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Heat transport and temperature boundary-layer profiles in closed turbulent Rayleigh–Bénard convection with slippery conducting surfaces

Published online by Cambridge University Press:  06 June 2022

Maojing Huang ,
Yin Wang Open the ORCID record for Yin Wang [Opens in a new window] ,
Yun Bao  and
Xiaozhou He Open the ORCID record for Xiaozhou He [Opens in a new window]
Maojing Huang
Affiliation:
School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen 518055, China
Yin Wang
Affiliation:
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA
Yun Bao
Affiliation:
Department of Mechanics, Sun Yat-Sen University, Guangzhou 510275, China
Xiaozhou He*
Affiliation:
School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen 518055, China Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
*
†Email address for correspondence: hexiaozhou@hit.edu.cn
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Abstract

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.

JFM classification

Boundary Layers: Boundary layer structure

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JFM Papers
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Journal of Fluid Mechanics , Volume 943 , 25 July 2022 , A2
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© The Author(s), 2022. Published by Cambridge University Press

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References

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