# Sail Area vs Keel Area - Stability Calculation

Discussion in 'Hydrodynamics and Aerodynamics' started by Dabrownone, May 28, 2012.

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### daiquiriEngineering and Design

It appears to me that the conditions of this thought experiment were changed from the opening post to the last one from Dabrownone.

In the OP, both the underwater and air-exposed plates were receiving fluid from the same side. In the last post, it appears that the fluid velocity is due to a rotational motion of the keel/sail body around a hull, which acts a a pivot. While the first case has nothing to do with any practical situation in the sailboating practice, the latter one vaguely resembles the operating principles of bilge-keels.

However, the equations form the OP model (rectlinear fluid motion) are no more valid for the modified model (fluid velocity due to angular velocity). In the latter case, the variation of velocity along the height/depth of the body surfaces have to be taken into account, and few integrals have to be performed in order to find the point on a plate on which the force ideally acts.

Once it has been sorted out, the next step should be a study and comprehention of fluid dynamics, because the model used here is too simplistic to get any practically useful result.

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### DCockeySenior Member

I don't see any more connection with bilge keels than a conventional single keel. For both types of keels the hull acts as a pivot point.

Now if you are talking about the system which uses a horizontal foil just below the surface fo the water to generate an counter-acting torque then that's a different case, but not a bilge keel.

For the torque to be balanced as in the OP's post #12 the velocity normal to the "keel" would need to be in the same direction as the "wind" which doesn't resolve the problem you identified earlier.

Before getting into fluid dynamics a simple but correct "free body diagram" and the associated force and moment balance equations are needed, along with overtly stating what the applied constraints are.

Last edited: May 29, 2012
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### DabrownoneJunior Member

The force is an instantaneous force applied when the upper plate applies torque to the plate as it rotates around the cylinder.

No reason to believe a dampening force wouldn't exceed the force of motion. If you don't believe me, go out and try and push a car from the side. You should be able to, except the dampening force afforded by friction. If the force you apply is not sufficient, the car will not move.

First of all, thank you for reading my post and providing feedback. I agree that it is quite difficult to read equations when they're written like this. Perhaps i'll attach a TEX document version of it.
The equation actually assumes no current in the water for simplicity. It can be added, but it doesn't clarify anything. The car example is useful here; if the torque applied by the wind is not great enough, it simply will not rotate the plate. That means you don't need motion through the water. Think also of apparent wind or motion against a current; if the apparent wind is 0, it doesn't mean that the true wind is at 0 knots. You could just be moving at the same speed and direction as the wind (same with motion through the water against the current).

Ah, here is important question. The usefulness of the derivation can be applied to a boat with anything being added. We're essentially looking at a minimum 'sail' area that can be maintained in high winds based on the area of the keel. The problems include the fact that a real life keel is not flat (except with a center/daggerboard) and so will damp less. I personally like long keels for their seakeeping ability and directional stability. Here is another advantage, basically stating that the boat will be able to stand up to heavier winds (although heavy winds mean big waves, and this is more critical). A greater 'keel' area (any surface that is at or less than parallel with the water) will mean you can carry more sail as well.
I'm sure there is more. Despite being an applied mathematician, I was less concerned with applications and more just sharing a cool property of boats that I stumbled over. I want to build a small model to test this theory and see if it works as expected. If anyone can point out any errors in the derivation (like not looking at torque) then let me know. Proofs like this usually take multiple passes to get them completely correct.

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### daiquiriEngineering and Design

Right so. The difference is purely functional. A keel is there mainly to provide a force opposing the action of sails (plus a roll damping as a by-product). A bilge keel's primary function is to damp the rolling motions. Since at some point it appeared to me that the idea behind Dabrownone's device was related to the roll motions, I have connected it to bilge keels. But it is quite possible that I have not understood at all what this is all about, as you'll find out from my following post...

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### daiquiriEngineering and Design

Ok, here we go again. You are again mixing up two different situations:
1. a static one - wind acting on sails, your flat-plate device countering that action and maintaining the hull in a given position (a keel)
2. a dynamic one - the hull is rolling, your device is acting as a roll damper or stabilizator (a keel or a bilge keel)
You will have to decide which case are we talking about, because the math is different in two cases.

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### DCockeySenior Member

What causes the counteracting force on the lower plate?

Frictional forces between solid objects such as tire tread and road surface are fundamentally different than the fluid mechanic forces. That's a simple fact of physics. A frictional force between solid objects exists whether there is motion or not. The frictional force equals the opposing force but in the opposite direction as long as the opposing force is less than the maximum static frictional force. Force on a solid object from a fluid only exists if there is relative motion. Fill a bath tub full of water. Put a kitchen spoon in the water and hold it still. No forces other than gravity on the spoon and the counteracting force of the person holding the spoon. Apply a additional force to the spoon. The spoon moves, no matter how small the force.

See response to the car example above which explains the fundamental error in applying it to the interaction of a sails and keels with the air and water.

Where does the v2 velocity of the water relative to the keel come?

If the apparent wind is zero then there will be no aerodynamic forces on the boat.

