Keel planform and elliptical loading

Hello to all in the know of this subject ,

I am a recent yacht design graduate, and it's amazing how one only really starts learning once real projects are in the process and your warm guiding lecturers are gone !

Referencing Larsson's "Princilpes of Yacht Design" and Fossati's "Aero-hydrodynamics" .

The Topic:
A keel shape for least induced resistance - is one that has an elliptical distribution of side force .
With a trapezoidal keel shape, this elliptical force distribution can be achieved by optimizing a taper ratio with a sweep angle, ,as per a graph derived from Lifting Line Theory . Ref : "Princilpes of Yacht Design" , Larsson, pg 109 .

My question:
The theory continues to say that for an 'unswept keel' with a taper ratio of 0.45 - "the area distribution in the vertical direction corresponds to the force distribution" , stating that this is a disadvantage . Why is this a disadvantage ?
Why can I not have an unswept keel, with a taper ratio of 0.45, giving me an elliptical force distribution ? I am not clear on the problem ?

Any insight into this topic would be greatly appreciated .

Regards
Dale Frahm

 

I don't have much time to write in detail now, but the quoted part (if correctly copied here) is clearly not true in general, because physically not possible. The shape of the lift (or pressure) distribution cannot follow the area distribution of the keel (for practical aspect ratios) because of the pressure-relieving influence of the free tip. If it followed the area distribution, you'd have a jump discontinuity of pressure right at the wing tip, where area drops from some finite value to zero, which just cannot occur.
At the upper side of the keel, the hull acts to some extent as a keel extension - if the heel angle is not too high. So, near the keel root a keel behaves grossly as mid-span part of a wing, where the pressure distribution can be approximated by the area distribution in case of an unswept keel.
The pressure can be assumed to follow the area distribution only for very high-AR unswept keels, and only far from the keel tip.

Cheers

 

A keel produces lift by deflecting the water to leeward, and for minimum induced drag you'd want the wake to be deflected uniformly all along the span of the keel. Achieving this is a complex function of the keel shape, interaction with the hull, and free-surface effects.

You can use this spreadsheet to experiment with different keel shapes, independent of the hull. You'll have to make some assumptions about the effect of the hull in carrying the lift past the root of the keel. That might be done by extending the planform of the keel, or by including the vertical profile of the hull but with a reduction in the lift curve slope for the hull portion. The free-surface approximation in this spreadsheet is appropriate for high-speed cases like hydrofoils, but probably isn't very good for keel boat Froude numbers.

What you'll find is the lift distribution lies in between an ellipse and a shape that is proportional to the planform. A taper ratio of 0.4 to 0.5 does a decent job of approximating an elliptical loading for an isolated wing, and this has been adopted for keels.

However an elliptical loading may not be what you want for a keel, as the surface effects mean it may be better to have more loading farther down. The spreadsheet will allow you to determine the planform for minimum drag and then see how much additional drag is associated with a practical planform.

More important than planform is the effect of span. A deeper keel with a more agricultural planform can have less drag than a shallower keel with an ideal planform.

 

Tom, thanks for sharing the spreadsheet. What do you mean by "agricultural" planform? This image qualifies?

AC72 rudders don't seem to have an elliptical or trapeze planform, are they more concerned on vortices than pressure distribution?

 

Quequen,

AC72 rudders have a winglet at the bottom which also acts as an end-plate for the rudder, in addition to the lift or altitude control it is supposed to address.

As a result of the end-plate effect, the induced drag is no more a concern, on the top you have the hull, at the bottom, the winglet, so the induced drag to be addressed is from the winglet's vertical lift.

Cheers

EK

 

Lifting line should give reasonable results as long as the
effective aspect ratio is not too small.
See, for example, the graph of Cl slope versus aspect ratio
for lifting line and some other theoretical calculations:
http://www.boatdesign.net/forums/at...t-ratio-surface-piercing-rudders-liftline.gif

 

As pointed out, AC72 rudders have wings on the bottom, so the span load on the rudder does not go to zero at the end.

The other factor is the immersion of the rudder varies over a wide range. When flying high, only the end of the rudder is in the water, so it needs area there to providing steering control power.