Betz limit for sailboats?

In the study of sustainable energy, there is a nice result on windmills, called the Betz limit: There is a finite limit to the fraction of the kinetic energy of the wind that a windmill can absorb or exploit. The reason is often stated as: If the windmill took all of the power in the wind, the wind would stop, and then there would be no flow of energy over the windmill. I'm not sure I exactly agree with that explanation, but let's leave that here.

On my travel home today I worked on the possibility that there is an equivalent to the Betz limit for sailboats. Is there an energetic way of looking at sailing that is useful?

One paradox is that a sailboat is sailing steadily when the net force on the boat is zero (just like when a windmill is turning at constant angular velocity). In the Betz limit, the windmill is thought of as having two different torques on it, one from the wind, and one from the turbine. Sailing has no turbine. So this problem has a conceptual component to it.

wind power

I met up with Matt Kleban (NYU) to discuss our dormant project on the physics of sailing. Our conversation ranged around many different things related to sustainable power. In particular, we discussed whether it was possible to take a energy or power point of view on sailing, which has to do with the work that the sailboat is doing on the water and on the air. I feel like there will be some symmetries in play there. We also discussed power generation with wind farms, including the Betz limit (which is a limit on how much power you can get out of the wind). Is there an equivalent of the Betz limit for a sailboat? Finally, Kleban made a remark that is simultaneously obvious and deep: If you have a propeller turning in a fluid (like air), it might be a turbine (generating power from the wind) or a fan (using power to make wind). The question of turbine or fan has a frame-independent (relativistically scalar) answer.

sailing

I gave a seminar at lunch today (black-board talk) about how sailboats work. I got lots of great comments and questions, especially about sailing down wind faster than the wind. I vowed to add a paragraph to my paper on sailing (with Matt Kleban) about how to sail this way. I think it is extremely hard to do, technically. So much so that some of the books on sailing say that it is impossible! It isn't, in principle.

a phase diagram for sailboats

Matt Kleban (NYU) and I are finishing up a paper on the theoretical basis for sailing (yes sailing). One of our conclusions is that (large) sailboats are described by three dimensionless ratios: The sail-to-keel ratio, the ratio of the sail working force to the air drag force, and the ratio of the keel working force to the water drag force. We imagined today a phase diagram that shows the space of dimensionless ratios that permit, for example, upwind sailing. And sailing downwind faster than the wind. And sailing cross-wind faster than the wind. And so on.

other kinds of sailing

In finishing up the first draft (yay!) of my paper on sailing, I thought about other kinds of sailing (for the discussion). One is solar sails: In principle if a spacecraft has a sufficiently large solar sail, of which it can change the size and shape, the spacecraft can navigate in arbitrary directions and perform arbitrary three-axis attitude adjustments, by working on a combination of radiation pressure and gravity. It's really very flexible. It makes me want to design a spacecraft!

I also thought about ice boats. An ice boat is like a sailboat with an extremely large keel and almost no water drag. That is, it sails like a sailboat in the limit that the keel gets large but the water drag gets small. This should make ice boats extremely fast at upwind sailing. Maybe I'll try to find an opportunity to sail on ice this winter?

sailing upwind; lift ratios; sail-to-keel ratio

In writing up a description of some of my physics-of-sailing results, I realized some things about sailboat design. For deep reasons you want the ratio of the area of the sail to the area of the keel to be roughly the ratio of the density of water to the density of air, or 700. That flows from the point that the sail and the keel have symmetric roles in sailing. But if you set this ratio to 700, you can only sail upwind if the sail lift ratio (the ratio of useful sail force to drag force) is very high. Since it is hard to make this lift ratio high on a commercial boat, the alternative is to make the sail smaller (relative to the keel). Looking at the data I can find on real sailboats, most have okay lift ratios but small sail-to-keel ratios (smaller than 700 anyway), so that they can sail quickly upwind. The cost of these design choices is that you can't go downwind faster than the wind. If you want to be able to sail both downwind faster than the wind—and also upwind—you have to have amazing sail and keel lift ratios.

optimal sizes of sails?

I spent some time in-between vacationing to look at the relative sizes of sails and keels on sailboats (yes, sailboats). I find that a boat sailing cross-wind sails fastest when the ram-pressure force prefactor (effective area times density) of the sail and the keel are comparable. That is, you want the effective area of the sail to be something like 800 times larger than the effective area of the keel! Strange, but maybe not false for the fastest competition sailboats?

physics-of-sailing literature

I sucked it up and read a bunch of the physics-of-sailing literature today (and on the weekend). Some of the books very correctly attribute the forces on sails and wings to momentum transport. Some of the books very incorrectly attribute them to differences of pressure calculable from Bernoulli effect alone. But in reading it all, I did come to the conclusion that no-one is working in precisely the space we want to work, so I do think there is a (correctly scoped) paper to write. Of course even if there weren't, I couldn't stop myself!

more sailing

I spent the weekend in an undisclosed location working on my ram-pressure model for a sailboat. I realized that there are multiple models, even if you decide that it will be ram pressure! I coded up multiple models, and also worked on writing text. I made figures like this one!

a model for sailing (yes, sailing)

I've had a lifetime of conversations with Hans-Walter Rix (MPIA) about the point that you could in principle sail with a sailboat with flat sails: Nothing about the curvature of the sails is integral or required by sailing. The curvature helps, but isn't necessary. I have had another lifetime of conversations with Matt Kleban (NYU) about the point that sailing depends on the relative velocity between the air and the water, and this leads to some hilarious physics problems involving sailing on rivers in zero wind (it's possible because a flowing river is moving relative to the dead air).

These worlds collided this weekend because—inspired by a twitter conversation—I finally built a proper ram-pressure model of a flat-sail, flat-keel sailboat and got it all working. It's sweet! It sails beautifully. Much more to say, but question is: Is there a paper to write?