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Penetrative turbulent Rayleigh–Bénard convection in two and three dimensions

Published online by Cambridge University Press:  14 May 2019

Qi Wang Open the ORCID record for Qi Wang [Opens in a new window] ,
Quan Zhou Open the ORCID record for Quan Zhou [Opens in a new window] ,
Zhen-Hua Wan Open the ORCID record for Zhen-Hua Wan [Opens in a new window]  and
De-Jun Sun
Qi Wang
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China
Quan Zhou
Affiliation:
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
Zhen-Hua Wan*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China
De-Jun Sun*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

Penetrative turbulent Rayleigh–Bénard convection which depends on the density maximum of water near $4^{\circ }\text{C}$ is studied using two-dimensional and three-dimensional direct numerical simulations. The working fluid is water near $4\,^{\circ }\text{C}$ with Prandtl number $Pr=11.57$. The considered Rayleigh numbers $Ra$ range from $10^{7}$ to $10^{10}$. The density inversion parameter $\unicode[STIX]{x1D703}_{m}$ varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thicknesses ($F_{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}_{t}^{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D706}_{b}^{\unicode[STIX]{x1D703}}$) increases with increasing $\unicode[STIX]{x1D703}_{m}$, and the relationship between $F_{\unicode[STIX]{x1D706}}$ and $\unicode[STIX]{x1D703}_{m}$ seems to be independent of $Ra$. The centre temperature $\unicode[STIX]{x1D703}_{c}$ is enhanced compared to that of Oberbeck–Boussinesq cases, as $\unicode[STIX]{x1D703}_{c}$ is related to $F_{\unicode[STIX]{x1D706}}$ with $1/\unicode[STIX]{x1D703}_{c}=1/F_{\unicode[STIX]{x1D706}}+1$, $\unicode[STIX]{x1D703}_{c}$ is also found to have a universal relationship with $\unicode[STIX]{x1D703}_{m}$ which is independent of $Ra$. Both the Nusselt number $Nu$ and the Reynolds number $Re$ decrease with increasing $\unicode[STIX]{x1D703}_{m}$, the normalized Nusselt number $Nu(\unicode[STIX]{x1D703}_{m})/Nu(0)$ and Reynolds number $Re(\unicode[STIX]{x1D703}_{m})/Re(0)$ also have universal relationships with $\unicode[STIX]{x1D703}_{m}$ which seem to be independent of both $Ra$ and the aspect ratio $\unicode[STIX]{x1D6E4}$. The scaling exponents of $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ and $Re\sim Ra^{\unicode[STIX]{x1D6FD}}$ are found to be insensitive to $\unicode[STIX]{x1D703}_{m}$ despite of the remarkable change of the flow organizations.

JFM classification

Convection: Convection in cavities
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 870 , 10 July 2019 , pp. 718 - 734
Copyright
© 2019 Cambridge University Press 

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