# General expression of wind force on a sail....

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• fog37
fog37
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General expression of wind force on a sail...
Hello everyone,

In trying to better understand how sailboats work, how they can sail upwind (not directly), how the go faster than the wind speed, I have been thinking about the magnitude of the wind force and its equation: $$F_{wind}= \frac {1}{2} \rho_{air} A v_{wind}^2$$
Instead of ## v_{wind}##, would the correct expression for the wind force include the magnitude of the apparent wind speed, which should be the magnitude of the difference between the true wind velocity and the boat velocity?

In general, I believe the expression$$F_{air}= \frac {1}{2} \rho_{air} A v^2$$ is found in physics book as the magnitude of the air drag due to air and experienced by an object moving through air (zero wind speed).

Thanks!

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Think of a sail like an airfoil, or an airplane wing. There are equations for lift on an airfoil, but they're pretty hairy.

But the bigger factor for sailboats is what is below the water line. The shape of the hull and the keel make it much easier for a sailboat to move forward than sideways. Therefore, a force which pushes mostly sideways but a little bit forward can make the sailboat move mostly in the forward direction.

Explanations usually make use of diagrams, more than formulas.

Here's an overly simplified version.

And another, a lot more complicated.

This may also be useful.
https://www.real-world-physics-problems.com/physics-of-sailing.html

I'm not sure what level you are seeking. A basic understanding or a wish to design racing boats.

Delta2, fog37, hutchphd and 2 others
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I will tell you all I know...it will not take long.
The other piece you need is that along a curved streamline exists differential pressure according to Euler $$\frac {dP} {dR}=\frac {\rho v^2} R$$ where R is the local radius of curvature of the streamline. The sail thereby creates lift (like an airplane wing). Note the similar dependence on v^2 and Area compared to drag force from the sail you have noted. There are many other forces on a boat from water drag and rudder and keel.
There you go.

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Here is a rule of thumb, which I am ashamed to say is in imperial units. Wind pressure on a flat plate = 40 psf at 100 mph.
So at any wind speed, pressure = 40 x (v/100)^2 pounds force per square foot.
To apply this to the sail, you multiply this force by the lift coefficient to find the force at right angles to the wind, and by the drag coefficient to find the force acting downwind. When you take the figure at right angles to the wind, it is not at right angles to the boat and so has a small forward component - this is the propulsion force for the vessel.
Typical figures for the maximum lift and drag coefficients for a dinghy sail are 1.5 and 0.2.
As the boat is moving then of course, apparent wind should be used.
As mentioned, the underwater foils have a vital role, so we can view the sail as an engine and the underwater foils as the wings of an aeroplane. The boat always sails at a yaw angle of a few degrees upwind relative to its track through the water, and this is equivalent to the angle of attack for aeroplane wings.

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But my intuition is that a sail is far from just a flat plat. The addition of a jib on the forestay does far more than would be explicable in terms of flat plate dynamics. Also one adjusts the belly of the sail. I will look it up later if I can.

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Well of course that is true, hence the lift coefficient is greater than 1.

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Typical figures for the maximum lift and drag coefficients for a dinghy sail are 1.5 and 0.2.
My apologies I misread this. I'm still not sure I like starting the explanation with the flat plate, however.

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My apologies I misread this. I'm still not sure I like starting the explanation with the flat plate, however.

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Yes, they really are like curved airfoils. But back in the day, when I was really into sailing, I read an excellent book "The Art and Science of Sails" by Tom Whidden. I really learned a lot by their description of how a flat airfoil can generate lift. The curvature is about optimization, not the basic physics, I think.

Rhetorical question: How do stunt pilots fly their airplane upside down in straight and level flight if the airfoil needs to be curved?

Delta2 and hutchphd
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The curvature is about optimization, not the basic physics, I think.
This.

If you want to understand how a boat can sail upwind, stop looking at the sails: look under the water.

sophiecentaur and tech99
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This.

