Torque measurements and numerical determination in differentially rotating wide gap Taylor-Couette flow

We investigate experimentally and numerically turbulent Taylor-Couette flow with independently rotating cylinders and radius ratio $\eta =0.5$. The torque acting on the inner wall is measured to analyze the transverse current of azimuthal motion $J_{\omega }$. The scaling of the torque with shear Reynolds number is determined for the outer cylinder at rest. For constant shear Reynolds number we investigate various ratios of angular velocities and find a torque maximum for counter-rotating cylinders that deviates from the prediction suggested by van Gils et al. [J. Fluid Mech. 706, 118 (2012)]. The direct comparison between the experiment and the numerical simulation shows a good agreement in the torques.

Figure 1

Drawing and image of the Taylor-Couette system, $R_1=35$ mm, $R_2=70$ mm, $\eta =0.5$, $\Gamma =20$, with a surrounding water bath. The inner cylinder is separated into three sections. The middle section measures the torque. The upper and lower sections are pivoted inside the end plates.

Figure 2

(a) The dependency of the quasi-Nusselt number ${\text{Nu}}_{\omega }=G/G_{\mathrm{lam}}$ on the shear Reynolds number ${\text{Re}}_S$ over the whole range of measurements. Strong counter-rotation ($\mu <-0.2$) is indicated by stars, counter-rotation of $-0.2<\mu <0$ by crosses, inner cylinder rotation $\mu =0$ by diamonds, corotation in the Rayleigh-instable regime $0<\mu <0.25$ by triangles, and corotation in the Rayleigh-stable regime $\mu >0.25$ by circles. The dashed line indicates a power law fit ${\text{Nu}}_{\omega }\sim {\mathrm{Re}}_S^{\alpha -1}$ with the exponent $\alpha =1.62$ for $3\times {10}^3\le {\text{Re}}_S\le 8\times {10}^4$ and $\alpha =1.78$ for ${10}^5\le {\text{Re}}_S\le {10}^6$ (solid line). (b) Local exponent for the quasi-Nusselt number $\alpha -1=\partial \left({\log }_{10}{\text{Nu}}_{\omega }\right)/\partial \left({\log }_{10}{\text{Re}}_S\right)$ in dependence on the shear Reynolds number, calculated from torque measurements for $\mu =0$ using a sliding least square fit over intervals of $\Delta \left({\log }_{10}{\text{Re}}_S\right)={\Delta }_{10}=0.2$ (triangles), $0.6$ (circles), 1 (squares).

Figure 3

The dependency of the quasi-Nusselt number ${\text{Nu}}_{\omega }=G/G_{\mathrm{lam}}$ on the ratio of rotation rates $\mu$ for constant shear Reynolds numbers (${\text{Re}}_S/{10}^4=0.5,1,1.5,2$). The circles represent experimental torque data for silicone oil, $\nu =8.41\times {10}^{-6}$ m${}^2$/s, and the crosses represent measurements for $\nu =22.81\times {10}^{-6}$ m${}^2$/s compared with direct numerical simulations (triangles). Dotted lines serve as a guide to the eye, and the black arrow indicates the maximum at $\mu =-0.2$.

Figure 4

The dependency of the quasi-Nusselt number ${\text{Nu}}_{\omega }=G/G_{\mathrm{lam}}$ on the ratio of rotation rates $\mu$ for constant shear Reynolds numbers (${\text{Re}}_S/{10}^5=1.00$, 2.00, 3.00, 4.00, 5.00).

Figure 5

Ratio of angular velocities of maximal torque ${\mu }_{\max }$ in dependence on the shear Reynolds number ${\text{Re}}_S$ from experimental measurements (circles) and numerical simulations (squares). The solid line shows $\mu$ along the angle bisector line by van Gils et al. [3] and is calculated using the Esser and Grossmann (E&G) stability calculations for viscous fluids [15].