What rotation rate maximizes heat transport in rotating Rayleigh-Bénard convection with Prandtl number larger than one?

The heat transfer and flow structure in rotating Rayleigh-Bénard convection are strongly influenced by the Rayleigh (Ra), Prandtl (Pr), and Rossby (Ro) numbers. For $\text{Pr}\gtrsim 1$ and intermediate rotation rates, the heat transfer is increased compared to the nonrotating case. We find that the regime of increased heat transfer is subdivided into low- and high-Ra-number regimes. For $\text{Ra}\lesssim 5\times {10}^8$ the heat transfer at a given Ra and Pr is highest at an optimal rotation rate, at which the thicknesses of the viscous and thermal boundary layers are about equal. From the scaling relations of the thermal and viscous boundary layer thicknesses, we derive that the optimal rotation rate scales as $1/{\text{Ro}}_{\mathrm{opt}}\approx 0.12{\text{Pr}}^{1/2}{\text{Ra}}^{1/6}$. In the low-Ra regime the heat transfer is similar in a periodic domain and cylindrical cells with different aspect ratios, i.e., the ratio of diameter to height. This is consistent with the view that the vertically aligned vortices are the dominant flow structure. For $\text{Ra}\gtrsim 5\times {10}^8$ the above scaling for the optimal rotation rate does not hold anymore. It turns out that in the high-Ra regime, the flow structures at the optimal rotation rate are very different than for lower Ra. Surprisingly, the heat transfer in the high-Ra regime differs significantly for a periodic domain and cylindrical cells with different aspect ratios, which originates from the sidewall boundary layer dynamics and the corresponding secondary circulation.

Figure 1

Simulated Ra and Pr combinations in a horizontally periodic domain. For each point a series of simulations for various $1/\text{Ro}$ is performed. Further details on the simulations can be found in Appendix pp2.

Figure 2

$\mathrm{Nu}/{\mathrm{Nu}}_0$ versus $1/\text{Ro}$ for different Ra as indicated in the legend. (a) Experimental (open symbols) [27] and simulation (solid symbols) results for $\text{Pr}=4.38$ and in a $\mathrm{\Gamma }=1$ cylinder. (b) and (c) show simulation results for $\text{Pr}=4.38$ and $\text{Pr}=100$, respectively, obtained in a periodic domain.

Figure 3

Comparison of Nu obtained in periodic and cylindrical domains with $\mathrm{\Gamma }=1/2$ and $\mathrm{\Gamma }=1$ for $\text{Pr}=4.38$ at (a) $\text{Ra}={10}^8$ and (b) $\text{Ra}={10}^9$. The $\mathrm{\Gamma }=1/2$ data are from Ref. [37].

Figure 4

The optimal rotation rate for different Ra for (a) $\text{Pr}=4.38$ and different geometries, i.e., a periodic domain and cylindrical domains with $\mathrm{\Gamma }=0.5$ and $\mathrm{\Gamma }=1$, and (b) different Pr in a periodic domain. The vertical line marks the transition from the low- to high-Ra regime. The dashed lines indicate the boundary layer scaling law $1/\text{Ro}\approx 0.12{\text{Pr}}^{1/2}{\text{Ra}}^{1/6}$. The experimental results are from Refs. [19, 21, 27], and the $\mathrm{\Gamma }=1/2$ simulation data are from Ref. [37].

Figure 5

(a) The thermal (${\lambda }_{\theta }$) and viscous (${\lambda }_u$) boundary layer thicknesses versus $1/\text{Ro}$ for $\text{Pr}=4.38$ and $\text{Ra}={10}^8$ obtained in a periodic domain. (b) $\text{Nu}/{\text{Nu}}_0$ as a function of ${\lambda }_{\theta }/{\lambda }_u$ for eight different combinations of Ra and Pr in the low-Ra regime; note that the optimal rotation rate is found at ${\lambda }_{\theta }/{\lambda }_u\approx 0.8$ (see the dashed vertical line). (c) Same as (b), but now for data obtained in a cylindrical domain.

Figure 6

$\mathrm{Nu}/{\mathrm{Nu}}_0$ as a function of (a) $1/\text{Ro}$ and (b) ${\text{Ro}}^{-1/2}{\text{Pr}}^{-1/4}{\text{Ra}}^{-1/12}$, which determines the locations of the optimal rotation rate in the low-Ra regime for eight combinations of Ra and Pr. The vertical line in (b) and (c) marks the location of the maximum heat transfer at ${(1/\text{Ro})}^{1/2}{\text{Pr}}^{-1/4}{\text{Ra}}^{-1/12}\approx 0.35$. (a) and (b) show the cases in a periodic domain, while (c) in a cylinder domain.

Figure 7

Volume renderings of the temperature field at the optimal rotation rate for simulations with $\text{Pr}=4.38$ at (a) $\text{Ra}={10}^8$ and (b) $\text{Ra}=2.3\times {10}^9$. The color map in both panels is identical. The plots show that the flow structure at the optimal rotation rate is very different in the low- and high-Ra regimes.

Figure 8

Correlation $C$ as a function of $\delta z$ for different rotation rates in periodic domains in the (a) low-Ra ($\text{Ra}={10}^8$ and $\text{Pr}=4.38$) and (b) high-Ra ($\text{Ra}=2.3\times {10}^9$ and $\text{Pr}=4.38$) regimes. The numbers indicate the value of $1/\text{Ro}$. The lines for each successive $1/\text{Ro}$ are shifted upward by 1 for visibility. The black line indicates the optimal inverse Rossby number.

Figure 9

The (a) location $\delta z_m$ and (b) magnitude $C_m$ for the peak of the correlation function $C\left(\delta z\right)$ versus 1/Ro for $\text{Pr}=4.38$ and two different Ra in periodic domains. The optimal rotation rate for $\text{Ra}={10}^8$ and $\text{Ra}=2.3\times {10}^9$ is indicated by the solid and dashed lines, respectively.

Figure 10

(a) The Reynolds number defined by the rms of the vertical velocity and (b) the ratio of the vertical and horizontal rms velocities as a function of 1/Ro for $\text{Pr}=4.38$ and two different Ra in periodic domains. The optimal rotation rate for $\text{Ra}={10}^8$ and $\text{Ra}=2.3\times {10}^9$ is indicated by the solid and dashed lines, respectively.

Figure 11

A comparison of (a) the location $\delta z_m$ and (b) magnitude $C_m$ for the peak of correlation function $C\left(\delta z\right)$ versus 1/Ro obtained from simulations performed in a periodic and cylindrical domain ($\mathrm{\Gamma }=1$) for $\text{Pr}=4.38$ and $\text{Ra}=2.3\times {10}^9$. For the cylinder domain the flow data from the region close to the sidewall are excluded from the analysis.