# Square top mains?

Discussion in 'Sailboats' started by kenwstr, Dec 6, 2005.

1. ### Doug LordGuest

My system as partially used on the model would allow complete adjustment of the camber at the top of the sail while sailing; not sure it's worth it though....

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### froshSenior Member

In any conditions that are less than fully powered up off the wind, an increase in camber near the head of the sail, if it was flatter there than in the main body of the sail, should translate to greater lift. The increase in drag would be vector that would be at right angles to motion (therefore only increase heeling) or maybe diagonally but somewhat in the direction of motion. Therefore the increase in drag would not counteract speed of the hull through the water.

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### Seafarer24Sunset Chaser

Tip vortices

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### froshSenior Member

Seafarer, why would Tip Vortices be a problem in this situation?

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### Seafarer24Sunset Chaser

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### froshSenior Member

Hi Seafarer, read the link suggested and it seems that rectangular planforms work quite well: QUOTE:
"A rectangular planform also has a pretty small tip vortex, and it can be made smaller, close to or better than that of an ellipse, if the tip of the sail is twisted to leeward. This is exactly how gaff rigs are shaped and why they are actually pretty efficient."

By increasing camber in the head of a rectangular planform OFF THE WIND, lift will be increased; tip vortices will be worse than if the sail was flat and twisted to leeward, but the drag force is not directed against the direction of motion of the hull. In all likelihood the net result will be greater forward force.

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### tspeerSenior Member

Actually, the optimum planform is not triangular, rectangular, or elliptical. It's egg-shaped, like a sailboard rig, with the maximum chord at roughly 40% of the luff length and a sizeable amount of roach in the head. If you allow the sail to twist off more than the optimum twist, you need to have more area in the head to make up for it. If you camber the head more than the foot, you need less roach in the head.

What matters to induced drag is the total lift distribution up and down the mast. You can tailor the lift distribution through planform shape (local area), twist (local angle of attack), or camber (local zero lift angle of attack). Vortices aren't only shed at the tip - the vorticity shed is proportional to the slope of the spanwise lift distribution. And by the total lift distribution, I mean the main plus jib.

There's not a unique solution - you can trade off twist, camber, planform shape, and main vs jib trim. If you achieve the same distribution of lift along the mast, you'll get the same induced drag. Actually, what you're shooting for is to have a linearly varying normal-wash along the span. The wake should come off the leech as though it were a smooth rigid sheet and not have wrinkes in it.

And even the spanwise lift distribution will change according to what you define as "optimum". There's always a limit to the heeling moment. You want to make the rig as tall as the heeling moment can stand. That's why 18' skiffs have tall, medium and short rigs. Once you've set the heeling moment, you can decrease the induced drag by making the spanwise lift distribution more tapered, but over a longer mast so as to maintain the same heeling moment.

A triangular sail is not a very good planform for its height. A square-top main is a much better fit to the optimum planform for the same size mast. A pin-head main might give the fat-head main a run for its money if it's fit to a mast that is taller so the two rigs have the same height of the center of area.

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### froshSenior Member

Tom, I liked the explanation you gave re planforms in your last posting. However, would you like to give your view on whether it is desirable to have any significant camber in the head region of a fat head sail (whichever type).
Would you consider the situation with the AW on the beam, and also when the AW is approaching from ahead of the beam? Also the effect of light and stronger wind strengths.

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### tspeerSenior Member

I don't think you can consider the camber at the head by itself. You have to know what the planform and twist are, too.

Compared to a pin-head main, you would need less camber (in terms of % chord) for a fat-head sail to have a comparable lift distribution, assuming that the twist is also comparable. The lift at a given section is the lift coefficient times the local chord:

Cl * c = dCl/dAlpha * (Alpha + twist - w/U - Alpha0 + WindShear) * c

Cl = local lift coefficient
c = local chord
dCl/dAlpha = 2D lift curve slope
Alpha = angle of attack of sail
twist = change in local incidence from section where Alpha is measured
w = local normal-wash induced by the wake
U = local apparent wind speed
Alpha 0 = zero lift angle of attack
WindShear = local change in apparent wind direction

Let's say you are shooting for a spanwise lift distribution like this:

because it minimizes the induced drag for a height of the center of effort around 45% of the mast height.

