Aspect Ratio Dependence of Heat Transfer in a Cylindrical Rayleigh-Bénard Cell

While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio $\mathrm{\Gamma }$ (diameter/height) for large $\mathrm{\Gamma }$, a small-$\mathrm{\Gamma }$ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given $\mathrm{\Gamma }$ follows ${\mathrm{Ra}}_{c,\mathrm{\Gamma }}\sim {\mathrm{Ra}}_{c,\infty }(1+C{\mathrm{\Gamma }}^{-2}{)}^2$, with $C\lesssim 1.49$ for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range $(1/32)\le \mathrm{\Gamma }\le 32$, the rescaling $\mathrm{Ra}\to {\mathrm{Ra}}_{\ell }\equiv \mathrm{Ra}[{\mathrm{\Gamma }}^2/(C+{\mathrm{\Gamma }}^2){]}^{3/2}$ collapses various OB numerical and almost-OB experimental heat transport data $\mathrm{Nu}(\mathrm{Ra},\mathrm{\Gamma })$. Our findings predict the $\mathrm{\Gamma }$ dependence of the onset of the ultimate regime ${\mathrm{Ra}}_{u,\mathrm{\Gamma }}\sim [{\mathrm{\Gamma }}^2/(C+{\mathrm{\Gamma }}^2){]}^{-3/2}$ in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for $\mathrm{\Gamma }=1$, $1/2$, and $1/3$) for the transition in OB RB convection and explains why, in small-$\mathrm{\Gamma }$ cells, much larger Ra (namely, by a factor ${\mathrm{\Gamma }}^{-3}$) must be achieved to observe the ultimate regime.

Figure 1

Critical ${\mathrm{Ra}}_{c,\mathrm{\Gamma }}$ for the onset of convection: Linear growth rates (colored vertically elongated boxes) from the linearized DNS approach (goldfish) compared to the neutral stability curves (blue lines) from the eigenvalue LSA for (a) 2D box with isothermal sidewalls, (b) 2D box with adiabatic sidewalls, and (c) cylinder with adiabatic sidewall. Black lines show ${\mathrm{Ra}}_{c,\mathrm{\Gamma }}=1708(1+C/{\mathrm{\Gamma }}^2{)}^2$ with a best-fit $C$ for the linearized DNS data (dashed lines) and with theoretical $C$ for isothermal sidewall (solid line). Pluses in (c) show ${\mathrm{Ra}}_{c,\mathrm{\Gamma }}$ from the nonlinearized DNS data (afid) [4]. Temperature contours near the onset of convection are shown for some $\mathrm{\Gamma }$, as obtained from the linearized DNS. See details in [5, 6, 7, 8] and the Supplemental Material [9].

Figure 2

(a) Compensated Nu vs Ra, as obtained in OB experiments and DNSs of RBC in a cylinder for $\mathrm{Pr}\approx 4.4$ (water) and different $\mathrm{\Gamma }$. Most data are for $\mathrm{\Gamma }=1$ and $1/2$, which form the shape of this dependence. The data for extremely small $\mathrm{\Gamma }$ show no discernible dependence. (b) Compensated Nu vs Ra based on the proper length scale $\ell$, for the same data as in (a). In the main plot, the theoretical value of $C=1.49$ is taken, while in the inset $C=0.77$, which corresponds to the best fit of the critical ${\mathrm{Ra}}_{c,\infty }$ for the onset of convection. Now the data for extremely small $\mathrm{\Gamma }$ follow the general trend.

Figure 3

(a) Compensated Nu vs Ra for OB RBC in a cylinder for Pr near 0.8 and in a 3D cell with periodic BCs for $\mathrm{Pr}=1$ and different $\mathrm{\Gamma }$. Vertical lines indicate the onset of the transition at high Ra, observed in Göttingen experiments (the onset moves to higher Ra with decreasing $\mathrm{\Gamma }$). (b) Compensated Nu vs Ra based on the proper length scale $\ell$, for the same data as in (a). Now the transition happens at the same location for all $\mathrm{\Gamma }$ [the vertical lines from (a) merge into one line]. The here presented experimental data from Chavanne et al. [42, 43] hold $\delta \rho /\rho <0.2$ for the density variation and $\delta \kappa /\kappa <0.2$ for the thermal diffusivity variation, as well as $\alpha \mathrm{\Delta }<0.2$ and $0.68\le \mathrm{Pr}\le 1$, i.e., similar almost-OB conditions as in [34, 39, 40, 41, 50] (however, in [34, 39, 40, 41, 50] the upper bounds for the fluid parameter variations are even slightly stricter). Data for $\mathrm{Pr}=0.74$ (gas ${\mathrm{N}}_2$) and $\mathrm{Pr}=0.84$ (gas ${\mathrm{SF}}_6$) were taken using the same apparatus as in [47] but were not published there. The inset shows an enlargement at the highest Ra in normal representation for both axes (see also Supplemental Material [9], which includes [10, 11, 12]).