Sailing Mechanics Modeling Part 2 - Forces of the Sails

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This is my 3rd blog in a series about computer modeling an 18th century ship for support of the world sailing mode of a game currently in development. I've covered hull resistances and now I am going over how I plan to model the sails. What I've learned about sails is from my reading of resources online and a few nuggets of info collected from several books. In fact I've only being on on a sailing ship once some 25 years ago in Galveston, TX while it was docked at that. But physics and math are the universal language, they still do their thing, even if you've witnessed it in person or not.

To model the performance of sails to then determine the forces exerted on a ship. This isn't meant to cover every facet of the topic, it is just scratching the surface really, but it is enough that implementation will be solid and done in a timely manner. The major variables which determine sail performance are covered here.

I want to model this because a ship of this era carries a lot of sails positioned about on 4 major points (for the rated ones). Each of these sails exerts a force and a moment about the ship's center of gravity. This in turn causes the ship to push against the water and the water to push back upon the ship (equal opposite reaction). Where this happens determines how the ship behaves (constantly drifting to starboard, helm adjustments needed?) Ultimately it is to determine how much of a top speed the ship can obtain and reward a captain that has trimmed his ship to catch that enemy frigate 10 miles away. And, honestly, I like a good study to dive in to.

[Thar she blows - A quick physics lesson]
Imagine a free stream of air traveling at some velocity at atmospheric pressure. Now take this stream and split it in half and call the left side of the stream Captain and the other half Crew.

This stream starts out together with the Captain and Crew air streams pacing along together. One day they happen upon a divide in the middle of their path and must take sides.

The Captain and Crew agree to meet back up again as they split and go their own way; with Captain going on the left side and Crew on the right side of the divide. The Captain didn't know this at the time but his side of the divide is longer due to a gradual curve. The Captain is annoyed by this because he agreed to meet back up with the Crew at the same time and place (i.e. after the divide). The Captain thus has to make up the distance by speeding up.

The Captain has an overall energy level that cannot change as energy is conserved. This energy level resides in "pools"; those mainly being height, temperature, pressure, and velocity. The Captain can control his energy distribution but, in doing so, must follow Admiral Bernoulli's instructions. To speed up the captain robs some energy from the "pressure pool" and adds it to the "velocity pool", in so doing, the effect being as pressure goes down the velocity goes up (in accordance with Bernoulli). This allows the Captain to rejoin the Crew at the time and place agreed upon earlier.

The pressure difference between the Captain (which is lower) and the Crew (higher) is the source of the force that propels the ship.

[The Model]
From the quick lesson above we know that air passing over a curved sail produces forces. Here is a sketch from reference 1.

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The take away from this sketch is that two forces are created by the sail (Lift and Drag) and that it happens perpendicular and parallel to the apparent wind angle (AWA).

The Lift and Drag forces vary with the angle of attack [AoA] of the apparent wind to the sails chord angle (along the yard). This AoA is what the crew changes by adjusting the yard orientation thru the bracing angle. The magnitude of the force and the direction of the total force changes with these angle adjustments, but, it does not follow the change of the angle of the sail linearly, it changes with the gain/loss of the individual Lift and Drag forces. The Lift and Drag forces are somewhat independent and follow this relationship. (Lift is solid line, drag is dotted line)
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[for a square rigged sail - comparing actual to computer model]

The left side of the graph shows the air stream passing sideways and to the back of the sail. As you move towards the right it represents the air stream moving more behind the sail (as if the ship is running with the wind with the yards braced square with the beam). At some point the sail's Lift force falls away and the sail is only producing a pure Drag force because the air is no longer flowing across.

These lift and drag coefficients are used in conjunction with the following formula to calculate the force generated by the sails:
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(Variables are air density, sail area, apparent wind speed).

The wind speed is not constant for a particular mast and generally increases with altitude.

[Some Model Results]
So these calculations haven't been fully vetted but I've simulated the frigate HMS Cleopatra traveling at 5.5knots with a true wind angle of 90degrees. Note the sail areas were taken from a model and the sail midpoint heights are not the real values, these were standin to get the calculation working. Negative is the north direction for speed.)
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This ends up being that the Main Mast is pushing the ship forward with a force of 8031 lbs force and exerting a heeling force (sideways) of 9784 lbs.

The total force due to all of the sails on fore, main, and mizzen is 15,583 lbs force forward and a heel of 19,268 lbs force.

At this point going forward I would take the 15,583 lbs of force and at a steady speed of 5.5 knots and acknowledge that it amounts to the same amount of resistance. I will then fine tune the ships hull resistances to match the value stated. From that point on as the wind changes, as sail brace angles changes, as hull fouling builds up, and sails get damaged the speed will adjust appropriately. Well that's the plan anyway.

That's it for now on modeling as I have work to do. Until next time.

1) Engineering Applications of Computational Fluid Mechanics Vol. 3, No. 1 (2009)
2) https://en.wikipedia.org/wiki/Forces_on_sails

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Updated 12-31-2017 at 08:05 by Aaron