August 23, 2011

The missing link

SOMETIMES I WONDER if there’s a small part missing from my brain. It’s the part that deals with heeling in a sailboat. Specifically, it’s the part that can understand why a boat with a low, spread-out sailplan should heel less than a boat with a tall, narrow sailplan.

I have read innumerable times that a gaff-rigged ketch with a bowsprit will heel much less in a wind of a certain velocity than will a high-aspect-ratio sloop. And for the life of me I can’t see why.

Now let me say straight away that I understand the extra leverage gained by a long tall mast of a sloop, compared with the stumpy masts of a gaff-rigged ketch. I accept, too, although with some cynicism and doubt, that the wind at the top of a mast blows harder than the wind at the bottom.

However, there can be no gainsaying the fact that it takes a certain amount of energy to move the mass of a sailboat through the water. That energy comes from the wind, and if the wind is abeam, or forward of the beam, it tends to make the boat heel to leeward.

Surely it takes the same amount of energy to make similar hulls heel over to the same number of degrees, no matter whether the sails are high or low, doesn’t it?

Obviously, the higher the center of effort, the less sail area is required to make the hull heel over. In the gaff-rigger, more sail area will be required to achieve the same amount of heel because the center of effort is lower, and the sails don’t have as much leverage.

In other words, the sails of the sloop are more efficient since they create the required amount of heeling energy with less sail area.

Thus, to say a low-lying rig will produce the same amount of energy (which translates to the same sailing speed) with a lesser amount of heel is nonsense. If the gaff rig doesn’t produce as much heel, it also lacks the ability to create energy to move the boat forward. In other words it lacks sail area. And that is surely not something to crow about as if it were the best thing since sliced bread. Whether your sail is tall and thin, or low and wide, it needs to produce the same amount of heeling energy.

If you were to take this questionable theory to its limit, a boat with sails a foot high and 100 feet long would cause no heeling at all. And very little forward motion.

Once a canard like this starts making the rounds, it gets repeated thoughtlessly by a thousand copycats eager to show off their new knowledge, and soon enough it becomes part of the accepted lore of the sea.

I really wish I, too, could accept it. I don’t like to stand out from the mob and be jeered at. But my sadly incomplete brain won’t let me. You won’t believe how I suffer.

Today’s Thought
The flying rumours gather’d as they rolled,
Scarce any tale was sooner heard than told;
And all who told it added something new,
And all who heard it made enlargements too.
—Pope, The Temple of Fame.

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Mike said...

Much as I am reluctant to give up my treasured status of the John Vigor fan club by this action, I feel I must try to respond to your “missing link” posting.
I think you are failing to differentiate between the heeling force and the drive force produced by the action of the wind on a sail.
Let’s assume (for simplicity) a wind square on the beam, and for completeness let’s say this is the resultant wind (in other words we’ll ignore the effect on the wind angle of the boat’s motion)
In simple terms, the wind force may be resolved into two forces acting at right angles – one along the axis of the hull (the driving force) and the other at right angles to it (the heeling force) The ratio of these forces is dependent on the effectiveness of the sail as an aerofoil, a perfectly flat board along the ships axis would produce zero driving force and a heeling force equal to the wind force. I cannot conceive of an aerofoil sufficiently efficient to produce 100% driving force and zero heeling force – let’s for argument’s sake say a really good aerofoil produces 70% driving force and 30% heeling force (I have no idea if this is realistic – for the purposes of this argument it doesn’t matter)
Now, the power of the sails – their ability to drive the boat forwards – is a function of their area (and of course other diverse elements)
The height of the centre of effort above the deck is not relevant to the ability of the driving force to propel the boat. The sail plan you describe as “a foot high and 100 feet long” would, in fact, drive equally well as a sail plan 100 foot high and a foot long, all other things being equal (though of course they are not, which is why racing boats have high, narrow sails – high aspect ratio)
The height of the centre of effort of the heeling force is, of course, very relevant to the amount of heel - as you have accurately described.
So – the basic error in your argument is expressed in your statement “In other words, the sails of the sloop are more efficient since they create the required amount of heeling energy with less sail area.”
Two points here – firstly the “heeling energy” isn’t “required” – it’s an unwanted but unavoidable effect. The ideal would be a 100% effective aerofoil with no heel at all.
Secondly energy and force aren’t the same thing. Time has to be considered when referring to energy – the more energy, or power, that is available the quicker the boat will go to the angle of heel at which the heeling force and the righting force exerted by the boat’s keel will balance – the higher the centre of effort, the less heeling force is required for a given angle of heel – but the same energy or power is needed to achieve it in a given time.
On re-reading this I feel I haven’t really clarified things – perhaps evidence that my thinking isn’t as clear as I would wish. Hopefully though it may help to ease your suffering!

