Steady free convection in a porous medium heated from below

Abstract

This is an experimental and numerical study of steady free convection in a porous medium, a system dominated by a single non-linear process, the advection of heat. The paper presents results on three topics: (1) a system uniformly heated from below, for which the flow is cellular, as in the analogous Bénard-Rayleigh flows, (ii) the role of end-effects, and (iii) the role of mass discharge. Measurements of heat transfer are used to establish further the validity of the numerical scheme proposed by the author (1966a), while the other flows allow a more extensive study of the numerical scheme under various boundary conditions. The results are very satisfactory even though only moderately non-linear problems can be treated at present.The main new results are as follows. For the Rayleigh-type flow, above a critical Rayleigh number of about 40, the heat transferred across the layer is proportional to the square of the temperature difference across the layer and is independent of the thermal conductivity of the medium or the depth of the layer. This result is modified when the boundary-layer thickness is comparable to the grain size of the medium. The investigation of end-effects reveals variations in horizontal wave-number and a pronounced hysteresis and suggests an alternative explanation of some observations by Malkus (1954).

Nonsteady End Effects in Hele-Shaw Cells

“…A new findin~ is the generation of vortices in the leading end corners of the cavity. This is a similar observation as reported by Elder (1967) After a regime with time~dependent convection that we discuss below, a reverse transition to a steady flow, exhibiting a hysteresis, was observed at high…”

Rock bed thermal storage: Concepts and costs

“…Technically, this concept may be higher risk than Concept 1, because thermocline destabilisation is a possibility as a consequence of natural convection caused by buoyancy effects, unless the rock is sufficiently small. According to Elder [11] natural convection and destratification should be negligible for a porous medium of depth L heated from below provided that the Rayleigh number is less than 40, a requirement which is fulfilled [12] …”

The Sensitivity of Thermohaline Groundwater Circulation to Flow and Transport Parameters: A Numerical Study Based on Double-Diffusive Convection above a Salt Dome

“…The dimensionless formulation based on the streamfunction has been used successfully for several different application cases of coupled flow and transport, such as for the classical saltwater intrusion test case defined by Henry (1960), and for the Elder problem (1967a;1967b), for convection rolls (Holzbecher 1998a), and some others. Simms and Garven (2004) use a streamfunction formulation in non-normalized formulation.…”