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Bounds on Rayleigh-Benard convection with general thermal boundary conditions. Part 1. Fixed Biot number boundaries | | Ralf W. Wittenberg
; | | Date: |
19 Nov 2008 | | Abstract: | We investigate the influence of the thermal properties of the boundaries in
turbulent Rayleigh-Benard convection on analytical bounds on convective heat
transport. Using the Doering-Constantin background flow method, we
systematically formulate a bounding principle on the Nusselt-Rayleigh number
relationship for general mixed thermal boundary conditions of constant Biot
number eta which continuously interpolates between the previously studied
fixed temperature ($eta = 0$) and fixed flux ($eta = infty$) cases, and
derive explicit asymptotic and rigorous bounds. Introducing a control parameter
R as a measure of the driving which is in general different from the usual
Rayleigh number Ra, we find that for each $eta > 0$, as R increases the bound
on the Nusselt number Nu approaches that for the fixed flux problem.
Specifically, for $0 < eta leq infty$ and for sufficiently large R ($R > R_s
= O(eta^{-2})$ for small eta) the Nusselt number is bounded as $Nu leq
c(eta) R^{1/3} leq C Ra^{1/2}$, where C is an eta-independent constant. In
the $R o infty$ limit, the usual fixed temperature assumption is thus a
singular limit of this general bounding problem. | | Source: | arXiv, 0811.3048 | | Services: | Forum | Review | PDF | Favorites | |
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