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Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh–Bénard convection

Published online by Cambridge University Press:  03 February 2017

Yang Zhang ,
Quan Zhou  and
Chao Sun
Yang Zhang
Affiliation:
Shanghai Institute of Applied Mathematics and Mechanics, and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
Quan Zhou*
Affiliation:
Shanghai Institute of Applied Mathematics and Mechanics, and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
Chao Sun
Affiliation:
Center for Combustion Energy and Department of Thermal Engineering, Tsinghua University, 100084 Beijing, China
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the statistical properties of the kinetic $\unicode[STIX]{x1D700}_{u}$ and thermal $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}$ energy dissipation rates in two-dimensional (2-D) turbulent Rayleigh–Bénard (RB) convection. Direct numerical simulations were carried out in a box with unit aspect ratio in the Rayleigh number range $10^{6}\leqslant Ra\leqslant 10^{10}$ for Prandtl numbers $Pr=0.7$ and 5.3. The probability density functions (PDFs) of both dissipation rates are found to deviate significantly from a log-normal distribution. The PDF tails can be well described by a stretched exponential function, and become broader for higher Rayleigh number and lower Prandtl number, indicating an increasing degree of small-scale intermittency with increasing Reynolds number. Our results show that the ensemble averages $\langle \unicode[STIX]{x1D700}_{u}\rangle _{V,t}$ and $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$ scale as $Ra^{-0.18\sim -0.20}$, which is in excellent agreement with the scaling estimated from the two global exact relations for the dissipation rates. By separating the bulk and boundary-layer contributions to the total dissipations, our results further reveal that $\langle \unicode[STIX]{x1D700}_{u}\rangle _{V,t}$ and $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$ are both dominated by the boundary layers, corresponding to regimes $I_{l}$ and $I_{u}$ in the Grossmann–Lohse (GL) theory (J. Fluid Mech., vol. 407, 2000, pp. 27–56). To include the effects of thermal plumes, the plume–background partition is also considered and $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$ is found to be plume dominated. Moreover, the boundary-layer/plume contributions scale as those predicted by the GL theory, while the deviations from the GL predictions are observed for the bulk/background contributions. The possible reasons for the deviations are discussed.

JFM classification

Convection: Convection Turbulent Flows: Turbulent Flows
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Journal of Fluid Mechanics , Volume 814 , 10 March 2017 , pp. 165 - 184
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© 2017 Cambridge University Press 

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