John Vigor said...

Mike, I was afraid this would happen. You're probably right and I'm probably wrong. I just wish I could understand why.
PS: Ivor Tunging-Cheaque, chairman of Vigor's Silent Fan Club, has granted you special dispensation to disagree with me (disagreeable as that is) in the interests of scientific enlightenment.

John V.

Aaron Headly said...

OK, John, because I admire and respect you I will try another example. You'll 'get' this eventually, I am certain.

I did a little drawing (here) showing two ways to try and careen a boat for a quick scrub (they do get weedy, especially in the tropics).

In the top version, the captain (always willing to help out) is pulling on a rope attached at the masthead. It's probably the halyard.

In the bottom version, the captain, convinced that Archimedes was all wet, has decided to rig another rope, this time half-way down the mast. (That was as high as he could reach standing on the boom. No, he didn't fall down, but it was touch-and-go for a minute there).

Which captain will have to pull harder to get the same heel?

John Vigor said...

Aaron I understand that perfectly. But to pull harder, and get the same angle of heel, a gaff-rigged boat would need more sail area. And that, in turn would make her heel to the same degree as a sloop-rigged boat.
I see no reason why, to drive the same deadweight, a boat with lower, wider sail area would heel less than a boat with a tall thin rig.

John V.

Mike said...


Thanks for your response. Please convey my best wishes to Ivor and thank him on my behalf for his much appreciated dispensation.

I have spent the last hour peparing a worked mathematical example to persuade you of the veracity of my argument. It involves geometry, algebra, formulae and the like, and on reflection I decided it will do nothing to convince you, so I will not bore you with it.

Instead, I will remind you of your dinghy racing days, when I am sure you worked very hard to sail the boat as "flat" as possible by sitting out - in other words no heel at all. Does this not show you that the angle of heel has nothing to do with the driving force? Conversely, if you flatten your sails to depower them, will this not increase the angle of heel whilst decreasing the driving force?

I suspect you will not be convinced, in much the same way that my father could not be convinced that the little fan at the front of his car engine was not what drove it along. Maybe he (and you) are right.



Adam said...

What an interesting puzzle!

I agree that there are some terminology issues with John's argument, like how he should be saying "lift" or "force" or "moment" instead of "energy". After a bit of rewording, I think it sounds like this:

All else being equal, two sails of different aspect ratios generating the same driving force must be generating the same lift, and therefore, the same heeling force.

I think this is probably true. The sail's lift is divided into a driving component and a heeling component, each of which is a force. For the most part, how quickly a boat accelerates is dependent only the driving force, not the driving moment (which would be larger, given the same force, for taller rigs), since all that moment serves to do is drive the bow down slightly. However, the angle of heel is a balance between moments---the righting moment generated by buoyancy and the keel, and the heeling moment generated by the mast---and this heeling moment is certainly larger (given the same lift), for taller rigs.

So while part of John's argument (same drive ==> same heeling force) is right, I think the mistake is in conflating forces with moments.

In reality I'm not so convinced that same drive ==> same heeling force, since there are probably a lot of factors that determine just how the lift (at each point along the luff) is divided into heeling force and driving force. Part of it is surely the angle between the sail's chord at that height and the boat's centerline, but maybe the placement of the draft has something to do with it as well. Then there's twist, and the fact that the chord length is less in region of higher winds, etc., which are basically beyond my ability to incorporate at this point.

Also, I came up with this argument about why low-aspect-ratio rigs might heel more, neglecting the difference in moment arm: if a high-aspect-ratio rig is more efficient, then for the same lift it should have less drag than a low-aspect-ratio rig. So if you imagine the two rigs generating the same heeling moment and driving force, more of the driving force of the low-aspect-ratio rig will be sapped by drag, meaning you'll need to generate more lift (and therefore more heeling force) to get the same net forward force.

