Boundaries of the conduction regime

The boundaries of the conduction regime for plane cavities were discussed in §5.4. Here, the possible effects of a finite spanwise aspect ratio are considered.

An investigation of the three-dimensional stability of the conduction regime flow is beyond the scope of the present project, but since the vorticity in the plane of transverse odd-symmetry--the plane of shear between the rising and falling fluid streams--only decreases with decreasing $\mathcal S$ (see §7.6.1), it is expected that the critical Grashof number for instability in any cavity of bounded section would be larger. This would be analogous with the linear stability analysis of finite rectangular boxes heated from below; Davis (1967) found that as the length of the rolls (i.e. the span of the box) decreased from infinity, the critical Rayleigh number increased. Thus while the conduction regime is somewhat restrictive--Gill and Davey (1969) calculated that the critical temperature difference for a 1cm air gap is around 40K, but only 5K for a 2cm gap--it may be less so once the third dimension is taken into account.

The importance of treating tall cavities as three-dimensional has been stressed by Chait and Korpela (1989) for another reason. They showed that for air in the $\mbox{$\mathcal A$}, \mbox{$\mathcal S$}\rightarrow\infty$ case, the secondary two-dimensional multiple cells are only stable up to $\mbox{\textit{Gr}}=8550$, beyond which the flow is no longer restricted to vertical-transverse planes.