Journal
JOURNAL OF FLUID MECHANICS
Volume 804, Issue -, Pages 370-386Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/jfm.2016.528
Keywords
Benard convection; variational methods
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Funding
- US National Science Foundation (NSF) [OCE-0824636, OCE-1332750]
- Office of Naval Research
- NSF's Institute for Pure and Applied Mathematics
- NSF [PHY-1205219, DMS-1515161]
- Simons Fellowship in Theoretical Physics
- John Simon Guggenheim Memorial Foundation Fellowship
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1515161] Funding Source: National Science Foundation
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We consider Rayleigh-Benard 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 -> infinity. 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 parallel to del h parallel to(2), where parallel to del h parallel to(2) is the mean squared magnitude of the boundary height gradients. Application to a particular geometry is illustrated for sinusoidal boundaries.
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