If you want to keep the same spanwise lift distribution for the two sails (to minimize the induced drag at the same height of the center of effort), you have to increase the local lift coefficient by ratio of the local chord for the two sails:

Cl_fat * c_fat = Cl_narrow * c_narrow
Cl_narrow = Cl_fat * c_fat/C_narrow

The 2D lift curve slope is basically the same regardless of camber or chord, so that affects the pin-head and fat-head sails the same. The apparent wind direction is the same, and the induced velocity will be the same. So the only two things you really have to play with are zero lift angle of attack, Alpha0, and the twist. Camber changes Alpha0.

Chances are, you'll do some of both. The pin-head main will probably use less twist, and it will need more camber in the head.

One of the reasons to use a fat-head main is because it does tend to twist off more, so it tends to help compensate for gusts. The optimum twist that goes with this spanwise lift distribution is

Notice that the optimum twist is proportional to the lift coefficient, CL, which means it's also proportional to the angle of attack. You need more twist as the angle of attack increases. A fat-head helps give you that effect.

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### froshSenior Member

Hi Tom, I am not sure if your last posting was intended to answer the questions asked in my posting re the effect of camber in the head of a fat head sail. You said at the outset that we should not look at the head in isolation. That might be true to a point but a modern sailboard sail can be considered as two independently behaving zones. The lower 60% approx. of the luff length is always driving and the camber is virtually locked in by outhaul position, camber inducers, and the tightness of the monofilm skin.
The upper 40% however is gust responsive and the upper mast lays off under wind pressure allowing varying degrees of twist and the accompanying reduction in camber.
Due to the speed of a sailboard relative to true wind, AW is always fairly far forward of the beam.
This is not necessarily the case for sail boats. Therefore this tendency for the head to readily depower in sailboards may be sacrificing usable power if a sailboard rig and sail was utilized on a sail boat.
This then was the rationale behind my original questions.

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### tspeerSenior Member

I think the ideal way to attack sail design is to work the problem backwards. Start with the loads you want to produce, then figure out the shape that will give you those loads. Finally, aproximate the ideal shape with something that is practical to build, and analyze how well it performs.

When you specify the spanwise loading, you'd said everything the structural designer needs to know to size the mast and rigging. You can match the loading to the boat's stability and vice versa. You know the induced drag corresponding to the ideal sail trim, and the rest of the drag components are largely independent of the sail shape and trim - so you have a pretty good handle on the performance as well. The spanwise loading becomes sort of a "contract" between the rig design and the hull design.

If you're designing a fat-head main and a flexible rig, I think it would be very useful to know what you're aiming for. How else are you going to trade off, say, sail shape vs the flexibility of the mast?

For example, let's say you picked a camber that had a zero lift angle of attack of -5 degrees, and designed a sail with an asppect ratio of 4 and a 5% gap to match the spanwise lift distribution above. When trimmed at 11 degrees angle of attack, the lift coefficient is 0.9 and the required twist is -8 degrees.

Now vary the camber linearly from foot to head and keep the same twist of -8 degrees, increasing the camber so that the zero lift angle of attack at the head is -7 degrees, and decreasing it so the zero lift angle of attack is -3 degrees. In order to keep the same lift distribution, the planform shape has to change, as shown by this plot:

With more camber at the head, the sail becomes more tapered, and with less camber at the head, the sail becomes fuller.

The change in camber is equivalent to changing the twist on the sail. So if you kept the same camber and allowed the sail to twist off, you'd have to shape it like the sail with reduced camber at the head. This is the kind of trade that a sail designer might make to engineer the rig, trying to match the tendency of the sail to twist with the desired twist so as to meet the specified load distribution.

Suppose that you modified the roach to make a fat-head sail, like this:

You can just about recover the design lift distribution by flattening the head so the zero lift angle of attack goes from -5 to -1 degree:

Allowing the head to fall off an additional 4 degrees while keeping the same camber would do the same thing.

The downside is you're not making full use of the sail area. The lift coefficient for the sections at the head drops off considerably for the fat head ("Analysis" curve).

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### froshSenior Member

Thanks Tom for your detailed and informative postings. You have covered a fair slab of sail aerodynamics in the last two postings. I'm sure all the theoreticians amongst the readers of this forum got a really good fix of the good stuff as well.

