Self-similar decay of high Reynolds number Taylor-Couette turbulence

We study the decay of high-Reynolds-number Taylor-Couette turbulence, i.e., the turbulent flow between two coaxial rotating cylinders. To do so, the rotation of the inner cylinder (${Re}_i=2\times {10}^6$, the outer cylinder is at rest) is stopped within 12 s, thus fully removing the energy input to the system. Using a combination of laser Doppler anemometry and particle image velocimetry measurements, six decay decades of the kinetic energy could be captured. First, in the absence of cylinder rotation, the flow-velocity during the decay does not develop any height dependence in contrast to the well-known Taylor vortex state. Second, the radial profile of the azimuthal velocity is found to be self-similar. Nonetheless, the decay of this wall-bounded inhomogeneous turbulent flow does not follow a strict power law as for decaying turbulent homogeneous isotropic flows, but it is faster, due to the strong viscous drag applied by the bounding walls. We theoretically describe the decay in a quantitative way by taking the effects of additional friction at the walls into account.

Figure 1

Schematic of the vertical cross section of the ${\mathrm{T}}^3\mathrm{C}$ facility. The laser beams are in the horizontal plane $(r,\theta )$ at midheight; $z=L/2$ (unless stated otherwise) for both the LDA and PIV measurements. The top right inset shows the horizontal cross section, showing the LDA beams (not to scale). The beams refract twice on the OC and intersect at the middle of the gap ($r=r_m$), giving the local velocity component $u_{\theta }$. The bottom right inset shows, for the PIV measurements, particles illuminated by a thin laser sheet. We use the viewing windows in the end plate to look at the flow from the bottom, thus obtaining the velocity components $u_{\theta }$ and $u_r$ in the $(r,\theta )$ plane.

Figure 2

Midgap azimuthal velocity as a function of time at midheight. PIV measurements with seven different interframe times $\mathrm{\Delta }t$ were performed, as shown in the legend, to produce accurate results over the entire velocity range. The PIV measurements are averaged azimuthally and radially; we average over $r_m-4\text{mm}<r<r_m+4$ mm, corresponding to 10% of the gap width. These results are confirmed by LDA measurements performed at several heights (dashed lines), which are shifted by one decade for clarity. The standard deviation ${\sigma }_{u_{\theta }}$, a measure for the spatial velocity fluctuations, is shown as the solid black line. The data are averaged azimuthally and radially as described above and binned using logarithmic bins of 0.1 decades. The measurements cover three orders of magnitude of the velocity, corresponding to six orders of magnitude in kinetic energy. The measurement uncertainty roughly corresponds to the width of the lines.

Figure 3

(a) Energy dissipation rate, normalized with ${\nu }^3/d^4$, as a function of time. Here $\epsilon$ is calculated from the PIV data as shown in Fig. 2; $\epsilon$ drops by more than 8 orders of magnitude. (b) Kolmogorov length scale ${\eta }_K$ as a function of time, normalized with the gap width $d$. As time progresses, the separation of scales becomes smaller, although ${\eta }_K$ remains small.

Figure 4

(a) The PIV results for ${Re}_{u_{\theta }}$ and ${Re}_{{\sigma }_{\theta }}$, binned using logarithmic bins of 0.1 decades: The scale on the $y$-axis refers to ${Re}_{u_{\theta }}$, whereas the one for ${Re}_{{\sigma }_{\theta }}$ is vertically shifted to show that the decay of the spatial fluctuations and the mean is the same. On the horizontal axis, both the real time and the nondimensionalized time (normalized by $\tau =d^2/\nu$) are shown. Included in the graph are the results for the HIT model of Ref. [29] and those of the Prandtl–von Kármán skin friction model (2), which includes the effects of the walls. Below the short line at $Re=500$, thermal effects set in. (b) Ratio between the measurements and, respectively, the HIT model (solid red line) and the skin friction model (solid blue line). The respective dashed lines show the ratio between the fluctuation decay and the two models.

Figure 5

Azimuthal velocity profiles $u_{\theta }\left(r\right)$ for $z=L/2$, averaged over $\theta$. The decelerating effects of the walls (left and right edges of the figure) can be seen clearly. (b) The velocity is normalized with the (spatial) mean azimuthal velocity ${\langle u_{\theta }\rangle }_{r,\theta }\left(t\right)$. The normalized velocity profiles overlap, indicating the self-similarity of the velocity profile during the decay.

Figure 6

(a) Measured velocity $u_{\theta }$ as a function of time, resulting from seven PIV measurements with changing $\mathrm{\Delta }t$ and averaged over $\theta$. (b) Same results as shown in (a) but now normalized with the mean velocity ${\langle u_{\theta }\left(t\right)\rangle }_{r,\theta }$. The normalized velocity is self-similar up to $t\approx 400$ s.