Extent of the effect of the end-walls

The representation (7.59) of $v_n$ suggests a means of finding a simple approximate answer to one of the principal questions of this chapter: how large $\mathcal S$ must be for the flow in the spanwise plane of symmetry to be essentially described by the Jones-Furry solution (7.24).

Near the front end-wall $(Z=1/2)$ it is clear that the second series can be neglected in comparison to the Jones-Furry solution and the first series if $\mathcal S$ is sufficiently large. In the first series, the effect of the $k^{\mathrm{th}}$ term has reduced in magnitude by a factor of e after a distance from the front end-wall of $\zeta=1/(2k\upi )$. Thus the first term, which also has the largest magnitude due to the pre-exponential factor with $k^{-3}$ dependence, acts over the longest distance. It has reduced to less than 0.5% of its influence at $\zeta=0$ for $\zeta\geq\ln (200)/2\upi$. Since the location of the plane of spanwise symmetry is $\zeta=\mbox{$\mathcal S$}/2$, the flow there may be expected to be sensibly independent of the presence of the end-walls for $\mbox{$\mathcal S$}\geq\ln (200)/\upi \doteq1.7$. This is a refinement of the prediction $\mbox{$\mathcal S$}\gg2$ based on Saint-Venant's principle given at the start of this section. Applying a similar procedure to $v_f$ leads to $\mbox{$\mathcal S$}\geq 2\ln(200)/\upi \doteq 3.4$.

To investigate this approximation, plots of the difference between the actual solution for $v_n$, calculated from (7.59) for each half of the section, and the Jones-Furry solution (7.24) are given in figure 7.4 for values of $\mathcal S$ spanning the estimate

Figure 7.4: The effect of the end-walls. Contours of the difference between the full solution (7.59) and the Jones-Furry limiting form (7.24). Contours at $\pm1(1)5\%$ of the maximum ( $=\max v_n^{\Vert} = \sqrt{3}/216$), as labelled in the rightmost plot.

$\mbox{$\mathcal S$}=1.7$. The figure shows that the departure from the two-dimensional Jones-Furry limit is indeed less than 1% for $\mbox{$\mathcal S$}\geq 1.7$.