Physics behind the force of wind on sails--is complicated

[Dave_R]

I’m trying to understand the physics behind the force of wind on sails. And the more I think about it, the more complicated it becomes.

It started with a simple question—Is the force of wind on an object a square function, or a cubic function? I figured it was one or the other. I know that water’s ability to carry rocks & stuff is a cube function. But power is a square function. Power=mass x velocity squared.

It appears that wind FORCE is a square function. Wind pressure in pounds per square foot is P = 0.00256 x V2 So, multiply the constant .00256 times the windspeed squared, and you have the total force of the wind on the sail in pounds per square foot. Multiply that times sail area, and you have the total force generated—but that’s only if the boat is not moving and the sails are sideways to the wind.

Once you start sailing, the math starts to get interesting….

When sailing downwind, it’s the apparent wind velocity, not the actual wind velocity, the counts. As your boat speed approaches true windspeed, your apparent windspeed falls.

When sailing upwind, the apparent wind is faster than the true wind, but the wind “sees” a smaller sail area. It sees kind of a wing instead of the actual sail area. (But if you fall off the wind without easing the sheets, the force increases quite quickly…)

When on a beam reach, the wind “sees” a bigger wing than when close hauled, I believe.

I don’t know how to quantify those aspects mathematically. It makes me appreciate a sail designer and a sailboat architect.

It’s a good thing that, all we as sailors need to know is “the time to reef is when you first start thinking about it.” And, if you see darker water ahead, that probably means an increase in windspeed.

[Sidebar: Incidentally, an article on designing wind turbines says the generating POWER of a turbine is a CUBE function of windspeed. Force on the blades is a square function, but the blades travel more rapidly as the wind increases, so they’re “grabbing” more air because of their increased speed. I don’t think this has an equivalent with sails…but I could be wrong…maybe wind ENERGY is a cube function, while wind FORCE is a square function????]

Please point out any incorrect assumptions or assertions. I’m sure many folks here understand more about this than I do.

[Bowdrie]

Peruse the books by C.A. Marchaj and you will get more in-depth learning about boats than can be imagined.
https://www.amazon.com/s?k=marchaj&i...l_34m79a6eze_e

[Don C L]

Hmmm I ain't no aeronautical engineer but I think you may be missing the force generated by the Bernoulli effect. i.e. the low pressure area created by the entire wing regardless wind direction, as long as the sail is trimmed as efficiently as possible to function as a wing, with as much laminar flow over the wing and a minimum of turbulence. Now, in looking at the equations describing a compressible flow, I hope someone well-versed in physics can explain it well here! I'd sure like to understand it better too! Per your post though, I believe your assumption that the wind "sees a different sized sail" is incorrect when we are talking about a wing. Keep in mind that sailboats are not pushed by the wind, but pulled (mostly) by the low pressure area created by the wing over the leading edge. I say "mostly" because there must be some push occurring on the trailing edge, deflecting the wind, that translates to some of the lift of the wing, as flaps do on a wing of an airplane. (I know my big long footed main does that!)

[catsketcher]

Interesting topic

First off - Power does NOT equal mv squared.

The kinetic energy of any body is related to mv^2. KE = 1/2mv^2

So to make our boats go two times faster requires 4 times the energy. This is why energy consumption graphs when motoring are shaped like a parabola. Going fast above a certain amount (where skin friction becomes less important) is really energy intensive. It is also why any car can do 100km/h but few can do 200km/h. Energy is measured in Joules

Power = energy over time, measured in watts.

So to go twice as fast requires four times the energy but your motor has to do this in half the time. This means that your motor has to put out 8 times the power (but you get there twice as quickly).

The same goes for sails and is seen in the equation airfoils have for lift

Lift = Lift = constant x Cl x density x velocity squared x area

I don't agree that the idea of apparent wind changing the size of the rig - it just changes the airspeed and angle. It is a simple two step force vector addition. You have to add the wind generated by the boat head to tail with the true wind. This gives the apparent wind. Apparent wind is no different from true wind - it is just greater than true wind and and further forward if you keep the apparent forward of the beam. It can be reduced by putting the apparent behind the beam, but the angles are still calculated by adding the vectors.