Keels are generally not more or less parallel with the surface of the water. But it is possible to generate a torque which counter acts the heeling moment of the boat with a foil just below the surface of the water with an appropriate angle of attack. If the foil is on the windward side then the force from the foil would need to be downward. Perhaps this is why daiquiri was trying to connect bilge keels to this duscussion.

Your use of the term "damp" appears to be non-standard. Could you elaborate on it?

Perhaps you should review some basic fluid dynamics first including force genration by fluids interacting with solids. Also relevant are the characteristics of airfoils and "wings". Review/learn about equations of Newtonian mechanics and associated free body diagrams. Then drawa free-body diagram of a boat under sail. You can find examples in most discussions in many books on the science of sailing as well as descriptions of Velocity Prediction Program methods, though be careful as sometimes unstated assumptions are made by those methods. Next write both the kinematic equations and force/accleration equations of motion with proper summations of forces and moments.

Currently this appears to be in the spirit of much 19th century "applied mathematics where a set of equations was derived and solved, and then the corresponding physical situation, if any, was identified.

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### DabrownoneJunior Member

If the sail area is greater than 5 times the size of the keel area, then you will have a dynamic situation, but that doesn't negate anything. It will still act as a roll dampener. I'm not sure what the issue is here
PS i'm putting together a pdf with the derivation, and i'll draw some quick diagrams when I get a moment.

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### daiquiriEngineering and Design

The issue here, imho, is the scientific terminology. As pointed out by DCockey, you are using an unconventional terminology, and it is complicating the comprehension of your ideas and intentions.

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### DCockeySenior Member

There also appears to be a fundamental confusion/misunderstanding of how forces are generated on solid objects in a air or water with frictional forces between solid objects. Such a misunderstanding would be consistent with statements such as:
If the sail area is greater than 5 times the size of the keel area, then you will have a dynamic situation, but that doesn't negate anything.

With a simple solid friction situation there will be no motion and therefore a static situation until the maximum static frictional force/moment is exceeded, then there will be motion and therefore a dynamic situation. As explained several times previously solid-fluid interaction is fundamentally different. However Dabrownone hasn't responded to questions regarding this issue.

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### DabrownoneJunior Member

The resistance of the water as the plate attempts to displace it so that it may rotate.

Frictional forces still exists in water, but that isn't what i'm getting at. There is always resistance when a solid body moves through a fluid. So a greater force is necessary to rotate a plate if it was entirely underwater than if it was entirely in the air. In this situation, part of the plate is under the water, and part of it is above the water, so the difference in their densities (and therefore the resistance they provide) is what is exploited to arrive at the result.

I think that this may be the actual problem. I have to reconsider this derivation, since I appear to have conflated the wind speed with the rotation speed v1. However, v2 is not the speed of the water but the speed of the center of the lower plate as it attempts to rotate.
After thinking about it some more, I think since we are dealing with torque, there may be around this error. I'll get back to you when I figure it out

I wasn't suggesting they were. I meant that any surface that projects outward from the axis of rotation and is parallel or less than parallel (in other words, below the water) will experience resistance as it moves through the water.

Absolutely! Since we can ignore Quantum effects, the physics necessary for this calculation can all arise from classical mechanics. I don't mind re-doing work already done, since part of what i'm doing is trying to understand aspects of sailing that I haven't found much material on.

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### DCockeySenior Member

But no resistance force when the solid body is not moving though the fluid, unlike a solid object against another solid object when there is a resistance force without the bodies in relative motion.

If the boat/keel is rotating then a static analysis is generally not valid. Either the boat will stop rolling or will capsize. Heel/roll angle is the integral of the roll rate with time.

Any reason not to start with one of the standard texts on the science of sailing? There are many available ranging from very simple and perhaps overly-simplified to ones using calculus and differential equations.

Last edited: May 29, 2012
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### PetrosSenior Member

Yes it IS ABSOLUTE! Yes, even a 2 knot wind will cause heel without the counter balance. the keel does not exist to counter the heeling moment!

There is NOTHING in your models I do not understand, I do this kind of thing for a living. Your models are flawed, the assumptions are flawed, you clearly do not understand how sails on sailboats work. The whole exercise is meaningless if it does not match reality, and you are long way from it. The model is meaningless because you fail to understand how a sailboat works.

You just waste your time and ours until you go learn something about sailboats. Everyone here is trying to be patient with you, taking time and explain why what you propose WILL NOT WORK. Yet you continue to assume no one here is as smart as you and do not understand your kooky theory.

If you will not listen and you will not educate yourself, no one will want to help you.

First: learn something about sailboats and the mechanics of how they operate, and than apply equations to it. Several good books were suggested, read them and than get back with us.

Second; realize that you are not the first person to apply mathematics to sailboat design. Lots of very knowledgeable and educated people have poured many millions of \$ into sailboat design, costly computer models using well established and proven methods. Until you understand it as good as them there is NOTHING meaningful you can contribute.

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### TeddyDiverGollywobbler

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### DCockeySenior Member

Only if he takes time to learn the appropriate physics.

15. ### Paul BPrevious Member

This is correct. Anyone can walk down to any marina in the world and watch the boats heel in their slips.

For the OP to have his calculation work the boat would have to be sitting broadside to a current that contributes a force on the keel that matches the force of the wind on the rig/sails.

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