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Steady Rayleigh–Bénard convection between stress-free boundaries

Published online by Cambridge University Press:  04 November 2020

Baole Wen Open the ORCID record for Baole Wen [Opens in a new window] ,
David Goluskin Open the ORCID record for David Goluskin [Opens in a new window] ,
Matthew LeDuc ,
Gregory P. Chini Open the ORCID record for Gregory P. Chini [Opens in a new window]  and
Charles R. Doering Open the ORCID record for Charles R. Doering [Opens in a new window]
Baole Wen*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI48109-1043, USA
David Goluskin
Affiliation:
Department of Mathematics & Statistics, University of Victoria, Victoria, BCV8P 5C2, Canada
Matthew LeDuc
Affiliation:
Department of Physics, University of Michigan, Ann Arbor, MI48109-1040, USA
Gregory P. Chini
Affiliation:
Program in Integrated Applied Mathematics, University of New Hampshire, Durham, NH03824, USA Department of Mechanical Engineering, University of New Hampshire, Durham, NH03824, USA
Charles R. Doering
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI48109-1043, USA Department of Physics, University of Michigan, Ann Arbor, MI48109-1040, USA Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI48109-1042, USA
*
†Email address for correspondence: baolew@umich.edu
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Abstract

Steady two-dimensional Rayleigh–Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios ${\rm \pi} /5\leqslant \varGamma \leqslant 4{\rm \pi}$, where $\varGamma$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\leqslant Ra\leqslant 10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\leqslant Pr\leqslant 10^2$. At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \rightarrow \infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$. The prefactor of this scaling depends on $\varGamma$ and is largest at $\varGamma \approx 1.9$. The Reynolds number $Re$ for large-$Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $\varGamma \approx 4.5$. All of these large-$Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini & Cox (Phys. Fluids, vol. 21, 2009, 083603). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $\varGamma$ is smaller.

Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , R4
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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