Double keel or single keel ?

[tspeer]

Originally Posted by rwatson

...I am leaning towards Ricks hint at using asymetric foil sections, and angled boards as Brent was talking about.

For a symmetrical section, the angle of attack for zero lift is zero. For a cambered section, the angle of attack for zero lift is negative, and typically on the order of minus three degrees. If you reference the alignment of the keels to their zero lift lines instead of the chord, there's no difference between a symmetrical section and a cambered section in this context.

Because of the orientation of the zero lift lines, when you mount a cambered section so the chord is parallel to the boat's center plane, it is effectively toed in already by the amount of the zero lift angle of attack.

...If a chord of 150mm in a fin can produce lift at 6 knots using Ricks example, I would expect a slight attack angle would help keep the flow attached, and also provide directional assistance, thereby reducing the leeway angle.

The leeway angle will be reduced, but the boat's heading will also change. If there were no difference in drag, then the boat's path through the water would be the same, but the bow would be rotated more off the wind to align with the direction of travel. So the most immediate effect of toe-in or camber has as much to do with changing the angle of the apparent wind to the boat's centerline as it does to changing the leeway angle relative to the hull.

Of course, the whole point of the exercise is not to keep the drag constant, but to reduce it. This is really the difference between the cambered section and the symmetrical section. Each section has a range of angles of attack at which its profile drag is minimized. A designer can make the low-drag region deep and narrow, or shallow and wide. For a symmetrical section, the low-drag region is necessarily centered on zero lift, by symmetry, and the maximum lift at which the section operates in the low-drag region is only half the width of the low-drag region. It's not only easy but probable that the lift coefficient going to windward will be outside the low-drag region for a symmetrical section. For a cambered section, the low-drag region can be centered on, or encompass, the intended operating condition. So camber should be viewed from the point of optimizing the profile drag rather than adding lift at the operating point.

For a twin-keel configuration, it would be advantageous if the low-drag region encompassed zero lift so the windward keel's drag was minimized, and the operating lift coefficient of the leeward keel when sailing to windward.

And I think your sentence "the zero lift line of the windward keel would be aligned with the incident flow at the leeway angle" means that one should try to keep the angle of the windward fin as close to the boats true heading (not steered heading) as possible.

A designer can come at this backwards, especially if a velocity prediction program (VPP) is available. A first cut using a non-toed in symmetrical keel will provide an estimate of the leeway angle. The zero lift line of the selected section (symmetrical or asymmetrical) can then be toed in by the amount of the leeway angle. Then the VPP rerun to see the difference in performance due to the better optimization of profile drag and induced drag (due to loading the deeper keel).

That is probably the answer to the mystery of the optional toe-in angle for a boat, and it sounds like it would vary a fair bit from one boat to another depending on hull/sail/size factors.
If I dont master the math, I might just incorporate adjustable foils and experiment in real life conditions.

A flap or trim tab on the keels would allow you to change the zero lift line very easily and home in on the best effective toe-in angle. It wouldn't even have to be adjustable from inside the boat, although that would be the most convenient. One of the main reasons for twin keels it to allow the boat to dry out, so the flaps could be adjusted at low tide.

[Brent Swain]

The hydro - aero dynamic forces are forces between the leeward keel and the sail. The position of the hull relative to either is somewhat irrelleveant, in the small anges we are talking about. I've built boats with assymetrical keels and symetrical ones . There was no noticeable difference in performance on any point of sail.
I can see problems with keels toed inward when running downwind in rough conditions. When the boat rolls ,the deepest keel points inward, steering the bow in that direction. Roll the other way and the other keel steers the boat in the opposite direction. Thus keels toed inward, while having no net benefit sailing to windward , can drastically reduce directional stability downwind ,and make a boat far more prone to broaching in rough seas.
Brent