Torque scaling in small-gap Taylor-Couette flow with smooth or grooved wall

The torque in the Taylor-Couette flow for radius ratios $\eta \ge 0.97$, with smooth or grooved wall static outer cylinders, is studied experimentally, with the Reynolds number of the inner cylinder reaching up to ${\mathrm{Re}}_{\mathrm{i}}=2\times {10}^5$, corresponding to the Taylor number up to $\mathrm{Ta}=5\times {10}^{10}$. The grooves are perpendicular to the mean flow, and similar to the structure of a submersible motor stator. It is found that the dimensionless torque $G$, at a given ${\mathrm{Re}}_{\mathrm{i}}$ and $\eta$, is significantly greater for grooved cases than smooth cases. We compare our experimental torques for the smooth cases to the fit proposed by Wendt [F. Wendt, Ing.-Arch. 4, 577 (1993)] and the fit proposed by Bilgen and Boulos [E. Bilgen and R. Boulos, J Fluids Eng. 95, 122 (1973)], which shows both fits are outside their range for small gaps. Furthermore, an additional dimensionless torque (angular velocity flux) $\mathrm{N}{\mathrm{u}}_{\omega }$ in the smooth cases exhibits an effective scaling of $\mathrm{N}{\mathrm{u}}_{\omega }\sim \mathrm{T}{\mathrm{a}}^{0.39}$ in the ultimate regime, which occurs at a lower Taylor number, $\mathrm{Ta}\approx 3.5\times {10}^7$, than the well-explored $\eta =0.714$ case (at $\mathrm{Ta}\approx 3\times {10}^8$). The same effective scaling exponent, 0.39, is also evident in the grooved cases, but for $\eta =0.97$ and 0.985, there is a peak before this exponent appears.

Figure 1

Grooved geometry used in this study. (a) Three-dimensional view. Fifty-four grooves perpendicular to the mean flow and equally spaced in azimuth. (b) Geometry perpendicular to rotating axis, and closeup of groove, characterized by depth scale $\delta$, width scale $b$, and radius scale $r_{\mathrm{g}}$.

Figure 2

Drawing (a) and image (b) of the Taylor-Couette system. The experimental apparatus comprises three parts: power source (motor), measurement part (torque-speed sensor), and test part (TC setup). The dashed lines indicate the working fluid level during measurements, ensuring the TC setup is filled with fluid.

Figure 3

Test compensated device to measure torques on sides of inner cylinder, the seals, and bearings.

Figure 4

Torque comparison for static outer cylinder for $\eta =0.97$, 0.985, 0.9925, 0.995, and 0.9975 (in this study with smooth and grooved outer cylinders), and $\eta =0.724$ (Lewis and Swinney [5], with smooth outer cylinder). ${\eta }_{\mathrm{s}}$ is for the smooth case, ${\eta }_{\mathrm{g}}$ for the grooved case, and ${\eta }_{\mathrm{L}}$ the Lewis and Swinney experiment result [5]. (a) A double-logarithmic plot of dimensionless $G$ versus ${\mathrm{Re}}_{\mathrm{i}}$. (b) The relative differences of torque between smooth cases and grooved cases.

Figure 5

Relative difference between smooth cases and past fits. The ${\eta }_{\mathrm{W}}$ represents the fit by Wendt [9] [Eq. (4))], and ${\eta }_{\mathrm{B}}$ the fit by Bilgen and Boulos [10] [Eq. (5))].

Figure 6

Global response $\mathrm{N}{\mathrm{u}}_{\omega }$ for smooth and grooved cases for different radius ratios as function of driving Ta. (a) Double-logarithmic plot of dimensionless $\mathrm{N}{\mathrm{u}}_{\omega }$ versus Ta. (b) Compensated plot of $\mathrm{N}{\mathrm{u}}_{\omega }/\mathrm{T}{\mathrm{a}}^{0.39}$ versus Ta. (c) Compensated plot of $(\mathrm{N}{\mathrm{u}}_{\omega }-1)/\mathrm{T}{\mathrm{a}}^{0.33}$ versus Ta. Dashed line represents transition to ultimate turbulent regime in smooth cases.