# Sail Boat

1. Jan 9, 2009

### Phrak

What limits the speed of a sail boat?

Give a sail boat that, for whatever reason, doesn't heal and dump wind, what would limit it's speed?

For the sail, the lift and drag are proportional the the square of the velocity. The same goes for the keel. So as the force of drag increases with speed to slow the boat, the lift increases proportionally to pull it along.

2. Jan 9, 2009

### HallsofIvy

When the wind strikes the sail, you can separate the winds velocity vector into two components: perpendicular to the sail and parallel to the sail. The component parallel to the just "slides" off and does not contribute any push to the sail. Further, unless you are going directly down wind, your sail will be at an angle to the motion of the boat. You now separate that "perpendicular to the sail" component into components that are perpendicular to the boat's motion and parallel to the boat's motion. The force perpendicular to the motion is blocked by the boats body and keel. It is, finally, the force parallel to the boats motion that accelerates the boat. As long as there is any non-zero component in the direction of the boat, the boat will accelerate. If, for example, the boat were sailing directly downwind, with the sail completely perpendicular to the boat, the entire force of the sail wind would be pushing the boat. As long as the boat is going slower than the wind there will be some non-zero component of wind accelerating the boat: the limit on the sail boat speed is the wind speed.

If the sail is set at all off the direction of the wind, the component of the wind perpendicular to the sail will be smaller that the full speed of the wind. Further, if the sail is not perpendicular to the boat, the component of that in the direction of the boat will be smaller yet- so the boat cannot go as fast as the wind. Contrary to what some people believe, a boat's fastest run is downwind and a sail boat cannot go faster than the wind. (On a tack, it may feel faster because there will be more wind crossing the boat than running with the wind where you feel very little wind at all.)

3. Jan 9, 2009

### cesiumfrog

Good grief, why are the mentors here still perpetuating this myth?

4. Jan 9, 2009

### cesiumfrog

HallsofIvy, here is a https://www.physicsforums.com/showthread.php?t=274996" to which I'd like to draw your attention. The second post both explains the speed limit of sail boats and links a very well presented and qualified source for further detail. (If you would reopen those discussions then I think it is a place where we could resume addressing valid misconceptions.)

Last edited by a moderator: Apr 24, 2017
5. Jan 9, 2009

### schroder

I think it would do everyone some good to reopen the thread in question, which was “locked pending moderation”. The locking was done even though there was no hostility or animosity in the thread, and some key questions were finally being addressed. NO evidence at all has ever been presented for DDWFTTW, none whatsoever! And yet, this myth continues to be perpetuated again and again all over the Internet, including this forum. A cart can go down wind, driven by the wind, and a cart can advance on a treadmill, driven by the treadmill. Nobody is seriously challenging that! But, a cart cannot go directly down wind, driven by the wind, at a faster velocity than the wind which is driving it! If it ever got close, it would be find itself pushing against that same wind as a headwind, in front of the cart! Can a tailwind push faster than its own headwind! Can a dog run faster than his own tail? It is high time to show that this DDWFTTW is nonsense, and I am glad to see that one of Physics Forums most qualified mathematicians, (HallsofIvy) has taken a position on this. Let the discussion begin!

6. Jan 9, 2009

### mgb_phys

I think that's a different question.
A sail cannot do downwind faster than the wind is true - but that does not prove that downwind is the fastest a sail can go.

I'm not sure there is an obvious speed limit to a boat with a side wind.
The wind causes a pressure difference across the sail so a forward force, the force depends on the area of the sail and the sideways velocity of the wind. The only limit I can think of is when the speed is high enough that the air pushed along in front of the sail causes a high pressure that cancels out the low pressure are caused by the curved sail.

Last edited: Jan 9, 2009
7. Jan 9, 2009

### schroder

Yes, you are right. The OP is a different question altogether. I was responding in a knee jerk reaction to the reference to the other (closed) thread, and that old saw about DDWFTTW. It is no secret that as long as the sailing skiff is angling to the wind, it can achieve a speed which is quite a bit faster than wind speed. But of course, this speed does have a limit! As I understand it, the drag of the hull, as well as the pattern of the waves created by the hull, determine what the limit is, and it is about twice the wind speed. Notice I use speed, not velocity, since speed is an absolute value and not a vector. Whether or not this speed can be translated into a velocity made good, along the original line of wind direction, which exceeds the wind velocity along that same line, is still a subject of debate. I have not seen any convincing evidence that it has ever been done. The skiff would need to accelerate at an angle away from the direct line of the wind, then decelerate and tack back at an angle towards the true line, and accelerate again to the finish. Even with a maximum burst speed of 2 times wind velocity within the legs, the skiff is covering a much greater distance, and it does not seem realistic that the overall velocity will ever exceed constant wind velocity along the true wind direction. Also, constant wind velocities and constant wind directions are very hard to find, and so the issue is hard to prove in any case. Sailing directly down wind, the skiff cannot ever achieve wind velocity, only reaching about 70% on average. As for those carts, the less said the better!

