Logarithmic Boundary Layers in Strong Taylor-Couette Turbulence

We provide direct measurements of the boundary layer properties in highly turbulent Taylor-Couette flow up to $\mathrm{Re}=2\times {10}^6$ ($\mathrm{Ta}=6.2\times {10}^{12}$) using high-resolution particle image velocimetry and particle tracking velocimetry. We find that the mean azimuthal velocity profile at the inner and outer cylinder can be fitted by the von Kármán log law $u^{+}=1/\kappa \ln y^{+}+B$. The von Kármán constant $\kappa$ is found to depend on the driving strength Ta and for large Ta asymptotically approaches $\kappa \approx 0.40$. The variance profiles of the local azimuthal velocity have a universal peak around $y^{+}\approx 12$ and collapse when rescaled with the driving velocity (and not with the friction velocity), displaying a log dependence of $y^{+}$ as also found for channel and pipe flows.

Figure 1

Sketch of the vertical cross section of the ${\mathrm{T}}^3\mathrm{C}$ [39]. The flow is illuminated from the side in the horizontal plane using a laser, and the flow is imaged from the top using a high-resolution camera. Top-right inset: schematic top view of different regions inside the gap: $\mathsf{I}\mathsf{C}$ (inner cylinder), $\mathsf{O}\mathsf{C}$ (outer cylinder), and $\mathsf{B}\mathsf{L}$ (boundary layer). Measurements were done at midheight.

Figure 2

Azimuthal velocity profiles for varying Re(Ta) across the gap of the TC apparatus. The legend indicates Re(Ta) of the experiments. Insets (a) and (b) show a zoom of the data of the inner and outer BL, respectively. Profiles in inset (a) are stacked top to bottom for increasing Ta. Individual data points are plotted in the insets showing the high resolution of the measurements. The gray solid line represents the exact laminar circular-Couette (nonvortical) solution of the Navier-Stokes equations. The dashed line is the profile for $\mathrm{Re}(\mathrm{Ta})=260({10}^5)$ obtained from DNS [21].

Figure 3

(a) Azimuthal velocity profile near the inner cylinder ($r^{\prime }\in [0,1/2]$) for varying Re(Ta). $y^{+}$ is the distance from the inner cylinder in units of the viscous length scale ${\delta }_{\nu }$. $u^{+}$ is defined as $[u(r_i)-u(r)]/u_{\tau }$, where $u(r)$ is the azimuthal component of the velocity, and $u(r_i)$ is the azimuthal velocity of the inner cylinder. (b) Azimuthal velocity profile near the outer cylinder $r^{\prime }\in [1/2,1]$. $y^{+}=(r_o-r)/{\delta }_{\nu }$ is the distance from the outer cylinder in wall units. For the case of the outer BL, the velocity is scaled as $u^{+}=u(r)/u_{\tau }$. Both figures include the logarithmic law of the wall $u^{+}=1/\kappa \ln y^{+}+B$ by von Kármán, with the typical values of $\kappa =0.40$ and $B=5.2$, the viscous sublayer $u^{+}=y^{+}$, and in gray the fitting domain $y^{+}\in [50,600]$. Each data set is accompanied by an arrow with caption indicating Re(Ta). Thin solid lines are DNS results [21].

Figure 4

The parameter $\kappa$ obtained by fitting $u^{+}=1/\kappa \ln y^{+}+B$ for $y^{+}\in [50,600]$ near the inner and outer cylinder. The color scheme is identical to Figs. 2, 3, the black symbols are DNS results [21]. The solid colored line shows the ratio of the upper edge $600{\delta }_{\nu }$ of the fitting regime to the radius of curvature of the cylinder, quantifying the relative influence of the curvature. Note that ${\delta }_{\nu ,i}/r_i={\delta }_{\nu ,o}/r_o$.

Figure 5

(a) The variance of the local azimuthal velocity is presented as a function of radial distance from the inner cylinder. The velocity is made dimensionless using the friction velocity $u_{\tau }$, and the distance is in wall units $y^{+}=(r-r_i)/{\delta }_{\nu }$. (b) Same as in (a) but the velocity has been rescaled using the driving velocity $u(r_i)$. Corresponding colors in (a) and (b) correspond to the same data set and are consistent with previous figures. The Re(Ta) are indicated with arrows in (a). For $y^{+}>50$, the five highest Ta cases are fitted with $[\sigma (u_{\theta })/u_{\tau }{]}^2=B_1-A_1\ln y^{+}$. The fitting parameters are listed inside (a) for increasing Ta and are colored accordingly.