Bounds for heat transport in a porous layer

Abstract

Strongly nonlinear heat transport across a porous layer is studied using Howard's (1963) variational method. The analysis explores a bifurcation property of Busse's (1969) multi-a solution of this variational problem and complements the 1972 study of Busse & Joseph by further restricting the fields which are allowed to compete for the maximum heat transported a t a given temperature difference. The restriction arises, as in the case of infinite Prandtl number convection studied by Chan (1971), from letting a parameter tend to infinity from the outset; here, however, the parameter which is assumed infinitely large (the Prandtl-Darcy number) is actually seldom smaller than O (10 7 ). The theoretical bounding heat-transport curve is computed numerically. The maximizing Nusselt number ( Nu ) curve is given a t first by a functional of the single-a solution; then this solution bifurcates and the Nusselt number functional is maximized for an interval of Rayleigh numbers ( R ) by the two-a solution. The agreement between the numerical analysis and recent experiments is striking. The theoretical heat-transport curve is found to be continuously differentiable but has piecewise discontinuous second derivatives. The results of an asymptotic ( R → ∞) analysis following Chan (1971) are in qualitative agreement with the results of numerical analysis and give the asymptotic law Nu = 0.016 R . This law is consistent with the result of the porous version of the well-known dimensional argument leading to the one-third power law for regular convection. The asymptotic results, however, do not appear to be in good quantitabive agreement with the numerical results.