Prandtl-, Rayleigh-, and Rossby-Number Dependence of Heat Transport in Turbulent Rotating Rayleigh-Bénard Convection

Experimental and numerical data for the heat transfer as a function of the Rayleigh, Prandtl, and Rossby numbers in turbulent rotating Rayleigh-Bénard convection are presented. For relatively small $\mathrm{Ra}\approx {10}^8$ and large Pr modest rotation can enhance the heat transfer by up to 30%. At larger Ra there is less heat-transfer enhancement, and at small $\mathrm{Pr}\lesssim 0.7$ there is no heat-transfer enhancement at all. We suggest that the small-Pr behavior is due to the breakdown of the heat-transfer-enhancing Ekman pumping because of larger thermal diffusion.

Figure 1

The ratio of the Nusselt number $\mathrm{Nu}(\Omega )$ in the presence of rotation to $\mathrm{Nu}(\Omega =0)$ for $\mathrm{Pr}=4.38$ ($T_m=40.00°\mathrm{C}$). (a) Results as a function of the rotation rate in $\mathrm{rad}/\sec$. (b) The same results as a function of the Rossby number Ro on a logarithmic scale. Red solid circles: $\mathrm{Ra}=5.6\times {10}^8$ ($\Delta =1.00\mathrm{K}$). Black open circles: $\mathrm{Ra}=1.2\times {10}^9$ ($\Delta =2.00\mathrm{K}$). Purple solid squares: $\mathrm{Ra}=2.2\times {10}^9$ ($\Delta =4.00\mathrm{K}$). Blue open squares: $\mathrm{Ra}=8.9\times {10}^9$ ($\Delta =16.00\mathrm{K}$). Green solid diamonds: $\mathrm{Ra}=1.8\times {10}^{10}$ ($\Delta =32.00\mathrm{K}$). Here and in Figs. 2, 3 experimental uncertainties are typically no larger than the size of the symbols.

Figure 2

The ratio $\mathrm{Nu}(\Omega )/\mathrm{Nu}(\Omega =0)$ for $\mathrm{Ra}=1.2\times {10}^9$ as function of Ro on a logarithmic scale. Red solid circles: $\mathrm{Pr}=3.05$ ($T_m=60.00°\mathrm{C}$). Black open circles: $\mathrm{Pr}=4.38$ ($T_m=40.00°\mathrm{C}$). Purple solid squares: $\mathrm{Pr}=6.41$ ($T_m=23.00°\mathrm{C}$). Blue stars: Numerical simulation for $\mathrm{Ra}=1.0\times {10}^9$ and $\mathrm{Pr}=6.4$ from Ref. 15.

Figure 3

The ratio $\mathrm{Nu}(\Omega )/\mathrm{Nu}(\Omega =0)$ as function of Ro on a logarithmic scale. Red solid circles: $\mathrm{Ra}=2.73\times {10}^8$ and $\mathrm{Pr}=6.26$ (experiment). Black open circles: $\mathrm{Ra}=2.73\times {10}^8$ and $\mathrm{Pr}=6.26$ (DNS). Blue solid squares: $\mathrm{Ra}=1\times {10}^9$ and $\mathrm{Pr}=6.4$ (DNS) [15]. Red open squares: $\mathrm{Ra}=1\times {10}^8$ and $\mathrm{Pr}=6.4$ (DNS). Green open diamonds: $\mathrm{Ra}=1\times {10}^8$ and $\mathrm{Pr}=20$ (DNS). Black stars: $\mathrm{Ra}=1\times {10}^8$ and $\mathrm{Pr}=0.7$ (DNS).

Figure 4

Numerical result for the ratio $\mathrm{Nu}(\Omega )/\mathrm{Nu}(0)$ as function of Pr for $\mathrm{Ra}={10}^8$ and $\mathrm{Ro}=1.0$ (red open diamonds), $\mathrm{Ro}=0.3$ (black open circles), and $\mathrm{Ro}=0.1$ (blue open squares).

Figure 5

3D visualization for $\mathrm{Ra}={10}^8$ of the temperature isosurfaces in a cylindrical sample with $\Gamma =1$ at $0.65\Delta$ (red) and $0.35\Delta$ (blue) for $\mathrm{Pr}=0.7$ (upper figures) and $\mathrm{Pr}=6.4$ (lower figures). Left: $\mathrm{Ro}=\infty$. Right: $\mathrm{Ro}=0.30$. The snapshots were taken in statistically stationary regimes.

Figure 6

The horizontally averaged vertical temperature gradient $d\theta /dz$ at the sample midplane as a function of Ro for $\mathrm{Ra}=1\times {10}^8$. Blue open squares: $\mathrm{Pr}=0.7$. Red open circles: $\mathrm{Pr}=6.4$. Black open diamonds: $\mathrm{Pr}=20$.