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Available potential energy in Rayleigh–Bénard convection

Published online by Cambridge University Press:  09 August 2013

Graham O. Hughes ,
Bishakhdatta Gayen  and
Ross W. Griffiths
Graham O. Hughes*
Affiliation:
Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, Australia
Bishakhdatta Gayen
Affiliation:
Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, Australia
Ross W. Griffiths
Affiliation:
Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, Australia
*
†Email address for correspondence: Graham.Hughes@anu.edu.au
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Abstract

The mechanical energy budget for thermally equilibrated Rayleigh–Bénard convection is developed theoretically, with explicit consideration of the role of available potential energy, this being the form in which all the mechanical energy for the flow is supplied. The analysis allows derivation for the first time of a closed analytical expression relating the rate of mixing in symmetric fully developed convection to the rate at which available potential energy is supplied by the thermal forcing. Only about half this supplied energy is dissipated viscously. The remainder is consumed by mixing acting to homogenize the density field. This finding is expected to apply over a wide range of Rayleigh and Prandtl numbers for which the Nusselt number is significantly greater than unity. Thus convection at large Rayleigh number involves energetically efficient mixing of density variations. In contrast to conventional approaches to Rayleigh–Bénard convection, the dissipation of temperature or density variance is shown not to be of direct relevance to the mechanical energy budget. Thus, explicit recognition of available potential energy as the source of mechanical energy for convection, and of both mixing and viscous dissipation as the sinks of this energy, could be of further use in understanding the physics.

JFM classification

Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulent convection
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , R3
Copyright
©2013 Cambridge University Press 

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