Numerical solutions

Solutions were obtained by the means described above (§5.1). The mesh was that of §5.1.5 and figure 5.3, except that for the narrowest cavity ( $\mbox{$\mathcal A$}=10$; see fig. 5.10 below), the number of elements in the vertical direction was doubled to 32. The results of two sample runs are plotted in figure 5.8.

Figure 5.8: Contours of stream-function (a,d), $T$ (b,e) and $m$ (c,f) from two Fastflo runs: $\varPhi =N=0$ in (a-c) and $\varPhi =0.69,\,N=0.5$ in (d-f). In both runs, $\mbox{$\mathcal A$}=5,\,\mbox{\textit{Gr}}(1+N)\mbox{\textit{Pr}}_r=10^{3},\,\mbox{\textit{Pr}}_I=\mbox{\textit{Sc}}=0.61$ and $\mbox{\textit{Pr}}_r=0.71$.

The first is for the zero mass transfer limit ($\varPhi =N=0$), for which the interfacial velocity (2.59) and interdiffusion energy flux (2.50) vanish. The second uses a rather large value of $\varPhi$, 0.69, to vividly illustrate the effects of a finite mass transfer rate.

That the mesh is sufficiently fine was demonstrated by the grid independence tests of §5.2.1. Further confidence can be derived from the agreement between the finite element and analytic solutions in the fully developed regions (§5.5; figs 5.9 - 5.11).

A third check on the accuracy of the solutions was obtained by comparing the results for several cases with those of the parallel FIDAP (Fluid Dynamics International 1996) solutions of Dr Jonathan Harris (McBain & Harris 1998). The solutions generated by FIDAP and Fastflo were found to agree closely. For example, for $\mbox{$\mathcal A$}=1,\,N=0.1,\,\mbox{\textit{Pr}}_r=0.71,\,\mbox{\textit{Pr}}_I=\mbox{\textit{Sc}}=0.61,\,\varPhi =0.105$ and $Gr(1+N)\mbox{\textit{Pr}}_r=10^{3}$, using similar 12$\times$12 nonuniform meshes of quadratic elements, the integrated mass and energy transfer rates agreed to within 0.6% and 0.02%, respectively.

That Gill's centrosymmetry properties do not apply when transpiration is introduced ( $\varPhi \neq 0$) is readily apparent in figure 5.8. Compare, for example, the stream-lines (figs 5.8a,d) in the `departure corners' (top-right and bottom-left). The most pronounced effect of the interfacial velocity in the temperature (e) and mass fraction (f) distributions is the reduction of the gradients normal to the hot wall (right) with a corresponding increase at the cold wall (left). The combined influence of the interdiffusion effect and the variation of specific heat (remembering that these cannot be separated--see §2.1.3) is most obvious in the dissimilarity of the temperature and mass fraction profiles across the fully developed region (temperature is more curved, since $\varPhi _T=1.49>\varPhi =0.69$) and the extreme stretching of the isotherms in the top-right corner of figure 5.8(e): in this region, the mixture is rich in vapour ($m$ large), so that, since $\mbox{\textit{Pr}}_I>0$, the effective Prandtl number,

$$\mbox{\textit{Pr}}_r+\mbox{\textit{Pr}}_I[1-\exp(-\varPhi )]m,$$

is comparatively high.

One of the most obvious features of figure 5.8 is that in (a), the entire boundary is a stream-line, whereas in (d), stream-lines intersect the vertical walls. Since, in general, a stream-line originating on the right wall does not intersect the left wall at the same height, $y$, there is a net vertical mass flow through the central portion of the cavity. The horizontal surfaces are impermeable, though, so that in the end-zones the net contribution of mass from the vertical walls at a particular level must be nonzero (positive in the lower portion, and negative above). The contribution in the fully developed portion vanishes since the horizontal component of velocity there is uniform, by (4.39). The reason for the departure in the end-zones is the shifting of the concentration isopleths by the convection cell. For example, the gradients are clearly steeper in the `starting corners' (bottom-right and top-left) in figure 5.8(f). Because the net vertical mass flow rate depends on the turning flow near the ends, no information about it can be gained from the fully developed solution; therefore, $c$ can only be evaluated a posteriori.