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### tspeerSenior Member

You're welcome.

A circular arc camber has a zero lift angle of attack that is approximately

alpha0 = -115 h/c

alpha0 = zero lift angle of attack, in degrees
h = maximum height of the camber line
c = chord

So a 10% thick circular arc section will have a zero lift angle of attack of about -11.5 deg.

Another rule of thumb is the zero lift angle of attack is parallel to the slope of the sail at the three-quarter chord point. So if you wanted to put black stripes on your sail to help visualize the twist, short stripes centered on the three-quarter chord would be a good way to do it.

One reason one might want to increase the camber of a fat-head main instead of flattening it is if the leech tension wasn't sufficient to hold the desired twist. More camber will increase the moment tending to twist off the head, but it will also result in a change in the zero lift angle of attack that is opposite the twist. Which is the more desireable approach I can't say. You would have to consider the elasticity of the sail material and bending of the mast as well as the sail shape, and do that for the design wind strength and off-design wind speeds.

One paper in the literature that looks at the effect of planform shape on a flexible sail is:

Greenhalgh, S. and Curthiss, H. C. Jr, "Aerodynamic Characteristics of a Flexible Membrane Wing", AIAA Journal, Vol. 24, No. 4, April 1986, pg. 545 - 551.

They examined triangular (chord varied as 1-y), parabolic (chord varied as sqrt(1-y)) and elliptical (chord varied as sqrt(1-y^2)) planforms for a sail behind a 41% thick wingmast-style leading edge that had a mast chord of 10% of the root sail chord. Aspect ratios ranged from 9.8 for the triangualr sail to 6.7 for the elliptical planform (due to the added area of the roach). The different planforms resulted in increasing amounts of roach, with the elliptical planform being nearly a fat-head shape. The root was mounted to a reflection plane, so although they tested a half-span model in the wind tunnel, it was representative of a hang-glider or a sail that went right down to the water.

The sail was given a conical shape as lines from the root chord were extended to a virtual apex past the head of the sail and the sail shaped to lie on that surface. They varied the height of the virtual apex to change the twist and camber, especially at the head. The triangular sections had a linear twist, while the fuller sections had less twist at the bottom, and progressively increasing twist at the head.

They got good agreement between lifting line theory and experimental data in the linear range.

Another very relevant paper is:

SUGIMOTO, TAKESHI (Tokyo, University, Japan), "Wing Design for Hanggliders Having Minimum Induced Drag", Journal of Aircraft 1992, 0021-8669 vol.29 no.4 (730-731).

Sugimoto looks at the design for minimum induced drag for a given root bending moment - the same problem that was solved by RT Jones for rigid wing - but Sugimoto comes up with a simple expression to represent the deflection of the trailing edge as a function of the trailing edge tension.

He then derives the optimum planform shapes for a range of trailing edge (leech) tension. As you might expect, the chord is greater for lesser tension because the twist is greater. All of his planforms are highly tapered, much like these planforms, because the taper allows a 1/3 greater span for the same bending (heeling) moment as an elliptical planform. The tapered planforms aren't as efficient for their span as an elliptical one would be, but the greater span allows them to cut the induced drag by 15% over an elliptical planform with the same center of effort.

I think his approach could be adapted to look at the tradeoff between planform shape, camber and leech tension for a fat-head sail.

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### brian eilandSenior Member

I have a question Tom. Could these more 'tapered planforms', 'tapered ellipses' and the 'pin-headed mains' be more parabolic in a sense than elliptical??

I may have mis-spoken when I included this statement on the DynaRig sailplan on my website:
"Very interestingly the overall profile of this sail plan almost perfectly matches that of the idealized semi-ellipitical/parabolic planform shape. The lift/drag factors for this optimized shape are so much superior to those for the triangular sail-shapes of the Bermuda rig."

I've always assumed a close relationship between the Ellipse and the Parabola?

And now the subject has been raised here recently:
http://www.boatdesign.net/forums/showpost.php?p=188310&postcount=72

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### Richard Atkinatn_atkin@hotmail.com

As I mentioned to Brian in the other thread, my question was poorly written. I was referring to the camber of the Dynarig sail...and I am thinking of how one would control it to optimise the foil shape.

- Richard

ps. this is a great thread....lots of juicy bits here...thanks Tom

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