8. Jan 9, 2009

### Carid

In a sailing boat the important wind is the apparent wind.
If you sail at right angles to the true wind the apparent wind is at a lesser angle due to the boat's own motion. (Sum of vectors).
For high performance catamarans this is the eventual limiting factor. The wind goes too far aft.

9. Jan 9, 2009

### mgb_phys

So the fastest sail would have to have a large area (more force) but be very narrow so the cross wind travel across it before there is significant forward motion - so something like a glider (sailplane) wing mounted vertically.
Presumably for a pure sidewind you wouldn't need a keel and so less drag - the boat would capsize immediately if you turned but that's somebody else's problem.

For a land yacht (lower friction than a hull) the speed record is 116mph
Cool looking attempt to beat it - http://news.bbc.co.uk/2/hi/technology/7610786.stm

Last edited: Jan 9, 2009
10. Jan 9, 2009

### rcgldr

In the downwind case, the sail diverts part of the apparent headwind directly upwind, so that the true wind is slowed down by interaction of the sail. In the above case where the VMG was 60.6 mph with an 18 mph wind, the component of 70 mph apparent headwind deflected directly upwind would need to be > (60.6 mph - 18 mph =) 42.6 mph.

I'm curious to find out if there's a way to approximate the air flow velocities in the vicinity of a moving sail (or wing), or a better mathematical model or explanation of what's going on.

Last edited: Jan 9, 2009
11. Jan 9, 2009

### Phrak

Hi Carid. This is about what I expected, but why?

12. Jan 9, 2009

### Phrak

In aeronautics, compontents are usually given in the lift-drag system of coordinates. It makes things simpler. Perpendicular vs. parallel velocitites WRT the sail would be obtained using angle of attack. Perhaps nautical people uses different methods?

Lift is defined as the force perpendicular to the relative free-air stream.
Drag is defined as the force oppositly directed from the free-air stream.

Last edited: Jan 9, 2009
13. Jan 9, 2009

### Phrak

The keel is a water foil. It does in the water what the sail does in the air. When tacking, both are needed. Exceptions might be a cat, where the hulls serve as (rather lousy aspect ratio) foils, I guess. Does a cat have a keel at all?

14. Jan 9, 2009

### Phrak

Jeff, I looked over the "Downwind Angles, Skeeter" drawing. If boat speed means speed over the ice, then the vectors are wrong. V_a and V_b are swapped.

To keep the vector labels unchanged on the drawing, then:

V_t = velocity of air with respect to the ice. (or just "wind velocity")
V_a = velocty of boat with resect to the ice. (or just "boat velocity")
V_b = velocity of air with respect to boat. (or "apparent wind")

Last edited: Jan 9, 2009
15. Jan 9, 2009

### rcgldr

The problem with a sailcraft is that it interacts with the air and the ground (or water), so you have two sets of components, one set relative to the wind, the other set relative to the ground.

Some cats have deployable "daggerboards", that can be raised or lowered. I've read that the daggerboards are mostly used for upwind tachs, but I also read that top speeds are faster with the daggerbords. The keel in a sailboat uses weight below the boat to counter the torque from the wind on the sail. Cat's don't require the weight, which is why they are faster than monohull sailboats.

Looks correct to me, and it's what I based my example numbers from. Ice boat speed 70 mph, apparent wind speed 55.15 mph, true wind speed 18 mph. This corresponds to the iceboats heading being 30 degrees offset from the true wind. The angle of the apparent wind (relative to iceboat) is arctan (9 mph / 54.4 mph) ~= 9.4 degrees (Beta). The angle of the sail (relative to sailcraft) would have to be smaller still.

Last edited: Jan 9, 2009
16. Jan 9, 2009

### Phrak

My Mistake! I read the drawing wrong. The craft is sailing downwind. I presumed it was sailing upwind.

Last edited: Jan 9, 2009
17. Jan 9, 2009

### rcgldr

Not my numbers, I'm just using the ones from that pdf file, filling in some digits to get the heading angle 30 degrees (adjusting 55 mph to 55.15 mph).

agreeed

The way I see the diagram, you have 3 vectors. A true wind vector, heading 180 (south) at 18 mph, the boat velocity vector, heading 150 (30 degrees east of south) at 70 mph, and the apparent wind, heading 320.6 (39.4 degrees west of north), 55.15 mph. Apparent wind vector = true wind vector - iceboat velocity vector.

18. Jan 9, 2009

### Phrak

Jeff. I hastily corrected my last post, when I discovered my mistake (I assumed the craft was sailing upwind.). But I was way late. Sorry about that.

19. Jan 9, 2009

### Phrak

I see this had roots in the 'DDWFTTW' business. (which is very much possible according to Newton, but not the way most people have in mind.)

