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].

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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... 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 ...
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... 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 ...

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... 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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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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