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Bounds on Rayleigh-Benard convection with general thermal boundary conditions. Part 2. Imperfectly conducting plates | | Ralf W. Wittenberg
; | | Date: |
19 Nov 2008 | | Abstract: | The effect of imperfectly conducting bounding plates on the heat transport in
turbulent thermal convection in the Rayleigh-Benard problem is considered in
the context of analytical upper bounds. Beginning with the evolution equations
in the fluid in the Boussinesq approximation, coupled through temperature and
flux continuity to identical upper and lower conducting plates with diffusive
heat flow, we derive an upper bound on the enhanced convective transport, as
given by the Nusselt number Nu, in terms of the Rayleigh number Ra measuring
the averaged temperature drop across the fluid layer; our formulation uses the
"background" variational bounding approach due to Doering and Constantin. We
find that the bounds depend on $sigma = d/lambda$, where d is the ratio of
plate to fluid thickness and lambda is the conductivity ratio. In particular,
for a fluid bounded by plates with finite thickness and conductivity, while the
bound on Nu scales as $Ra^{1/2}$ in terms of the usual Rayleigh number Ra, for
sufficiently large Ra (depending on sigma) we show that $Nu leq c(sigma)
R^{1/3}$, where the control parameter R is a Rayleigh number defined in terms
of the full temperature drop across the entire plate-fluid-plate system. In the
asymptotic $Ra o infty$ limit, the usual fixed temperature situation forms a
singular limit of the general bounding problem, while fixed flux conditions
appear most relevant to the Nu-Ra scaling even for highly conducting plates. | | Source: | arXiv, 0811.3051 | | Services: | Forum | Review | PDF | Favorites | |
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