Development of boundary layers in transient buoyant convection about a vertical plate in a porous medium

Abstract

Transient natural convection over a vertical plate in a porous medium is considered. The Brinkman-extended Darcy flow model is adopted. The Rayleigh number is large to render a boundary layer-type flow pattern. An order-of-magnitude analysis is performed. Numerical solutions are secured over broad ranges of nondimensional parameters. The results indicate that, for high Darcy and Rayleigh numbers, the porous system shows a double boundary layer structure (delta(m) > delta(T)), similar to the pure fluid system. In the opposite limit of low Darcy and Rayleigh numbers, the Darcy term effect is notable in the thermal and momentum boundary layers, and the present results are in accord with the Darcy model. In the intermediate parameter region, epsilon(3/2)J(1/2)Pr < Ra-1/2 Da < epsilonJ(1/2)sigma(1/2)Pr and (epsilon/J)(1/2) < Ra(1/2)Da < epsilon(3/2)J(1/2)Pr, the Darcy term effect is seen only in the outer momentum boundary layer for transient periods. At very small times (t < JKsigma/epsilonkappa(e), K/epsilonnu(f)), the Darcy term effect is very small in both the thermal and momentum boundary layers even for low Darcy numbers. The early-time boundary layer structure of the porous system is akin to that based on the pure-fluid model. In all of the parameter regions, with high heat capacity ratio sigma, the ratio of delta(m) to delta(T) for small times is larger than that of the steady state.