Introduction

In the investigations of vapour transport across cuboids described in previous chapters, it has been assumed that all dependent variables were independent of $z$ (see fig. 2.1a for the geometry and axes), but since the confining walls cannot physically be of infinite horizontal extent, the flow in such a geometry is inherently three-dimensional. This assumption is common to the vast majority of previous studies of both the present and the analogous single fluid heat transfer problems. This is true not only of analytical (Batchelor 1954) and numerical (Elder 1966) investigations, but also many experimental studies. The two main questions about this assumption are whether or not the flow at any particular location is affected by the presence of the end-walls, and what effect the finite spanwise aspect ratio, $\mathcal S$, has on global parameters, such as the overall flow and vapour and energy transport rates.

Nonintrusive techniques for visualizing the density disturbance field, such as the shadowgraph method (Schöpf, Patterson & Brooker 1996), schlieren method (Han & Kuehn 1991) and Mach-Zehnder (Eckert & Carlson 1961) and schlieren (Sernas & Fletcher 1970) interferometry, all involve an integration through the fluid along the path of a light ray. With rare exceptions (Schöpf & Stiller 1997), these rays have been directed horizontally and parallel to the hot and cold walls, leading to images more or less comparable with the results of planar analyses. The problem with this methodology is that such images provide no information on variations along the line of sight, whereas this information is required for their interpretation, e.g. for the back-calculation of the temperature field. The typical remedy is the assumption of uniformity, consistent with the two-dimensional hypothesis.

Some common methods of investigating the flow field, such as thermocouple probes and laser-Doppler and hot wire anemometry do not have this intrinsic planarity, since the measurements are strictly local. Nevertheless, experiments are often set up with large $\mathcal S$ and measurements restricted to the plane of spanwise symmetry ($z=0$). An example of this is the work of Aung, Fletcher and Sernas (1972) on flow in vertical ducts, where $8<\mbox{$\mathcal S$}<32$. Though these figures are doubtless large enough for the midplane flow to be independent of $\mathcal S$, it is necessary to know the minimum value of $\mathcal S$ for which this is true in order to apply the results. Further, at the lower values of $\mathcal S$ employed, the end-walls would certainly have reduced the net vertical flow rate. In their study of a fully enclosed flow, Ozoe et al. (1983) did vary the spanwise location of the point of intersection of the Doppler laser beams, but only within a restricted range about the centre: sufficient to demonstrate local two-dimensionality of the flow, but providing only an upper bound on the domain of influence of the end-walls.

Early investigations of the analogous single fluid heat transfer problem, such as those of Mull and Reiher (described by Jakob 1949, p. 537), focused on measurements of the overall heat transfer rate rather than the flow or temperature fields. After a test in which $\mathcal S$ was halved from 25.7 while the other parameters were kept constant resulted in only a 1.3% reduction in the heat transfer coefficient, $\mathcal S$ was dropped as a governing dimensionless group. For the laminar boundary layer regime, however, $\mathcal S$ must become increasingly important as it decreases, as demonstrated by ElSherniby, Hollands and Raithby (1982) who on halving $\mathcal S$ from 15 found a 3% change in the heat transfer coefficient. In the Handbook of Heat Transfer Fundamentals, Raithby and Hollands (1985) state that very little information exists on the effect of $\mathcal S$ on the overall heat transfer rate for moderate to high $\mathcal A$.

No single approach can lead to the complete solution of this problem, since there are several possible flow regimes; depending on the existence of multiple cells, boundary layers and/or turbulence; all of which will interact differently with the end-walls. The purpose of the present chapter is to provide exact answers to these questions for the conduction-diffusion regime; defined in chapter 5, and here generalized to three-dimensions for the low mass transfer rate limit.

There have been several three-dimensional numerical treatments of the analogous single fluid heat transfer problem; the special case of the cube $(\mbox{$\mathcal A$}=\mbox{$\mathcal S$}=1)$ being recently proposed as a bench-mark computational fluid dynamics problem (Leong, Hollands & Brunger 1998, 1999). Most studies have dealt with values of $\mathcal A$ near unity (e.g. Viskanta, Kim & Gau 1986; Mallinson 1987; Fusegi, Hyun & Kuwahara 1993). This is probably because the greatest accuracy for a given number of grid points is achieved if the solution domain is a cube, according to Mallinson and de Vahl Davis (1977), who did treat cases with $\mbox{$\mathcal A$}=5$, but only for a very few values of the Rayleigh number, Ra, the least value being 10000. Batchelor's (1954) two-dimensional criterion ( $\mbox{\textit{Ra}}<500\mbox{$\mathcal A$}$) indicates that a fully developed flow could only be expected in a cavity of this height if the Rayleigh number were less than about $2500$, and, indeed, the effects of the horizontal surfaces were felt throughout the height of the cavity.

The only treatment, then, of the effects of the end-walls on a fully developed buoyant flow uncovered in my search of the literature was the recent theoretical study of Bühler (1998). There, however, the fluid is almost ideally thermally and electrically conducting and is subjected to a strong magnetic field, so that the buoyancy force and pressure gradient are mostly balanced by the Lorentz force. Since, under these conditions, viscous effects are only important in boundary layers near the walls, there is little connection between Bühler's work and the present problem.

In this chapter, unique analytic solutions of the governing equations are derived for the fully developed mass fraction, temperature and velocity profiles which satisfy the boundary conditions at the vertical walls, for cavities and ducts of rectangular or elliptic section. These solutions are then analysed to reveal how large $\mathcal S$ must be for the plane-flow assumption to be accurate for the central region and to quantify the retarding effect of the end-walls on the vertical flow.