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Available potential energy in Rayleigh–Bénard convection

  • Graham O. Hughes (a1), Bishakhdatta Gayen (a1) and Ross W. Griffiths (a1)

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.

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Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503537.
Ahlers, G. & Xu, Xiaochao 2001 Prandtl-number dependence of heat transport in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 86, 33203323.
Gargett, A. E. & Holloway, G. 1984 Scaling in thermal convection: a unifying theory. J. Mar. Res. 42, 1527.
Gayen, B., Griffiths, R. W., Hughes, G. O. & Saenz, J. A. 2013a Energetics of horizontal convection. J. Fluid Mech. 716, R10.
Gayen, B., Hughes, G. O. & Griffiths, R. W. 2013b Completing the mechanical energy pathways in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett.  (submitted).
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 2756.
Hughes, G. O., Hogg, A. Mc C. & Griffiths, R. W. 2009 Available potential energy and irreversible mixing in the meridional overturning circulation. J. Phys. Oceanogr. 39, 31303146.
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169184.
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 323.
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335364.
Lorenz, E. N. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7, 157167.
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135167.
Prastowo, T., Griffiths, R. W., Hughes, G. O. & Hogg, A. Mc C. 2009 Effects of topography on the cumulative mixing efficiency in exchange flows. J. Geophys. Res. 114, C08008, doi:10.1029/2008JC005152.
Scotti, A. & White, B. 2011 Is horizontal convection really non-turbulent? Geophys. Res. Lett. 38, L21609.
Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42, 36503653.
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137168.
Tailleux, R. 2009 On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy. J. Fluid Mech. 638, 339382.
Winters, K. B., Lombard, P. N., Riley, J. J. & D’Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115128.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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