A numerical study of non-Darcian natural convection in a vertical enclosure filled with a porous medium is performed. The flow is modeled using the Brinkman-Forchheimer-extended Darcy equations. The governing equations are solved with the SIMPLER algorithm and good agreement with previously reported numerical and experimental results is found. An order of magnitude analysis and the numerical results demonstrate the importance of non-Darcian effects. For high Darcy numbers (Da > 10−4), both extensions are of the same order of magnitude and must be used simultaneously. In addition, Forch-heimer's extension must be included for Pr ≤ 1.0 and all Darcy numbers. Finally, Nusselt number correlations are presented for three different ranges of the Darcy number covering wide ranges of the governing parameters.

A numerical study of non-darcian natural convection in a vertical enclosure filled with a porous medium

The nondarcy regime for vertical boundary layer natural convection in a porous medium

Publisher Summary This chapter describes the convective and radiative heat transfer in porous media. Convective and radiative heat transfer and multiphase transport processes in porous media, both with or without phase change, have gained extensive attention. Several applications related to porous media require a detailed analysis of convective heat transfer in different geometrical shapes, orientations, and configurations. Based on the specific application, the flow in the porous medium may be internal or external. Forced convection over external boundaries in the presence of a porous medium constitutes a very important subject area. Analysis of convective heat transfer from this type of external boundary embedded in a porous medium has many important applications. It is shown that for the flow field the boundary effect is confined within a thin momentum boundary layer, which often plays an insignificant role in the overall flow consideration. The models developed for simultaneous heat and mass transfer processes in multiphase porous systems may have differences due to the various assumptions made in developing each model. The thermal radiation characteristics of porous beds are also elaborated.

Convective and Radiative Heat Transfer in Porous Media

Natural-convection flow in an enclosure with adiabatic horizontal walls and isothermal vertical walls maintained at a fixed temperature difference has been investigated. At high values of the natural-convection parameter, the Rayleigh number, a recirculating pocket appears near the corners downstream of the vertical walls, and the flow separates and reattaches at the horizontal walls in the vicinity of this recirculation. There is also a considerable thickening of the horizontal layer. In some previous studies by different authors, this corner flow was considered to be caused by an internal hydraulic jump, and the jump theory was used to predict bifurcation of the steady flow into periodic flow. The present work examines the corner phenomenon closely to decide if it is indeed caused by a hydraulic jump. The results of the analysis reveal the oversimplification of the problem made in the previous studies: there is no connection of the corner phenomenon with a hydraulic jump. The separation of flow at the ceiling is not a feature of hydraulic jumps, and the essential energy loss associated with hydraulic jumps is not observed in the corner flow. It is shown that the corner structure is caused by thermal effects. Owing to the temperature undershoots in vertical boundary layer, which are known to be caused by the stable thermal stratification of the core, relatively cold fluid reaches the upper corner. This cold fluid detaches from the ceiling like a plume at high Rayleigh numbers, and causes the separation and recirculation.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112094000522

On the high-Rayleigh-number structure of steady laminar natural-convection flow in a square enclosure

Si le nombre de Prandtl est suffisamment faible il apparait des cellules transversales comme resultat de l'instabilite hydrodynamique. On considere le cas de frontieres inferieure et superieure, conductrice et isolante

Natural convection in a shallow cavity

Cavity flows driven by an applied horizontal temperature gradient are considered in the high Rayleigh number limit for a fluid-saturated porous medium. The analysis is concerned with the behaviour of the vertical boundary layer equations near the corners of the cavity. Implications for the structure of the core flow are discussed. The present results, which are new, compare well with a recent numerical solution. Although the results are consistent with the standard hypothesis that the vertical boundary layers empty into the core, they are not in agreement with the corner behaviour previously suggested in the literature. The analysis of the vertical boundary layer structure is also applicable to cavity flows in electrically conducting fluids in the limit when magnetic drag is the dominant force.

Thermally Driven Cavity Flows in Porous Media. I. The Vertical Boundary Layer Structure Near the Corners

Thermal convection in a cavity filled with a porous medium: A classification of limiting behaviours

Shallow cavity flows, driven by horizontal thermal gradients, are analysed over a range of Rayleigh numbers R and Prandtl numbers σ. Emphasis is placed on the limit in which R is comparable in size to the cavity aspect ratio L . Departures from two-dimensional steady Hadley cells, defined by the limit L → ∞ at fixed R and σ, are shown to be associated with nonlinear end effects. Eigenvalue calculations indicate the existence of a critical Prandtl number σ c , below which the core structure is not necessarily parallel with the horizontal boundaries. For σ c the parallel flow core is destroyed at Rayleigh numbers R > R c (σ). Results for the stability boundary R c (σ) are presented.

Onset of Multicellular Convection in a Shallow Laterally Heated Cavity

An analysis is given of the structure, at high Rayleigh numbers, of thermally driven cavity flows in fluid-saturated porous media. Particular emphasis is placed on the description of the diffusive layers near the horizontal surfaces. Solutions consistent with the core behaviour determined in part I of this series are obtained. These solutions, which are new, indicate that the horizontal boundary layers have a double structure in which the outer layer has a thickness O(R$^-\frac{1}{4}$), where R is the Rayleigh number. Corner interactions between the horizontal and the vertical boundary layers are also discussed, and the present theory provides a first-order description of the entire flow field.

Thermally driven cavity flows in porous media II. The horizontal boundary layer structure

Cavity flows driven by an applied horizontal temperature gradient are encountered in a variety of industrial and environmental situations. Determination of the core structure, away from the flow boundaries, depends strongly on the corner behaviour of the vertical boundary layer equations. An analysis of these corner zones is given for flows in which both the Rayleigh number and the Prandtl number are large. It is shown that both corner flows have a double structure with the outer layers being convection dominated. The consequences of the analysis for the core mass flux, a much debated question in the literature, are discussed. Numerical integration, with spectral decomposition, is shown to lead to values of the core mass flux that are in good agreement with existing experimental data.

P. G. Daniels

Papers

Thermally Driven Cavity Flows in Porous Media. I. The Vertical Boundary Layer Structure Near the Corners

Thermal convection in a cavity filled with a porous medium: A classification of limiting behaviours

Onset of Multicellular Convection in a Shallow Laterally Heated Cavity

Thermally driven cavity flows in porous media II. The horizontal boundary layer structure

Thermal convection in a cavity: the core structure near the horizontal boundaries