green, and blue, respectively. All the data shown is averaged over the region $0.23\mathrm{m}\le r\le 0.25\mathrm{m}$, and measured at midheight. All quantities with tildes are standardized (shifted and scaled such as to have zero mean and unit variance). (a) PDF of the standardized angular velocity. (b) PDF of the standardized radial velocity. (c) Standardized normalized local convective angular velocity flux PDF. (d) Cross-correlation coefficient of the angular velocity and the radial velocity, the dimensionless decaying time (in number of rotations) is found to be 0.07. The corresponding length scale can be found by multiplying this number with the circumference of the inner cylinder giving $\delta =88\mathrm{mm}$, which is of the same order of magnitude as the gap width $d=80\mathrm{mm}$.