If you want to understand how a boat can sail upwind, stop looking at the sails: look under the water.
Or get out on the water and try to sail a dinghy upwind with the centerboard raised!

anorlunda
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Yes, they really are like curved airfoils. But back in the day, when I was really into sailing, I read an excellent book "The Art and Science of Sails" by Tom Whidden. I really learned a lot by their description of how a flat airfoil can generate lift. The curvature is about optimization, not the basic physics, I think.

Rhetorical question: How do stunt pilots fly their airplane upside down in straight and level flight if the airfoil needs to be curved?
A modern aeroplane has wings which have a convex surface top and bottom, so it can work upside down, provided the pilot flies with the nose pitched up when inverted.
Incidentally, a flat surface can provide lift but stalls at a fairly small angle of attack. It is suitable for a centre board or keel, where the boat is sailed at only a small angle to the water. On the other hand, a rudder has to turn to large angles, and in that case a properly curved aerofoil provides a greater stall angle.

DaveE
fog37
Think of a sail like an airfoil, or an airplane wing. There are equations for lift on an airfoil, but they're pretty hairy.

But the bigger factor for sailboats is what is below the water line. The shape of the hull and the keel make it much easier for a sailboat to move forward than sideways. Therefore, a force which pushes mostly sideways but a little bit forward can make the sailboat move mostly in the forward direction.

Explanations usually make use of diagrams, more than formulas.

Here's an overly simplified version.
View attachment 305012
And another, a lot more complicated.
View attachment 305013

This may also be useful.
https://www.real-world-physics-problems.com/physics-of-sailing.html

I'm not sure what level you are seeking. A basic understanding or a wish to design racing boats.
Thank you. My summary:

I am first considering a saiboat sailing upwind at an angle. The wind pushes on the sail producing a force ##F_{wind}## that is perpendicular to the sail surface (we assume the sail is smooth enough that there is not force of friction parallel to the surface). The force #F_{wind}# can be decomposed into two components, one parallel to the boat direction of motion and one perpendicular to it. The perpendicular force is balanced by the resistive force jointly produces by the keel+hull. All that is left is the propulsive for pushing the boat forward.

The wind true speed has an important role since it is squared: the larger ##v_{wind}## the larger the wind force magnitude ##F_{wind}##. I think sailing upwind at 45 degrees gives the largest wind force because it maximizes the wind force component parallel to the boat direction...

As far as the boat going faster than the wind, i.e. ##v_{boat} > v_{wind}##, this is only possible when the wind direction and boat direction are not parallel so the wind force continues to accelerate the boat (up to a point). This happens because, at an angle, the wind impacts with the sail producing the force. When sailing straight downwind, the max boat speed is the same as the wind speed.

In essence, the difference between vectors representing wind velocity and boat velocity are the wind apparent velocity. When the wind apparent velocity is zero, the wind force ##F_{wind} = 0##...

I think that a more correct expression for the lift force (the wind produces a lit force parallel to the water instead of upward) should include the magnitude of the apparent wind velocity: $$F_{wind} = \frac {1}{2} \rho A v_{relative} ^2$$

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Strictly speaking, my understanding is the lift force acts at 90 degrees to the wind and not 90 degs to the sail. In a general case we can find lift from all manner of objects, including spinning cylinders, which might not have a flat side to use as a reference. I think you will find that this simplifies the vector diagram. Of course, you can decompose the sail forces into other directions as wished.

fog37
Staff Emeritus
The wind pushes on the sail producing a force Fwind that is perpendicular to the sail surface
Remember that the sails are not flat. So the force is a function of position on the sail. The total force results from an integration of force across the entire sail area. The relationships are nonlinear, and can't be simplified to the degree you made in your post.
As far as the boat going faster than the wind, i.e. vboat>vwind, this is only possible when the wind direction and boat direction are not parallel
There are two boat directions parallel to the wind, directly upwind and directly downwind. Going downwind, the apparent wind speed becomes smaller, the faster the boat goes. Going upwind, the apparent wind speed increases the faster the boat goes. No sailboat can go directly into the wind.
No sailboat doing directly downwind can go as fast as the true wind (assuming zero water current).

