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Bounds for convection between rough boundaries

Published online by Cambridge University Press:  09 September 2016

David Goluskin  and
Charles R. Doering
David Goluskin*
Affiliation:
Department of Mathematics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA
Charles R. Doering
Affiliation:
Department of Mathematics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
*
†Email address for correspondence: goluskin@umich.edu
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Abstract

We consider Rayleigh–Bénard convection in a layer of fluid between rough no-slip boundaries where the top and bottom boundary heights are functions of the horizontal coordinates with square-integrable gradients. We use the background method to derive an upper bound on the mean heat flux across the layer for all admissible boundary geometries. This flux, normalized by the temperature difference between the boundaries, can grow with the Rayleigh number ($Ra$) no faster than $O(Ra^{1/2})$ as $Ra\rightarrow \infty$. Our analysis yields a family of similar bounds, depending on how various estimates are tuned, but every version depends explicitly on the boundary geometry. In one version the coefficient of the $O(Ra^{1/2})$ leading term is $0.242+2.925\Vert \unicode[STIX]{x1D735}h\Vert ^{2}$, where $\Vert \unicode[STIX]{x1D735}h\Vert ^{2}$ is the mean squared magnitude of the boundary height gradients. Application to a particular geometry is illustrated for sinusoidal boundaries.

JFM classification

Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 804 , 10 October 2016 , pp. 370 - 386
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Copyright
© 2016 Cambridge University Press 

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