As for pulls and pushes - remember there is no such thing as suction or pulling in a fluid - fluids can't pull (try pulling some water or air). They can only be pushed. The how and whys are far beyond mere mortals but the basics are that the leeward side of a sail develops a lower pressure than the windward side and therefore there is a pressure differential created on the sail which gets forced mostly outwards but slightly forwards. We use the mostly sideways vector (when we minimise the sideways component) to make our boats move forward.)

cheers

Phil

[Lee Jerry]

Originally Posted by Dave_R

I’m trying to understand the physics behind the force of wind on sails. And the more I think about it, the more complicated it becomes.

It started with a simple question—Is the force of wind on an object a square function, or a cubic function? I figured it was one or the other. I know that water’s ability to carry rocks & stuff is a cube function. But power is a square function. Power=mass x velocity squared.

It appears that wind FORCE is a square function. Wind pressure in pounds per square foot is P = 0.00256 x V2 So, multiply the constant .00256 times the windspeed squared, and you have the total force of the wind on the sail in pounds per square foot. Multiply that times sail area, and you have the total force generated—but that’s only if the boat is not moving and the sails are sideways to the wind.

Once you start sailing, the math starts to get interesting….

When sailing downwind, it’s the apparent wind velocity, not the actual wind velocity, the counts. As your boat speed approaches true windspeed, your apparent windspeed falls.

When sailing upwind, the apparent wind is faster than the true wind, but the wind “sees” a smaller sail area. It sees kind of a wing instead of the actual sail area. (But if you fall off the wind without easing the sheets, the force increases quite quickly…)

When on a beam reach, the wind “sees” a bigger wing than when close hauled, I believe.

I don’t know how to quantify those aspects mathematically. It makes me appreciate a sail designer and a sailboat architect.

It’s a good thing that, all we as sailors need to know is “the time to reef is when you first start thinking about it.” And, if you see darker water ahead, that probably means an increase in windspeed.

[Sidebar: Incidentally, an article on designing wind turbines says the generating POWER of a turbine is a CUBE function of windspeed. Force on the blades is a square function, but the blades travel more rapidly as the wind increases, so they’re “grabbing” more air because of their increased speed. I don’t think this has an equivalent with sails…but I could be wrong…maybe wind ENERGY is a cube function, while wind FORCE is a square function????]

Please point out any incorrect assumptions or assertions. I’m sure many folks here understand more about this than I do.

The force of the wind is a square function. It's odd (to me) to talk about the power of sails.

Sails work the same way as wings, propellers and hydrofoils. We usually try to get them to work as airfoils, where they generate way more lift than drag (i.e. at "low" angles of attack), but can also create force as drag device (i.e. at "high" angles of attack, like a parachute). So if you understand or learn about wings, then you basically understand sails.

The easiest, classical method (although somewhat lacking in some respects) is using Bernoulli, as you and others have done above. More modern (accurate?) methods use circulation theory or lifting-line theory (and maybe others), but are way more complicated.

Airfoil forces are generally broken down into lift and drag components for calculation, and the total resultant found by Pythagoras or vector math. The component (L or D) is calculated as the product of the dynamic pressure (1/2 rho v^2) x area x coefficient. The characteristic or reference area for a foil is the planform area (not the frontal area). The coefficient is usually determined experimentally (historically) or more recently by calculation.

The first problem with calculating the performance of sails is that there is no general repository (that I know of) for lift and drag coefficient data for sails (or similar soft shapes) as there is for hard foils (see NACA and many others). If anyone knows of one, please share with the group. The second problem is that the sail "airfoil" is constantly changing - as the sheet is eased (say from a tight, upwind trim as the boat falls off) the camber increases a lot and the cord length(s) decrease a little (i.e. the planform area decreases a little). This probably contributes to the first problem.

At high angles of attach, when the sail is no longer a foil but just a drag device, the force calc is similar but there is only the drag component and the coefficient to use will be different.

It then gets more complicated when you get into details like "the slot" (flaps), the mast in front of the main sail, the changing "apparent wind" as the boat pitches and rolls, etc.

(All of these comments apply to "white" headsails; spinnakers behave a little differently, especially symmetric spin on a run or deep reach, which actually have a large component of flow originating at the head and flowing down the sail toward the foot rather than along it from luff to leach.)


