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# Critical Ra c;Γ 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 Ra c;Γ ¼ 1708ð1 þ C=Γ 2 Þ 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 Ra c;Γ from the nonlinearized DNS data (AFiD) [4]. Temperature contours near the onset of convection are shown for some Γ, as obtained from the linearized DNS. See details in [5-8] and the Supplemental Material [9].

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While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ, a small-Γ 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...

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Context 1
... LSA.-We have verified the estimate (14) for the Γ dependence of the critical Ra c;Γ for the onset of convection with linearized DNSs for the 2D and 3D cases and with the eigenspectrum LSA for the 2D case. The growth rates obtained with both methods are in a very good agreement, see Figs. 1(a) and 1(b). The numerically obtained Ra c;Γ as function of Γ [Eq. (14)] for the isothermal sidewalls are in excellent agreement with the analytical estimates. Equation (14) captures the trend and reflects well also the shape of the neutral curve for the case of adiabatic sidewalls. The best-fit constants C (C ≈ 0.52 for the 2D domain and C ≈ 0.77 ...
Context 2
... neutral curve for the case of adiabatic sidewalls. The best-fit constants C (C ≈ 0.52 for the 2D domain and C ≈ 0.77 for the cylinder) are, however, smaller than the theoretical predictions for the isothermal sidewalls, see Figs. 1(b) and 1(c). Isosurfaces of the temperature of the flow fields near the onset of convection are shown for some Γ in Fig. 1 as well. The azimuthalmode transition found for the cylinder between Γ ¼ 1 and 2 is consistent with the experiments ...

## Citations

... In this regime, the heat transport is mostly insensitive to Γ −1 and adapts the value from the unconfined case. By reducing the horizontal extent the flow first enters a plume-controlled regime for moderate confinement (Γ −1 1), in which the heat transport is enhanced, before, in the severely confined regime (Γ −1 1), the heat transport is strongly reduced (Chong et al. 2015;Chong & Xia 2016 Ahlers et al. 2022). Similar to the rotation-controlled regime in rotating RBC, vertically coherent structures form within the plume-controlled regime in confined RBC (Chong et al. 2015;Hartmann et al. 2021). ...
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Moderate rotation and moderate horizontal confinement similarly enhance the heat transport in Rayleigh–Bénard convection (RBC). Here, we systematically investigate how these two types of flow stabilization together affect the heat transport. We conduct direct numerical simulations of confined-rotating RBC in a cylindrical set-up at Prandtl number $\textit {Pr}=4.38$ , and various Rayleigh numbers $2\times 10^{8}\leqslant {\textit {Ra}}\leqslant 7\times 10^{9}$ . Within the parameter space of rotation (given as inverse Rossby number $0\leqslant {\textit {Ro}}^{-1}\leqslant 40$ ) and confinement (given as height-to-diameter aspect ratio $2\leqslant \varGamma ^{-1}\leqslant 32$ ), we observe three heat transport maxima. At lower ${\textit {Ra}}$ , the combination of rotation and confinement can achieve larger heat transport than either rotation or confinement individually, whereas at higher ${\textit {Ra}}$ , confinement alone is most effective in enhancing the heat transport. Further, we identify two effects enhancing the heat transport: (i) the ratio of kinetic and thermal boundary layer thicknesses controlling the efficiency of Ekman pumping, and (ii) the formation of a stable domain-spanning flow for an efficient vertical transport of the heat through the bulk. Their interfering efficiencies generate the multiple heat transport maxima.
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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 Shishkina et al. ( Phys. Rev. Lett. , vol. 114, 2015, 114302) and Wang et 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.
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We report on the presence of the boundary zonal flow in rotating Rayleigh–Bénard convection evidenced by two-dimensional particle image velocimetry . Experiments were conducted in a cylindrical cell of aspect ratio $\varGamma =D/H=1$ between its diameter ( $D$ ) and height ( $H$ ). As the working fluid, we used various mixtures of water and glycerol, leading to Prandtl numbers in the range $6.6 \lesssim \textit {Pr} \lesssim 76$ . The horizontal velocity components were measured at a horizontal cross-section at half height. The Rayleigh numbers were in the range $10^8 \leq \textit {Ra} \leq 3\times 10^9$ . The effect of rotation is quantified by the Ekman number, which was in the range $1.5\times 10^{-5}\leq \textit {Ek} \leq 1.2\times 10^{-3}$ in our experiment. With our results we show the first direct measurements of the boundary zonal flow (BZF) that develops near the sidewall and was discovered recently in numerical simulations as well as in sparse and localized temperature measurements. We analyse the thickness $\delta _0$ of the BZF as well as its maximal velocity as a function of Pr , Ra and Ek , and compare these results with previous results from direct numerical simulations.