Log law of the wall revisited in Taylor-Couette flows at intermediate Reynolds numbers

We provide Reynolds averaged azimuthal velocity profiles, measured in a Taylor-Couette system in turbulent flow, at medium Reynolds (7800 $<$ Re $<$ 18000) number with particle image velocimetry technique. We find that in the wall regions, close to the inner and outer cylinders, the azimuthal velocity profile reveals a significant deviation from classical logarithmic law. In order to propose a new law of the wall, the profile of turbulent mixing length was estimated from data processing; it was shown to behave nonlinearly with the radial wall distance. Based on this turbulent mixing length expression, a law of the wall was proposed for the Reynolds averaged azimuthal velocity, derived from momentum balance and validated by comparison to different data. In addition, the profile of viscous dissipation rate was investigated and compared to the global power needed to maintain the inner cylinder in rotation.

Figure 1

Diagram of the setup.

Figure 2

Reynolds averaged azimuthal velocity $\overline{U_{\theta }}$ profile versus the radial distance to the wall $\delta$ for different impeller rotational speeds $N$: $\circ$ 114 rpm, $▹$ 90 rpm, $\diamond$ 70 rpm, $◃$ 50 rpm; the velocity scale is the inner cylinder velocity and the length scale is the gap between the cylinders.

Figure 3

Reynolds averaged azimuthal velocity $U^{+}$ plotted in semilogarithmic scale and in local wall units against the inner cylinder radial distance ${\delta }^{+}$ from the inner cylinder, for different impeller rotational speeds $N$: $\circ$ 114 rpm, $▹$ 90 rpm, $\diamond$ 70 rpm, $◃$ 50 rpm; dotted line: viscous sublayer; dashed line: logarithmic law.

Figure 4

Logarithmic diagnostic function ${\delta }^{+}\frac{d{U}^{+}}{d{\delta }^{+}}$ plotted in local wall units against the inner cylinder radial distance ${\delta }^{+}$, for different impeller rotational speeds $N$: $\circ$ 114 rpm, $▹$ 90 rpm, $\diamond$ 70 rpm, $◃$ 50 rpm; solid line: von Karman $\kappa$ value.

Figure 5

Turbulent mixing length profile plotted against the wall distance $\delta$ from the (a) inner and (b) outer cylinder; $\circ$ numerical data; dashed line: van Driest model; solid line: model corresponding to Eq. (6).

Figure 6

Reynolds averaged azimuthal velocity $U^{+}$ plotted in semilogarithmic scale and in local wall units against the inner cylinder radial distance ${\delta }^{+}$ from the inner cylinder (a) = ($\circ$ $N=114\mathrm{rpm}$, $\text{Re}\cong 18000$, $C_U=5.1$) and (b) = ($▹$ $N=90\mathrm{rpm}$, $\text{Re}\cong 14000$, $C_U=3.9$); dotted line: viscous sublayer; dashed line: logarithmic law.

Figure 7

Turbulent mixing length profile plotted against the wall distance $\delta$ from the inner cylinder, $\circ$ Twente turbulent Taylor-Couette (T3C) data ($\mathrm{Re}=2\times {10}^5$), dashed line: linear expression based on $\kappa =0.6$ and solid line: quadratic model based on $C_l=0.0125$.

Figure 8

Reynolds averaged azimuthal velocity $U^{+}$ plotted in semilogarithmic scale and in local wall units against the inner cylinder radial distance ${\delta }^{+}$ from the inner cylinder, $\circ$ Twente turbulent Taylor-Couette (T3C) data (a) $(N=120\mathrm{rpm}$, $\text{Re}\cong 200000$, and $C_U=6.75$, (b) ($N=60\mathrm{rpm}$, $\text{Re}\cong 100000$, and $C_U=3.2$) and (c) ($N=30\mathrm{rpm}$, $\text{Re}\cong 50000$, and $C_U=1.55$); dotted line: viscous sublayer; dashed line: logarithmic law.

Figure 9

Viscous dissipation of kinetic energy profile divided by its inner wall value ${\epsilon }_i$ against the inner cylinder radial distance ${\delta }^{+}$ from the inner cylinder ($N=114\mathrm{rpm}$): $▹$ TKE production; $◃$ mean KE dissipation; $\circ$ total KE dissipation.