My final question to you is this: what is the proper way to hyphenate "low aspect ratio" and "high aspect ratio"?

Aaron Headly said...

Uh, ok. I might be seeing where you're thinking has gone.

Let's rehash some things:
In rough terms, the amount of 'push' you get from the sails is directly related to the square footage of the sails. Heeling is a side effect of the 'push' you get from the sails (both figuratively and literally, which I find amusing), it is not directly connected to the amount of forward 'push' you get.

You've experienced this on the Sound: Over-sheeting doesn't add speed or power, it reduces speed and power and increases heel. More heel does not mean more power, even though more power often come with more heel.

The sail is imparting, mainly, two forces on the mast: sideways (heel) and forwards ('push', or propulsion).

The boat responds to the 'forwards' force by schooning along (and, to a certain, rather minor, extent, burying the prow). A well designed hull puts up very little fight when being pushed forwards.

Hulls are, on the other hand, designed to fight the heeling force. They bend, but they're not supposed to break (capsize) or move sideways (excessive leeway).

One way to picture these two forces is as two ropes pulling on the mast; one straight ahead and one directly abeam. More wind, more pull. More sails, more pull. In both directions.

Pretty much regardless of how high up the mast the forward rope is hitched, the boat speeds up the same amount.

But if the athwart-ship rope is mounted higher, it has more leverage, and there is more heel.

I need a drink.



John Vigor said...

Adam, the only thing I seem to be competent to comment on is your last question. It appears (he said cautiously) to be a compound adjective and you are hyphenating it exactly right.

As far the rest of it, the news is disheartening. I wrote to well-known naval architect Ted Brewer, designer of many gaffers and Bermudian rigs. He says quite categorically that gaffers heel less.

Mind you, he didn't claim that yacht designers never make mistakes. And there are still niggling doubts in the reptilian
core of my grey matter.

It's an interesting conondrum.

John V.

Aaron Headly said...

OK, then.

By the way: I trust Ted Brewer so much that I bought one of his gaffers.



Matt Marsh said...

All else being equal, it is true that a low, spread-out sail plan will result in less heel than a tall sail plan on the same boat.

I am hard-pressed, though, to explain why without resorting to (a) math, or (b) jargon-heavy discussion of airfoil performance, induced drag, sail twist control, potential flow theory, or the splicing of fish DNA into tomato seeds. (OK, that last one's not that relevant.) But I will try...

Consider two identical hulls travelling at the same speed in identical conditions. They have the same total resistance (wave drag + skin friction + a few minor factors). Therefore, they require the same driving force to maintain their speed.

One hull has tall, high aspect ratio sails. The other has a low, spread-out, low aspect ratio sail plan. The forward (driving) component of the total sail force is the same in both cases.

For reasons I'd better not go into here, a large span, small chord airfoil produces more lift per unit area than a small span, large chord one. (Span ~= luff length.) So the boat with the tall rig will have a bit less sail area than the one with the low-slung rig, if they are to make the same speed in the same wind.

The forward component of the sail force may be the same, but the sideways component is not. The gaffer has lower aspect ratio sails (thus a lower lift to drag ratio for each sail), and more sail area. So the gaffer will produce a bit more heeling force, for the same boat speed.

But heeling force isn't what determines the angle of heel, heeling moment is. (Moment = force * distance from pivot point.) And the tall rig's much higher centre of effort more than overwhelms the short rig's slightly higher heeling force. Assuming, of course, that the sails on both boats are properly cut and are trimmed appropriately.

There's another important factor I've left out so far: the lift/drag ratio of an airfoil (sail) is not the same on all parts of the sail. It's at its maximum/best near the middle of the sail, and very close to zero at the headboard. The top few feet of a tall triangular sail are producing a lot of side force and almost no driving force. And that side force is acting near the top of the mast, on a very long heeling moment arm. You can make the top of the sail more efficient (square-topped or elliptical mains, for example, with a small gaff or batten to control twist) but doing that brings the centre of sail area higher, again increasing heeling.

The short version, then: The higher centre of sail area in the tall rig has a much more dramatic effect on heeling than the slightly greater sail area of the short, spread-out rig.

Clear as last Sunday's fog now, yes?

(Not an NA, but an engineering physicist with some aero background who dabbles in small-craft design.)

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