An explanation for the multivalued heat transport found experimentally for convection in a porous medium

Abstract

Convection experiments were conducted in a porous slab 3 m in diameter and 30 cm thick, using two quite different media: a filling of rubberized curled coconut fibre and of clear polymethylmethacrylate beads. The second experiment involved the successful use of a visualization scheme for the flows at the upper boundary. Convection began in a hexagonal pattern with a slight tendency to form into rolls, but became very complex, irregular and three-dimensional at higher Rayleigh numbers, without developing any obvious temporal instabilities. Above a Rayleigh number of 1000 a significant number of dendritic downwellings appeared, where smaller downwellings seemed to feed into larger areas such that the whole complex may have converged into a single downwelling plume. The visualization provides direct confirmation that the lateral scale of the convection decreases with increasing Rayleigh number, approximately as ([Ascr ] + C)−0.5.Nusselt number versus Rayleigh number curves were obtained for both experiments. The only feature they have in common is a central section where the slope on a log/log graph is slightly over 0.5. On the graph from the first experiment, this section is preceded by a slope close to 1 and followed by a slope close to 0.33. The temperature measured at a point in the fill 25 mm below the top boundary was unsteady at conditions representative of the upper two segments of the graph; sensitivity was insufficient to measure fluctuations at lower temperature differences. The Nusselt number for the bead fill jumps upward just above onset (where [Ascr ] = 4π2), rapidly settles to a slope of 0.52, and then gradually breaks upward again to a slope of greater than 1. Increases in conductivity and permeability close to the boundary are not a large enough fraction of the boundary-layer thickness to cause this. A new phenomenon, lateral thermal dispersion, appears to be responsible. It occurs because there is no constant separation distance between adjacent channels in a bead fill. Thermal exchange in the junction pores exceeds the average if the flow is fast enough, especially when the fluid is more conductive than the beads.A simple boundary-conduction theory can be matched to the uncontaminated results. It is based on relative scaling of the residence time of fluid in the boundary layer, and predicts Nusselt number growth as the 0.55 power of Rayleigh number toward the high values typical of major geothermal areas.