So to address a few "themes" in your post, which I've highlighted:

The sails will generate force whether the boat is moving or not (corresponding to the current inflow angle and velocity). That's how the boat starts moving as you trim in.

It's always apparent wind velocity that matters. The sail doesn't know or care what direction the boat is pointed (or how fast it's moving), only the velocity and angle of attack of the incoming flow.

The change in area (planform area) is relatively small as described above compared to other changes / issues; it's not switching between frontal and planform area or anything like that (until it is no longer acting as an airfoil).

[psk125]

Bringing up Bernoulli reminds me of Coriolus. Don't forget that everything in the Northern Hemisphere is reversed when you go south of the equator, so you have to re-draw all the diagrams the other way around.

[Nicholson58]

Add to your thoughts. Cold air is denser than warm. This has a marked effect on the forces at equal wind speeds.

What you get from the wind does increase by the square of wind speed.

The V squared calculations, formulas give wind pressure but the numbers are irrelevant since we do not stop the wind with sails but rather deflect it using our soft wings.

Sails work together, jib increases the wind speed over the Lee side of the main.

It’s complicated.

[AnsleyS]

The best discussion that I have seen was done by Wallace Ross in his book Sail Power. The book is no longer in print but you may be able to find a copy. My suggestion is to just read the first three chapters where he does the physics, after that he gets into sail making design.

[Dr. D]

Great articles on this subject and more:

Gentry Sailing | Theory and Practice

[Tedd]

You're venturing into the territory of mathematical modeling. Engineers break down these things into individual factors that can be analytically represented by simple relationships, such as your model of aerodynamic force being proportional to the square of relative air speed. However, as you're discovering, things can quickly get complicated when you start trying to put all these separate models together to model the entire system because they are interdependent: changing one changes the others.

Traditionally, this was dealt with graphically. For example, with a powered vessel you could make a plot of propeller thrust versus water speed at a given power output and then overlay on top of that a plot of hull drag versus water speed. The speed you'd actually attain at that power setting would be the point where those two curves intersect, called the "operating point." You could then add additional curves for prop thrust at different power settings, and now you'd have a "map" of operating speed relative to power setting.

However, even that becomes cumbersome for more than a handful of variables. This is where computers come in. Since the "operating point" can be found mathematically (the point of intersection of two curves is high school algebra), a computer can easily find it. And, unlike a person, the computer cares little about how many different factors (dimensions) you add to your model. It can still find those intersections. Typically, for a complex model, the computer will use numerical methods to find the answers, rather than solving the absurdly complex algebraic expressions that would result from an analytical model. But, conceptually, it's doing the same thing you did with your power and drag plot.

You might find it valuable to try a fairly simple example, such as your example of sail force and relative wind. You could simplify the problem by making assumptions about some the variables that, while not strictly true or accurate, are also not completely outrageous. For example, you could assume that the boat's speed through the water is directly proportional to sail force. That's obviously completely wrong but, for a fixed true wind angle and a narrow range of speeds well below "hull speed," it's not miles and miles away from the truth. Then, start with an assumed boat speed--let's say 3 knots--and calculate the relative wind at that speed. Then calculate the sail force component in the forward direction at that relative wind speed and angle. Then apply that to your assumed model of speed-through-the-water and sail force. The resulting speed isn't going to be three knots. You then put that resulting speed back into the sail force equation and start again. If you've done it properly, once you repeat this iterative process long enough the results will start to "converge"--i.e., the difference between the speed at the end of the calculation will be hardly different from the speed at the start. And that's your answer.

The answer will be wrong, of course. But it won't be outrageously wrong and you will learn a lot about what's going on from the process of building the model. This is essentially what engineering education is: building a bunch of models of different scenarios until you're reasonably competent at modeling any scenario that's relevant to your branch of engineering. I'm a mechanical engineer so I've built many models of the type I described above. Electrical engineers will build models the same way to analyze circuits. Civil engineers will build similar models of complex structures and loads. Getting the "right" answer to a complex problem, such as the performance of a sailboat, involves refining and combining these models into giant models (such as computational fluid dynamics models). It's simply not feasible to design something as complex as a foiling race boat any other way, if you hope to win.