Sailing upwind at a nonzero angle to true wind, the upper speed limit is not defined. It is determined by drag. Iceboats have much less drag than sailboats. Iceboats claim to be able to go five times faster than true wind speed. My guess is that at their top speed, their angle to apparent true wind is 10-15 degrees.

One key factor you may have missed, is that boat speed is not constant. If the forces cause the boat to go 1 knot faster into the wind, the apparent wind speed increases by nearly 1 knot, and the angle to apparent wind reduces.

This article may help. Despite the title, it talks about both forces on the sails and forces on the hull. As you will see, even a simplified treatment takes much more space than we typically put in a forum post. The effort to really understand it is roughly equivalent to a college engineering course.

https://en.wikipedia.org/wiki/Forces_on_sails

This diagram may also help.

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fog37, berkeman, hutchphd and 1 other person
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I think naval engineers discovered the first empirical laws of aerodynamics and hydrodynamics long ago before they were developed as full theories at the end of the 19th- beginning of 20th century,

fog37
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Here's one more thing that may help. Some of the discussion here seems to visualize wind hitting the sail nearly perpendicular. When sailing into the wind, it is closer to parallel than perpendicular, and the sail acts like a wing. Never mind that a wing has thickness that a sail doesn't, the wind streamlines around a sail are very similar to streamlines around a wing.

fog37, DaveE and Delta2
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Incidentally, a flat surface can provide lift but stalls at a fairly small angle of attack. It is suitable for a centre board or keel, where the boat is sailed at only a small angle to the water.
Edit: (what was I thinking). This is not correct, upwind the keel has an angle of attack of 45°. High performance keels are shaped to reduce this (and in some classes are allowed to adjust to reduce the angle of attack), nevertheless I don't know of any design where laminar flow (i.e. not stalling) is maintained throught the operating range.
Yes a boat is sailed at only a small angle to its direction of movement through the water; this is called leeway and when sailing close to the wind can be about 10°-15°. Modern keels often have an airfoil shape to reduce this, and some high performance classes permit adjustment of the yaw angle of the 'keel' (or more specifically daggerboard) to reduce this further, in some cases even providing negative leeway.

Remember also that while there are similarities between aerodynamics and hydrodynamics, the fact that air is compressible whereas water is not means there are also significant differences.

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Sailing upwind at a nonzero angle to true wind, the upper speed limit is not defined. It is determined by drag. Iceboats have much less drag than sailboats. Iceboats claim to be able to go five times faster than true wind speed. My guess is that at their top speed, their angle to apparent true wind is 10-15 degrees.
No, high performance craft achieve their maximum speed at 120-150° from the true wind. I haven't got a calculator in front of me but at 5x wind speed I think this is less than 10° from apparent wind.

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Going downwind, the apparent wind speed becomes smaller, the faster the boat goes.
Only up to the limit where the apparent wind is at 90°, after that the apparent wind increases the faster the boat goes (and the apparent wind angle becomes even less). Note also that in practice it doesn't happen like that: you don't sail at a constant angle to the true wind and keep accelerating through that point, you 'head up' (sail at a closer angle to the true wind, sometimes almost 90°) to increase speed above the wind speed and then 'bear away' (turn downwind) once you have established flow over the sails and foils.

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What does the ellipse of clockwise arrows represent?

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What does the ellipse of clockwise arrows represent?
I don't know. I got that image from Google images searching for sail streamlines. I also wondered about the ellipse, but it was the best image I found.

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What does the ellipse of clockwise arrows represent?
If memory serves, there's vorticity associated with airflow over a wing. I guess that's a sketch of the sense of it, though not magnitude or distribution.

