the amount of ballast carried near the tip is highly dependent on the tip chord.
Fig 6.8 is obtained from the lifting line theory, as is Fig 6.9, which shows the penalty if the force distribution is not elliptical. Only the zero sweep angle case is presented. The vertical axis shows the percentage increase in drag for the trapezoidal keel as compared to the elliptic one. It may be seen that the penalty is smallest at a taper ratio of about 0.45, as expected. In this case very small drag increases are noted, for practical aspect ratios less than 1%. If the taper is far from the optimum the increase may be up to 3-4%. It is interesting to note that the importance of a correct force distribution is quite dependent on the aspect ratio. For long keels with ARe smaller than 1.0 the penalty is practically insignificant.
Tip shape The lifting line theory is a useful tool in explaining the most important features of planform design. The most important conclusion to be drawn is that the primary parameter is the aspect ratio, whose influence on the forces can be computed with good accuracy using the formulae above. Sweep angle and taper may contribute a few per cent to the efficiency of the keel, but there are other factors not included in the theory which could also have some influence. We discuss one of them here, namely, the shape of the tip.
In the theory the wing is replaced by a vortex system, which is appropriate for the major features of the flow. However, in reality, the detailed shape of the wing tip will have some influence on the velocity distribution. One effect is that the trailing, free vortices, which are aligned with the local flow, may be positioned slightly differently, depending on the tip shape. This is important, since the effective span of the wins in Fis 6.5 is determined from the distance between the two trailing vortices far behind the wing. For a sailing yacht this means that it is the depth of the trailing vortex that defines the effective aspect ratio, rather than the actual keel depth.
Fig 6.10 shows the measured results of a series of tips. Both the planform view and a front view are shown, and the change in aspect ratio relative to the theory is given for each configuration. The location of the tip vortex is also indicated. It may be seen that the best design is the simplest one with a square cut off in both views. The worst one is a tip that is rounded in both directions. In the former case the geometric aspect ratio is reduced by 0.04, while in the latter case the reduction is 0.20.
The reason why the square tip is better, considering first the planform view, is that the flow along the tip is guided backwards by the flat ending. It will not tend to move upwards as much as it would if the tip had been rounded. So the vortex stays further down. In fact, it would be possible to improve the tip shape even further by rounding the forward part in such a way that the flow approaches the tip smoothly, but the important thing is to keep the aft part straight.
The square shape in the front view is better than the rounded one,
Fig 6.10 Influence of tip shape and aspect ratio w
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