[Chris31415]

Originally Posted by Dave_R

I’m trying to understand the physics behind the force of wind on sails. And the more I think about it, the more complicated it becomes.

It is complicated. This is a good article from 2008 that covers the physics of sailing. A good overview of a complicated subject.

https://pubs.aip.org/physicstoday/ar...and-keels-like

[Don C L]

Originally Posted by Chris31415

It is complicated. This is a good article from 2008 that covers the physics of sailing. A good overview of a complicated subject.

https://pubs.aip.org/physicstoday/ar...and-keels-like

What a cool shot!
https://pubs.aip.org/view-large/figu...03/38_1_f3.jpg

[Dr. D]

Originally Posted by Chris31415

It is complicated. This is a good article from 2008 that covers the physics of sailing. A good overview of a complicated subject.

https://pubs.aip.org/physicstoday/ar...and-keels-like

The first part of the article, on Wind Power, is a bit misleading. As pointed out in several of the Gentry articles (URL above), there are two components to the air moving past a sail. The first is the straight line flow, as often shown in diagrams. The second is a circulation of air around the sail: Aft on the leeward side, forward on the windward side. Gentry provides a simple experiment one can do at home to visualize this circulation. (I have done the experiment and it is fun.) The difference in the speed of the airflow on the two sides creates the pressure difference; this is stated in the Physics Today article, though the article simply states the air moves faster on one side without saying why.

[Bowdrie]

Youse guys make it too complicated.
You just need a generator on your boat to power a big fan.

[Charlie Baker]

Originally Posted by Dave_R

I’m trying to understand the physics behind the force of wind on sails. And the more I think about it, the more complicated it becomes.

It started with a simple question—Is the force of wind on an object a square function, or a cubic function? I figured it was one or the other. I know that water’s ability to carry rocks & stuff is a cube function. But power is a square function. Power=mass x velocity squared.

It appears that wind FORCE is a square function. Wind pressure in pounds per square foot is P = 0.00256 x V2 So, multiply the constant .00256 times the windspeed squared, and you have the total force of the wind on the sail in pounds per square foot. Multiply that times sail area, and you have the total force generated—but that’s only if the boat is not moving and the sails are sideways to the wind.

Once you start sailing, the math starts to get interesting….
Vector algebra, trigonometry, and differential calculus must be used to describe these aspects mathematically. I heard that https://mysupergeek.com/math-assignment-help-service can explain in more detail. There are various models and formulas that allow you to calculate this force. When sailing downwind, it’s the apparent wind velocity, not the actual wind velocity, the counts. As your boat speed approaches true windspeed, your apparent windspeed falls.

When sailing upwind, the apparent wind is faster than the true wind, but the wind “sees” a smaller sail area. It sees kind of a wing instead of the actual sail area. (But if you fall off the wind without easing the sheets, the force increases quite quickly…)

When on a beam reach, the wind “sees” a bigger wing than when close hauled, I believe.

I don’t know how to quantify those aspects mathematically. It makes me appreciate a sail designer and a sailboat architect.

It’s a good thing that, all we as sailors need to know is “the time to reef is when you first start thinking about it.” And, if you see darker water ahead, that probably means an increase in windspeed.

[Sidebar: Incidentally, an article on designing wind turbines says the generating POWER of a turbine is a CUBE function of windspeed. Force on the blades is a square function, but the blades travel more rapidly as the wind increases, so they’re “grabbing” more air because of their increased speed. I don’t think this has an equivalent with sails…but I could be wrong…maybe wind ENERGY is a cube function, while wind FORCE is a square function????]

Please point out any incorrect assumptions or assertions. I’m sure many folks here understand more about this than I do.

Interesting topic. Once upon a time, I came across a recording of a lecture by some physicist, who at first puzzled me if I didn’t understand the process. This is approximately what you are talking about, that the sail in this case is a wing, and makes it possible to go faster than the speed of the wind itself. Here, to a greater extent, it is necessary to take into account not the wind pressure, but the lift coefficient and drag coefficient, which depend on the shape and orientation of the sail. For me it becomes more intuitive until the formulas appear