DaveE and Delta2
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If you want to understand how a boat can sail upwind, stop looking at the sails: look under the water.
It seems to me that an iceboat (or the wheeled version thereof) is perhaps a more approachable entry into the dynamics of sailing upwind. The hydrodynamics part is daunting but the incompressability of water makes it different, and perhaps less obtuse. The two problems (above and below water) are largely separable so conflating them from the outset seems counterproductive.
Of course who would have predicted the hydrofoil wingsail yachts that inhabit racing (most predominantly the America's Cup). Truly amazing.

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I agree about separating the above and below water situations. For the underwater action, the sails can be replaced by an engine which is pushing at an angle to the track. Then the foils are equivalent to the wing and tail plane of an aeroplane. The weight of the aircraft is equivalent to the resolved lateral force of the sail(s), and the resolved forward force of the sail is equivalent to the aircraft engine. The leeway angle is equivalent to the angle of attack, or nose-up, angle. It is usual with an aeroplane to have the weight (CG) located in front of the centre of lift of the wing, which gives the aircraft nose-down bias, and then to have the tail pressing downwards to obtain balance. For a sailing craft this corresponds to sailing with lee helm i.e. the tiller must be held slightly downwind, which is not favoured by sailors.

hutchphd
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If memory serves, there's vorticity associated with airflow over a wing. I guess that's a sketch of the sense of it, though not magnitude or distribution.
Yes, the airflow can be decomposed into different flow solutions (like superposition), one of which circulates around the sail as shown.
http://www.gentrysailing.com/pdf-theory/How-a-Sail-Gives-Lift.pdf

Ibix
vinnie78
I agree about separating the above and below water situations. For the underwater action, the sails can be replaced by an engine which is pushing at an angle to the track. Then the foils are equivalent to the wing and tail plane of an aeroplane. The weight of the aircraft is equivalent to the resolved lateral force of the sail(s), and the resolved forward force of the sail is equivalent to the aircraft engine. The leeway angle is equivalent to the angle of attack, or nose-up, angle. It is usual with an aeroplane to have the weight (CG) located in front of the centre of lift of the wing, which gives the aircraft nose-down bias, and then to have the tail pressing downwards to obtain balance. For a sailing craft this corresponds to sailing with lee helm i.e. the tiller must be held slightly downwind, which is not favoured by sailors.
Well I agree all things being possible. Yet viewing the dynamics like aeronautics. The water is the drag on the craft. Of course the best way to overcome drag is to avoid it. Like using a Hydrofoil body that allows for as little drag as possible. It lifts the craft out of water as much as possible during travel. Allowing for minimum drag and maximum propulsion.

vinnie78
Lift as far as aviation: disruption over the top of the wing moving faster than below. Lifting the wing from above. Same as in a helicopter Rotor Blade design. Disruption of this flow, turbulence lift is lost and periods of intermediate drops may occur.

In nautical understanding: all the Physics applied the vessel, are explored to reduce drag. The surface area the vessel makes contact with the surface of the water.

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For the underwater action, the sails can be replaced by an engine which is pushing at an angle to the track. Then the foils are equivalent to the wing and tail plane of an aeroplane.
I think "equivalent" is a bit strong, given that the aeroplane's medium is compressible but water is not, but I see the analogy.

For a sailing craft this corresponds to sailing with lee helm i.e. the tiller must be held slightly downwind, which is not favoured by sailors.
It's not that lee helm is "not favoured", lee helm leads to potentially catastrophic positive feedback during a gust:
1. gust increases lateral force
2. turning moment about centre of lateral resistance increases
3. boat turns away from wind
4. lateral force increases, further increasing 2
5. sail is let out to reduce lateral force
6. centre of effort moves further forward
7. turning moment about centre of lateral resistance increases, further increasing 2
This can only be avoided by attempting to correct with the rudder, however as the boat heels due to the increased lateral force the authority of the rudder is diminished so that this is not always sufficient.

