Bounds on Rayleigh - Bénard convection with general thermal boundary conditions . Part 2 . Imperfectly conducting plates By Ralf

The effect of imperfectly conducting bounding plates on the heat transport in turbulent thermal convection in the Rayleigh-Bénard 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 σ = d/λ, where d is the ratio of plate to fluid thickness and λ 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 in terms of the usual Rayleigh number Ra, for sufficiently large Ra (depending on σ) we show that Nu ≤ c(σ)R, 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 → ∞ 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.