Bounds for heat transport in a porous layer

Abstract

Bounds on convective heat transport in a porous layer heated from below are derived using the background field variational method (Constantin & Doering 1995a, b, 1996; Doering & Constantin 1992, 1994, 1996; Nicodemus, Holthaus & Grossmann 1997a) based on the technique introduced by Hopf (1941). We consider the infinite Prandtl–Darcy number model in three spatial dimensions, and additionally the finite Prandtl–Darcy number equations in two spatial dimensions, relevant for the related Hele-Shaw problem. The background field method is interpreted as a rigorous implementation of heuristic marginal stability concepts producing rigorous limits on the time-averaged convective heat transport, i.e. the Nusselt number Nu, as a function of the Rayleigh number Ra. The best upper bound derived here, although not uniformly optimal, matches the exact value of Nu up to and immediately above the onset of convection with asymptotic behaviour, Nu[les ]9/256Ra as Ra→∞, exhibiting the Howard–Malkus–Kolmogorov–Spiegel scaling anticipated by classical scaling and marginally stable boundary layer arguments. The relationship between these results and previous works of the same title (Busse & Joseph 1972; Gupta & Joseph 1973) is discussed.