# Relationship between angle of attack of keel and heel angle

Discussion in 'Sailboats' started by stelios, May 18, 2006.

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### steliosNew Member

Hello there!

Peter van Oossanen writes in his paper "Predicting the Speed of Sailing Yachts", that the angle of attack (aoa) of the keel is not the same with the leeway angle (lee), because it is affected by the heel angle (f).

He claims that the real aoa of the keel is:
tan(aoa) = tan(lee) cos(f)

I did some calculations and I conclude to a similar but difrent equation:
sin(aoa) = sin (lee) cos(f) !!

Because it does not seem to be a typografic error I ask if anyone can confirm the right equation.

thanks

Stelios.

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### SailDesignOld Phart! Stay upwind..

At the angles you will be seeing for leeway, the difference is less than negligible. Which does not answer your question.

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### Raggi_ThorNav.arch/Designer/Builder

And at small angles Sin and Tan is almost the same

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### FAST FREDSenior Member

The angle of attack is not the same as the keel angle because their are so many variations in keels.And of course all lift has a sideslip component.

The racy set may have 12ft draft and a foot wide keel , so they can out point their competators and force a situation.

The cruising folks are far more interested in shallow draft , driving over logs and sea land boxes ,.

The cruisers KNOW it takes good HP to pull a heavy boat up a series of hills , so are not as concerned with a few deg of sideslip as they cant point really high because they NEED the hp.

The racer is more interested in Tactics , and is willing to give up speed for the ability to point really high, and rapid manuverability.

FAST FRED

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### SailDesignOld Phart! Stay upwind..

Raggi,
We have got to stop meeting like this - people are talking....
Steve

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### Brent BJunior Member

It looks to me that the sin(aoa) version is correct based on the extreme situation where the heel angle is 90deg. The aoa should go to zero.

Brent

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### MikeJohnsSenior Member

Stelios
Particularly with modern beamy hulls, you must also factor the angle in induced by the trimming of the hull with heel.
When the bow trims down the boat is skewed in the water (unless you have a balanced hull) this provides its own lift factor.

So we have Leeway, induced keel angle due to longitudinal trim, and then you can start looking at the effective lift drag relationships. The above equation looks a little too attractively simple.

Cheers

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### tspeerSenior Member

It may depend on your definition of angle of attack. If you resolve the velocity components into the keel coordinate system, angle of attack is typically defined as

Ukeel = chordwise component
Vkeel = component normal to plane of keel
Wkeel = spanwise component

alpha = arctan(Vkeel/Ukeel)

To go from the boat's course through the water to the keel orientation, you need to rotate through the leeway angle, and then through the heel angle to get to the keel coordinate system. Taking a positive leeway angle (lambda) as slipping to starboard and positive heel angle (phi) as rolling to starboard, the velocity components in the keel coordinate system are:

Ukeel = Vb cos(lambda)
Vkeel = Vb cos(phi) sin(lambda)
Wkeel = Vb sin(phi) sin(lambda)

tan(alpha) = Vkeel/Ukeel = cos(phi) tan(lambda)

I'm with Peter van Oossanen.

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### steliosNew Member

As some of you already mentioned, the two equations are practically the same. But my question was about the theory.

If we assume that trim is zero, I think Tom Speer has cleared the whole problem. (Thanks Tom): Both equations are correct, but they assume a different definition of angle of attack.

In my calculations, angle of attack was defined as the angle between the velocity of the fluid and the projection vector of the velocity on the symmetry surface of the keel.

How I do calculate it? :
I assume the normal vector to the symmetry surface of the keel (Nkeel) and I calculate the angle between this vector and the velocity (Vb). The angle of attack is the calculated angle minus 90 degrees.

To find Nkeel in the boat's course coordinate system, you need to rotate vector {0,1,0} through the heel angle first, and then through the leeway angle.
Thus,
Nkeel = {-cos(phi) sin(lambda), cos(phi) cos(lamblda), sin(phi)}

I am very happy, because I cleared this issue, but now I face a new question: Which definition of angle of attack is more appropriate?

Peter van Oossanen definition is consistent with the chord and span of the keel as originally defined, but I am worrying about the spanwise velocity component (wkeel), which is non zero. I think that the classic empirical formulas and theories, we use to calculate Lift, assume this component to be zero.

To be consistent with zero wkeel, we have to redefine the chord of the keel in the heeled position and define the angle of attack as I mentioned previously. The consequent of this approach is that the foil section we study is a bit different than the originally designed in upright condition.

I think that the approach I propose is more precise and more consistent with the physics. But I have to accept that it is too complex and it looks that practically in the design process, gives almost the same results.