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I'd like to create a basic model of the forces acting on a sail (wind sail, like a tail ship)

A couple of things I was thinking about:

1) Can create a very simple model where wind is 'one' force acting on the a uniform body.

2) Model wind as vectors. This is where I am a little confused on how to start.

3) Adding in multiple sails.

4) I know that this would be a differential equation but after that I can't really see how it would be modelled.

Any pointers?

I'm not looking for someone to actually do the modelling for me, just a place to start. Like maybe some Wikipedia articles, etc.


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+1 I think this is a good use of this website, and a fairly well thought out question. However, I don't have much help to lend you. :) – BBischof Aug 16 '10 at 22:43
@BBishof, lol, thanks for the feedback :) – cbrulak Aug 16 '10 at 22:46
The post Sailing into the wind, or faster than the wind by Professor Terence Tao may be of interest to you.… – Américo Tavares Aug 16 '10 at 23:11
@BBischof, I can never detect sarcasm on the Internet. What were you implying? – Vortico Aug 17 '10 at 0:16
@Vortico, I was sincere! I hope the OP realizes that I wasn't being sarcastic and snide! As Katie said, I like to let posters know when a question exemplifies good uses of the site, that haven't been fleshed out yet. Especially when the user is new to the site. I do this to provide encouragement to that user, and so that other users can look and say "ah that question is a good model, I should strive for something similar." Again, I hope I didn't come across sardonic. :/ – BBischof Aug 17 '10 at 14:18
up vote 6 down vote accepted

Assuming no edge effects/turbulence, we can calculate the force of wind on a single sail using the drag equation.

$$\vec{F_D} = \tfrac12 \rho \vec{u}^2 C_D A$$

where the drag coefficient $C_D$ is going to be fairly high, for a concave sail.

In general, however, the wind may not be hitting the sail orthogonally, but rather at some angle. We can generalise the drag equation as such:

$$\vec{F_{sail}} = \tfrac12 \rho \vec{u}^2 C_D (\hat{\vec{u}} \cdot \vec{S})$$

where $S% is the vector area of the sail. We take the dot product of this vector with the unit velocity vector to resolve the wind force in the direction of the sail. This is of course an approximation, as the concave shape of the sail will come into play, but probably a good enough one.

That pretty much explains the basic situation for a single sail. Now, for multiple sails, you would of course simply combine the forces on the individual sails, which will all have their own values of $C_D$ and $S$.

$$\vec{F_{sails,total}} = \sum \vec{F_{sail}}$$

I suggest you familiarise yourself with the mechanics and specific equations used here, and use that as a basic model to start.

For some general information on the physics of sailing, you may find this page helpful.

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Awesome! Thanks so much. Lots to work with. – cbrulak Aug 17 '10 at 13:07
No problem. As a physicist, I like it when these questions come along! Feel free to ask for clarification if needed. – Noldorin Aug 17 '10 at 13:20

The post The Sailing into the wind, or faster than the wind by Professor Terence Tao and the comments to it may be of interest to you. (See my comment above and cbrulak's reply)

To explain "the power of the wind to sail in a direction against that of the wind or to sail with a speed faster than the wind itself" the author describes the one-, two- and three-dimensional sailing models.

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What about $F = \sum_i W \cdot S_i$, where $W$ is the wind velocity vector and $S$ is a list of all the sail unit vectors?

I don't see this at all being a differential equation, but if you further specify your idea of this model, I can help you a little more.

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Thanks, that's good starting point. But can I ask you to dig a bit deeper? How would I integrate the size of the sails (surface area) into the equation? Is there an example/tutorial I could look at for generating the sail unit vectors? – cbrulak Aug 17 '10 at 3:58
This equation is dimensionally incorrect... – Noldorin Aug 17 '10 at 10:22

Since you are asking for just a place to start:

Try Yakov Perelman, "Physics can be fun".

The book has two volumes. It has an article somewhere, the precise location of which I am unable to locate as I do not have the book handy, on how to explain the nautical method of "tacking", ie sailing against the wind. It explains the forces on the sails using diagrams, etc.. It has also many other articles on nautical themes. I think this book is the best place to start for an amateur.

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  1. When your analysis is on a simple basis where the sail can be modeled as a plane, the force will be perpendicular to the plane. If you need a 3-D model of sail, it's more complex.

  2. If you take what might be called the 20th century model in sailboat design, the force on the sail is resolved into a vector perpendicular to the approaching wind (lift) and a force parallel to the wind (drag). Likewise the forces on the hull are resolve to lift perpendicular to the direction of travel, and drag parallel to the direction of travel. You can find diagrams like this going back to Manfred Curry.

  3. I don't know about square riggers, but for the normal sloop rig, the jib and main interact too closely to consider them as independent.

  4. One math professor who wrote about sailing was c. stanley ogilvy. I wouldn't rush out to buy his books, but if you can find them in a library, you might find them interesting. The most widely available heavy duty treatments are by Marchaj (

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