A Numerical Solution for Natural Convection in Cylindrical Annuli

Abstract

The results of a finite-difference solution for natural convection within horizontal cylindrical annuli for Prandtl numbers near 0.7 (air) are presented. The ranges of Rayleigh number and inverse relative gap width over which such a solution yields valid results are investigated. It is shown that this solution, though formulated for steady flow, can be used to obtain an indication of the Rayleigh number at which transition from a steady to an unsteady flow will occur for a wide range of inverse relative gap widths (2.8–12.5). A recent experimental investigation has shown that steady secondary cellular flows occur immediately prior to transition for this range of inverse relative gap widths, and the Rayleigh number at which these secondary flows occur is accurately predicted by the numerical solution, thereby yielding a good indication of the Rayleigh number at which transition to an unsteady flow occurs. Flow patterns predicted by the numerical solution near transition are compared with photographs of the flow, and excellent qualitative agreement is noted. It is thus shown that this numerical technique gives, at least qualitatively, valid results for all Rayleigh numbers at which the flow is steady for the range of inverse relative gap widths under consideration. Heretofore unavailable data on temperature profiles, radial and angular velocity components, and local Nusselt numbers for Rayleigh numbers near the transition value are presented. It is found that the foregoing parameters (velocity, temperature, and Nusselt number) are little affected by the appearance of the secondary flow for the smaller inverse relative gap widths considered, whereas for the larger inverse relative gap widths a pronounced increase in magnitude of temperatures and velocity components is noted. The overall or mean Nusselt number is not affected by the appearance of these secondary flows for any of the inverse relative gap widths considered.