Torque Scaling in Turbulent Taylor-Couette Flow with Co- and Counterrotating Cylinders

We analyze the global transport properties of turbulent Taylor-Couette flow in the strongly turbulent regime for independently rotating outer and inner cylinders, reaching Reynolds numbers of the inner and outer cylinders of ${\mathrm{Re}}_i=2\times {10}^6$ and ${\mathrm{Re}}_o=\pm 1.4\times {10}^6$, respectively. For all ${\mathrm{Re}}_i$, ${\mathrm{Re}}_o$, the dimensionless torque $G$ scales as a function of the Taylor number Ta (which is proportional to the square of the difference between the angular velocities of the inner and outer cylinders) with a universal effective scaling law $G\propto {\mathrm{Ta}}^{0.88}$, corresponding to ${\mathrm{Nu}}_{\omega }\propto {\mathrm{Ta}}^{0.38}$ for the Nusselt number characterizing the angular velocity transport between the inner and outer cylinders. The exponent 0.38 corresponds to the ultimate regime scaling for the analogous Rayleigh-Bénard system. The transport is most efficient for the counterrotating case along the diagonal in phase space with ${\omega }_o\approx -0.4{\omega }_i$.

Figure 1

(a) Explored phase space (${\mathrm{Re}}_o$, ${\mathrm{Re}}_i$) of TC flow with independently rotating inner and outer cylinders. To the right of the horizontal axis the cylinders are corotating, to the left of it they are counterrotating, and a log-log representation has been chosen. Experimental data by Wendt [12] (pluses), Taylor [21] (left triangles), Smith and Townsend [22] (open circles), Andereck et al. [23] (grey box), Tong et al. [13] (upward triangles), Lathrop et al. [6] (stars), Ravelet et al. [14] (hexagrams), Borrero-Echeverry et al. [24] (upward solid triangles), and simulations by Pirro and Quadrio [25] (solid squares), Bilson and Bremhorst [26] (downward solid triangles), and Dong [27, 28] (solid circles). The dashed lines are Esser and Grossmann’s [29] estimate for the onset of turbulence with $\eta =0.71$. The many data points in the small Reynolds number regime of pattern formation and spatial temporal chaos (see e.g. 23, 30, 31) have not been included in this phase diagram. Our data points for this publication are the black diamonds. (b) Our data points in the phase diagram on a linear scale. (c) Our data points in the phase space (Ta, $a$); note that Ta also depends on $a$. (d) Our data points in the phase space (Ta, $R_{\Omega }$) [see Eq. (7)].

Figure 2

The dimensionless torque $G({\mathrm{Re}}_i)$ for counterrotating TC flow for four different fixed values of ${\mathrm{Re}}_o=-1.4\times {10}^6$, $-0.8\times {10}^6$, $-0.4\times {10}^6$, and 0 (top to bottom data sets); see inset for the probed area of the parameter space.

Figure 3

(a) ${\mathrm{Nu}}_{\omega }$ vs Ta for various $a$; see Fig. 1b for the location of the data in parameter space. A universal effective scaling ${\mathrm{Nu}}_{\omega }\propto {\mathrm{Ta}}^{0.38}$ is revealed. The compensated plots ${\mathrm{Nu}}_{\omega }/{\mathrm{Ta}}^{0.38}$ in (b) show the quality of the effective scaling and the $a$-dependent prefactor of the scaling law.

Figure 4

Prefactor of the effective scaling law ${\mathrm{Nu}}_{\omega }\propto {\mathrm{Ta}}^{0.38}$ (shown in Fig. 3) as a function of $a=-{\omega }_o/{\omega }_i$. The inset shows the effective exponents $\gamma$ which results from an individual fit of the scaling law ${\mathrm{Nu}}_{\omega }\propto {\mathrm{Ta}}^{\gamma }$.