Compare this to the negative feedback with weather helm (i.e. centre of effort behind centre of lateral resistance):
1. gust increases lateral force
2. turning moment about centre of lateral resistance increases
3. boat turns towards wind
4. lateral force decreases, mitigating 2
However this is slow; the quick alternative:
1. sailor observes incoming gust
2. as gust arrives, sail is let out
3. effects of gust increasing lateral force, eased sail decreasing lateral force and moving centre of effort forward exactly cancel

fog37 and DaveE
fog37
Sailboats generally have multiples sails of different sizes while windsurf boards have only one sail connected to the board via a mast (hard to hold more than one sail at a time). I have also seen hydrofoils with wing sails (no mast) which seem a lot of fun. I guess multiple sails help with multiplying the lift force.

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It's not that lee helm is "not favoured", lee helm leads to potentially catastrophic positive feedback during a gust:
I understand this mechanism when sailing with lee helm, but another mechanism giving negative feedback is this:-
Boat turns to leeward.
Angle of attack of rudder is increased.
Boat turns to windward.
In a similar way, an aeroplane having CG ahead of CP, tail plane pushing down, is stable and behaves like this:-
Nose goes down
Angle of attack of tailplane increased
Nose comes up.
One difference I can see from the aeroplane case is that for the 'plane, weight is constant. For a sailing boat, the lateral force is variable.
I have tried by experiment to see if a weather helm or lee helm condition gives better course stability and the results have been so far uncertain.

vinnie78
As for understanding and calculating lift force. It is essentially a calculation of gravitational forces applied. An objects gravitational weight, must be equalized and overcome for lift to take place. Sails can be used in such a manner to assist in this. A sail tilted drastically could catch air and help raise the vessel. Usually however a hydrofoil has been designed with a large enough surface are to provide required lift on its own. The wind force can be focused as a propulsion force.

If added power is required. Rather than a traditional engine. An inlet hole on the forward hydrofoil can feed large gpm pumps. The pumps then can direct the flow through water outlets on the back of the rear hydrofoil. This powered propulsion acts like a jet boat.

However wind power is our current focus.

DaveE
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As for understanding and calculating lift force. It is essentially a calculation of gravitational forces applied. An objects gravitational weight, must be equalized and overcome for lift to take place. Sails can be used in such a manner to assist in this. A sail tilted drastically could catch air and help raise the vessel. Usually however a hydrofoil has been designed with a large enough surface are to provide required lift on its own. The wind force can be focused as a propulsion force.

If added power is required. Rather than a traditional engine. An inlet hole on the forward hydrofoil can feed large gpm pumps. The pumps then can direct the flow through water outlets on the back of the rear hydrofoil. This powered propulsion acts like a jet boat.

However wind power is our current focus.
I thought lift was the force normal to a foil caused by mass flow over/around that foil, without regard to forces from other phenomena.

It isn't common in physics to define forces (and other things) and include multiple mechanisms, although that's what's necessary to solve most problems. For example, no one says Newton's laws of force are ##\vec F = m \vec a + m \vec g##.

Gravity certainly isn't essential in calculating the lift from a fin keel when the boat isn't healing, yet there is a real hydrodynamic force there that most everyone calls "lift".

vinnie78
Well lift force is the force of air surrounding the foil for aviation. So if pressure force is measured as psi. Lift is calculated as Lift per square inch of surface area. The Lift needs to counteract gravitational weight for Lift of an object to occur.

vinnie78
There are a number of forces that are not recognized. Yet we deal with the reactionary result. Centrifugal force for example. It isn't an actual explainable force at the moment. Yet put an object in the middle of a spinning Marry go Round and you see the result.

In aviation the concept of lift changes. As a wing design changes. So does the value of lift change. When a design is chosen. Then the size required to provide proper lift can be made. Required Lift for a kite is next to nothing, compared to required lift for a 747. The variable beyond adjustable lift force. Is always the gravitational weight of the object. If gravitational weight isn't counteracted. An object sits in the wind rather than being able to fly in wind.