Non-Darcian effects on natural convection in porous media confined between horizontal cylinders

The inertia, boundary and velocity-square terms, normally not included in the flow analysis, are included in the study of natural convection between isothermal, concentric cylinders (inner cylinder heated) filled with saturated, porous media. The results show that all of these effects reduce the heat transfer rate with the boundary term being the most significant. It is shown that since at high Rayleigh numbers the flow adjacent to the confining walls becomes of boundary-layer type, with a very thin sublayer over which the velocity reaches its maximum value, then as long as the contribution of the velocity-square term is small, Darcy's model holds for very large Rayleigh and Prandtl numbers. A flow regime diagram showing the pseudo-conduction, Darcy and non-Darcy regimes, is given. FLUID llow in saturated, porous media with high ~~~bilities manifests some of the characteristics of the flow in the absence of a rigid matrix, i.e. the inertia and boundary effects not included in Darcy's model may become significant (l-4). Also, at high velocities the rigid matrix resistance is no longer given by Darcy's law, i.e. the velocity-square term becomes significant (SJ. The inertia or development term which may be significant in the leading edge region (external flows) (l-3) and in the entrance region (internal flows) (4) becomes less significant in the presence of a solid matrix because the lower the permeability the shorter is the development region (l, 3,4). This is due to the proportionality of the flow resistance to the local velocity which tends to make the velocity field uniform. In boundary-layer flows, the relative influence of the boundary term on the heat transfer rate is manifested in the ratio of the thermal to momentum boundary-layer thickness ( 1,3,4). In general, for a given flow driving force (pressure gradient or buoyancy~ the results have shown that each of the three effects, i.e. inertia, boundary and velocity-square, reduces the heat transfer rate. In natural convection between isothermal, horizontal cylinders-as in other enclosure convec- tions-as the buoyancy potential (Rayleigh num- ber) increases, three regimes are observed. First is the regime of no significant convection, i.e. conduction re- gime, then the non-boundary-layer convection or the conduction+onvection regime is observed. Finally the boundary-layer flow regime is reached. At still higher Rayleigh numbers the flow becomes unstable and various transitions have been observed. Convection between isothermaf, horizontal cylinders has been studied in detail in the presence of a rigid matrix in Darcy's regime Cd-91 and also in the absence of any rigid matrix (lo, 111. It is found that when no rigid matrix is present, and for air, unsteady three- dimensional flows begin above Ra = 3 x lo6 and 10' for R = 2 and 4, respectively fll), where R is the diameter ratio and Ra is the Rayleigh number. When the solid matrix is present and Darcy's law holds, it is found that for R = 2 and Raye2 greater than 134, unsteady three-dimensional convection will take place (6), where y2 is proportional to the inverse of the ~~eability. In this study, the non-Darcy effects on natural convection between two concentric, isothermal, horizontal cylinders (inner cylinder heated) filled with a fluid-saturated, porous medium, are studied by solving the two-dimensional conservation equations numeri~lly. Some of the Ra and Ray-' values considered in this study are larger than the critical values given above for very small and very large permeabilities. However, since no definite statement can yet be made for moderate permeabilities the numerical results are assumed to be for stable flows. Figure 1 gives a schematic of the problem considered.