20. Jan 9, 2009

### rcgldr

The basic principle of any wind powered device is that it has to "slow" down the wind in order to use the wind as a power source. A sailcraft diverts enough of the apparent wind upwind that the net effect of the sailcraft's interaction with the air is to slow it down with respect to the ground the sailcraft travels on, so that even though the sailcraft's component of downwind speed may be greater than the true wind, the true wind is slowed down by interaction with the induced upwind wash off the sailcraft's sail.

The same principle applies to a DDWFTTW cart, the cart generates an upwind thrust speed at a fraction (< 1) of it's forward ground speed, allowing the cart to go DDWFTTW, although it's upper limit will be lower than a sailcraft, in my opinion (one issue is that the "operating" ground force opposes motion for the cart, but is perpendicular to the motion for a sailcraft).

21. Jan 10, 2009

### Phrak

In the attachment there's a force-velocity diagram that seems to confirm your statement.

The difficult part has been to determine what should remain constant, and what should vary.

The terminal speed of the boat through the water is reached when the sum of the forces of lift and drag sum to zero—the acceleration is equal to zero.

Apparently, the best speed of the boat through the water, for any given tack, is obtained when the speed of the boat through the water is equal to the speed of the boat through the air. This is the scenario examined to analyse terminal velocity. (Does it hold true for any tack?)

There are a few assumptions that are made, such as the angle of attack of sail and keel remain constant—so that the L/D of both sail and keel are each constant.

I've concluded that when the magnitude of V_{B-W} is held equal to the magnitude of V_{B-A} , then V_{B-W} will reach its terminal velocity when alpha+beta=gamma. (see attached file)

#### Attached Files:

• ###### MaxSpeed.doc
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29.5 KB
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Last edited: Jan 10, 2009
22. Jan 10, 2009

### rcgldr

Do you mean the overall speed or only the component of speed parallel to the wind? Note that in the example from the link to the pdf file the iceboats speed was 70mph, with an apparent wind of 55.15 mph, not the same. The wiki article includes an unverified claim that catamarans can go 1.5 times wind speed. Assuming this is a near perpendicular tack, then the water speed is 1.5 times the true wind speed, while the apparent wind would be 1.8 times the true wind speed.

Ignoring drag, the best tack for maximum total speed, is Beta (the smallest angle between apparent wind and sailcraft direction that the sailcraft can maintain) degrees downwind of perpendicular. One of the other threads included the best tack for maximum downwind speed:

23. Jan 10, 2009

### Phrak

One of the assumptions I've made is that the L/D is the same in air and water. D includes parasitic drag in both cases. For instance the drag of the hull through the water would be included in the force I've labeled K_D. For an ice craft the L/D of the skates will be a great deal better than the sail--but it's not zero.

As an example, in my attachment where the two L/D's are equal, I used L/D=12/5, true wind was 10, speed of the boat through the water was 13, and the apparent wind speed was also 13. With the assumption of equal L/D's, maximum speed of the boat through the water is obtained when the boat progresses atan(5/12) degrees downwind from directly across the wind.

I don't follow uarts terminology, so I can't comment.

I fixed-up the drawing slightly to make these things more apparent.

#### Attached Files:

• ###### Sail Boat, Terminal Velocity, Gull Wing Drawing.doc
File size:
32 KB
Views:
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Last edited: Jan 11, 2009
24. Jan 11, 2009

### rcgldr

There's a smallest angle between the direction a sailcraft can sustain versus an apparent near headwind, and this is called Beta. Beta = tan(apparent crosswind / apparent headwind). Uart called it $\phi$. Does this help?

25. Jan 12, 2009

### Phrak

Great. I got it. By best convention, the wind comes from the north. Best speed according to uart is at 90 degrees plus (1/2)phi. This is what I got, eventually, so I'm happy.

The only cavet is that we assume the L/D of sail and keel don't change substancially around this critical direction. Ideally, if speed is your thing, you want to the sail in the low drag pocket, if there is such a thing for two sided sails. You'd want the pocket to cover the region where you expect beta will fall.

Where you've said the best direction is beta degrees downwind from perpendicular to the wind, you should have said half of beta. No big thing.

This beta angle is really quite a useful value. I hadn't taken it seriously enough. Beta and and wind speed will tell you all you need for any given heading, given you keep in mind that beta is a function of the L/D's of both sail and keel.

Given all this, the maximum speed of the boat over the water is about

$$\frac{V_{true.wind}}{2} = V_{boat} \ tan{\left( \frac{\beta}{2} \right)$$

$$\beta = arctan(LD.sail) + arctan(LD.keel)$$

(The reality is something of a disappointment, but expected. I guess I can't fly a sail over a keel arrangment and expect to approch the sound barrier anytime soon. But in a hurrican and a beta=8 degrees...)

Last edited: Jan 12, 2009