We report results for the temperature profiles of turbulent Rayleigh-Bénard convection (RBC) in the interior of a cylindrical sample of aspect ratio $\Gamma \equiv D/L=0.50$ ($D$ and $L$ are the diameter and height, respectively). Both in the classical and in the ultimate state of RBC we find that the temperature varies as $A\times \ln (z/L)+B$, where $z$ is the distance from the bottom or top plate. In the classical state, the coefficient $A$ decreases in the radial direction as the distance from the side wall increases. For the ultimate state, the radial dependence of $A$ has not yet been determined. These findings are based on experimental measurements over the Rayleigh-number range $4\times {10}^{12}\lesssim \mathrm{Ra}\lesssim {10}^{15}$ for a Prandtl number $\mathrm{Pr}\simeq 0.8$ and on direct numerical simulation at $\mathrm{Ra}=2\times {10}^{12}$, $2\times {10}^{11}$, and $2\times {10}^{10}$, all for $\mathrm{Pr}=0.7$.

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.

We report a direct numerical simulation (DNS) study of the mean velocity and temperature profiles in turbulent Rayleigh–Bénard convection (RBC) at low Prandtl numbers ( RBC.

We report a systematic study of the collective effect of thermal plumes on the probability density function (p.d.f.) $P(\delta T)$ of temperature fluctuations $\delta T(t)$ in turbulent Rayleigh–Bénard convection. By decomposing $\delta T(t)$ into four basic fluctuation modes associated with single and multiple warm and cold plumes and a turbulent background, we derive an analytic form of $P(\delta T)$ based on the convolutions of the five independent modes. To test the derived form of $P(\delta T)$ in the multiple-plume regions, where the thermal plumes are heavily populated, we conduct time series measurements of temperature fluctuations in two convection cells; one is a vertical thin disk and the other is an upright cylinder of aspect ratio unity. For a given normalized position in most regions of the convection cell, all of the measured p.d.f.s $P(\delta T)$ for different Rayleigh numbers fall onto a single master curve, once $\delta T$ is normalized by its root-mean-square (r.m.s.) value $\sigma _T$ . It is found that the measured $P(\delta T/\sigma _T)$ at different locations along the symmetric horizontal and vertical axes of the convection cells can all be well described by the derived form of $P(\delta T/\sigma _T)$ . The fitted values of the parameters associated with the number of plumes in multiple plume clusters and their relative strengths and degrees of intermittency are closely linked to the spatial distribution of thermal plumes and local dynamics of the large-scale circulation in a closed convection cell. Our work thus provides a unified theoretical approach for understanding scalar p.d.f.s in a turbulent field, which is very useful not only for the present study but also for the study of many turbulent mixing problems of practical interest.

We report direct numerical simulations (DNS) of the Nusselt number $Nu$ , the vertical profiles of mean temperature $\varTheta (z)$ and temperature variance $\varOmega (z)$ across the thermal boundary layer (BL) in closed turbulent Rayleigh–Bénard convection (RBC) with slippery conducting surfaces ( $z$ is the vertical distance from the bottom surface). The DNS study was conducted in three RBC samples: a three-dimensional cuboid with length $L = H$ and width $W = H/4$ ( $H$ is the sample height), and two-dimensional rectangles with aspect ratios $\varGamma \equiv L/H = 1$ and $10$ . The slip length $b$ for top and bottom plates varied from $0$ to $\infty$ . The Rayleigh numbers $Ra$ were in the range $10^{6} \leqslant Ra \leqslant 10^{10}$ and the Prandtl number $Pr$ was fixed at $4.3$ . As $b$ increases, the normalised $Nu/Nu_0$ ( $Nu_0$ is the global heat transport for $b = 0$ ) from the three samples for different $Ra$ and $\varGamma$ can be well described by the same function $Nu/Nu_0 = N_0 \tanh (b/\lambda _0) + 1$ , with $N_0 = 0.8 \pm 0.03$ . Here $\lambda _0 \equiv L/(2Nu_0)$ is the thermal boundary layer thickness for $b = 0$ . Considering the BL fluctuations for $Pr>1$ , one can derive solutions of temperature profiles $\varTheta (z)$ and $\varOmega (z)$ near the thermal BL for $b \geqslant 0$ . When $b=0$ , the solutions are equivalent to those reported by Shishkinaet al.(Phys. Rev. Lett., vol. 114, 2015, 114302) and Wanget al.(Phys. Rev. Fluids, vol. 1, 2016, 082301(R)), respectively, for no-slip plates. For $b > 0$ , the derived solutions are in excellent agreement with our DNS data for slippery plates.