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 ft 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 residts 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 angidar 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.Transactions of the ASME Copyright © 1971 by ASME Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 06/08/2015 Terms of Use: http://asme.org/terms

A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus

“…For other than very small Prandtl numbers, only a few numerical studies of posttransitional flow in a horizontal annulus have been published. In nearly all of these studies, a two-dimensional model was employed (Powe, Carley & Carruth 1971 ;Fant, Rothmayer & Prusa 1991 ;Cheddadi et al 1992 ;Yoo 1996) and therefore neither the spiral secondary flows which are observed experimentally in annuli of moderate R nor the influence of solid endwalls which bound a finite-length annulus could be analysed. Rao et al (1985) carried out steady three-dimensional calculations for a case of moderate R. However, only a fluid with Prandtl number of 5000 was studied which limits the range of practical applicability of their results.…”

The numerical and analytical study of bifurcation and multicellular flow instability due to natural convection between narrow horizontal isothermal cylindrical annuli at high Rayleigh numbers

“…From numerical results obtained in this study, and results discussed in Powe et (1971) and Rao et (1985), it can be seen that the narrow-gap solutions for flows prior to multicellular transition (and even beyond to a certain extent) behave in a seemingly steady-state manner. Assuming the steady-state condition and using the leading-order solution of Eq.…”

“…Therefore, the multicellular regime that is associated with the narrow type gaps (at high Rayleigh number) most likely reverts back to two cells and then, finally to a pure conductive mode when the velocities tend to zero as G -»• 0 (or G + 0 for a specified Rayleigh number). Contrary to this result, many investigators (Walton, 1980;Powe et , 1971;Liu et£!_., 1962) believed that as the gap approached zero, a true Bénard type instability would evolve, where the typical critical Rayleigh number (Ray_a) of about 1,700 is approached. Instead, it appears that after a certain point, the transitional Rayleigh number increases with decreasing gap size (to well above 1,700), until eventually, a conductiondominated flow results as the gap approaches zero (see Chapter 6).…”

Investigations of Natural Convection Heat Transfer in Nanofluids Filled Horizontal Semicircular-Annulus using Nonhomogeneous Dynamic Model

“…There are several researches on both numerical and experimental investigation of natural convection heat transfer between two concentric circular cylinders. Powe et al (1971) studied numerical solution for natural convection in cylindrical annuli and classified the flow patterns conferring to suitable combinations of the Rayleigh number and the radius ratio. Kuehn and Goldstein (1976) conducted an experimental and theoretical study of natural convection in an annulus between horizontal concentric cylinders.…”