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Vertical natural convection: application of the unifying theory of thermal convection

Published online by Cambridge University Press:  06 January 2015

Chong Shen Ng Open the ORCID record for Chong Shen Ng [Opens in a new window] ,
Andrew Ooi ,
Detlef Lohse  and
Daniel Chung
Chong Shen Ng*
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia
Andrew Ooi
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia
Detlef Lohse
Affiliation:
Physics of Fluids Group, Faculty of Science and Technology, J. M. Burgers Center for Fluid Dynamics and MESA+ Institute, University of Twente, 7500 AE Enschede, The Netherlands
Daniel Chung
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia
*
†Email address for correspondence: chongn@unimelb.edu.au
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Abstract

Results from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^{5}$–$1.0\times 10^{9}$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann–Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh–Bénard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl–Blasius–Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the $-1/2$-power of a wind-based Reynolds number, where the ‘wind’ of the GL theory is interpreted as the maximum mean velocity. Away from the walls, the dissipation of the turbulent fluctuations, which can be interpreted as the ‘bulk’ or ‘background’ dissipation of the GL theory, is found to obey the Kolmogorov–Obukhov–Corrsin scaling for fully developed turbulence. In contrast to Rayleigh–Bénard convection, the direction of gravity in vertical natural convection is parallel to the mean flow. The orientation of this flow presents an added challenge because there no longer exists an exact relation that links the normalised global dissipations to the Nusselt, Rayleigh and Prandtl numbers. Nevertheless, we show that the unclosed term, namely the global-averaged buoyancy flux that produces the kinetic energy, also exhibits both laminar and turbulent scaling behaviours, consistent with the GL theory. The present results suggest that, similar to Rayleigh–Bénard convection, a pure power-law relationship between the Nusselt, Rayleigh and Prandtl numbers is not the best description for vertical natural convection and existing empirical relationships should be recalibrated to better reflect the underlying physics.

JFM classification

Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 764 , 10 February 2015 , pp. 349 - 361
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© 2015 Cambridge University Press 

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