- Armet 1 Author: David Paul Armet Mentor: Jerrold Marsden Editor: Gilliain Pierce
The Physics of Sailing
Sailboats have been one of the most important inventions ever made by human beings. They enabled ancient peoples to travel long distances without relying on the energy of their own legs or of pack animals; instead the free energy of the wind powered their boats. They could then discover far away continents and strange new lands, and sailing led to trade and exchange of ideas across broad oceans around the world. Sailboats are ancient, invented over 5000 years ago by the Babylonians in Mesopotamia. Since then, sailors and enthusiasts have been continually using and building new boats, always trying to better an old design. As a result of these years of improvement, modern sailboat designs is very sophisticated and refined. Modern designers do not attempt to find a completely new way to build a yacht; instead, they use modern scientific methods to better understand and optimize the ancient, already very good designs.
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The Concept of Lift
Contrary to the common conception about how sailboats sail, sails are not just big bags of air that drag the boat along. While this is true when the boat is going completely with the wind (see figure 2) when traveling perpendicular to the wind (see figure 1), all the forward force that drives the yacht comes from the lift of the keel and sails, just as lift keeps airplanes up in the air. Lift can occur in any
Fig 1: (Above) Two Americas cup yachts race into the wind. Fig 2: (Left) The same two yachts race with the wind both taken from the Associated Press, 2001. fluid that is moving around an object like a wing or a sail and it is simply a force that acts perpendicular to the flow of the air. Lift can be thought of two different ways. One way to think about lift is the science behind an airplane in flight, also explains a sailboats movement. The principles that keep the plane aloft also keep the yacht in motion. An eighteenth century mathematician Daniel Bernoulli described the concept of lift in terms of fluid (air or water generally) movement around a wing. Because the shape and angle that the wing is to the wind, wind travels a farther distance across the top of
Armet - 3 the wing and a shorter distance across the bottom, so it is moving relatively faster above the wing than below it. This difference in speed then creates a pressure difference because as air velocity increases pressure decreases, so this creates high pressure, pushing the airplane Fig 3: High pressure below the wings of an airplane and low pressure above creates the lift force that keeps an airplane in the air. Taken from Mr. Hand's 8th Grade Science Site
up, on one side of the wing and low pressure, pulling the airplane up, on the other, thus keeping the airplane aloft. This vacuum cleaner and hair dryer effect, as shown in figure 3, creates a net force upwards that counteracts the net force downwards made by gravity. Bernoulli formulated this idea into a law1, which describes a conservation of energy. Just as a roller coaster's potential energy when it is high at the top is quickly turned into fast speed (kinetic energy), air also loses potential energy when it gains kinetic energy; But in the airplanes case this cannot occur through a change in height but instead by a decrease in pressure so that the total energy of the air remains constant. But this is not the only way of thinking about lift. Lift can also be thought about using Newton third law "For every action, there is an equal and opposite
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Armet - 4 reaction." The force to keep the airplane up is equal and opposite to the force of accelerating air downward. A similar process is involved in sailing a yacht. Yachts are like airplanes with two wings, one that points up Fig 4: As air is deflected downward the wing is pushed upward, Also taken from Mr. Hand's 8th Grade Science Site.
(the sail) and another that points down (the keel). When airspeed increases on one side of the sail, the pressure decreases, and with the higher pressure on the opposite side of the sail the whole sail produces a net force propelling the yacht through the water. Just as lift keeps airplanes aloft it is the lift that drives the yachts through the water. When sailing into or perpendicular to the wind, lift is an incredibly powerful force that would push the yacht to the side instead of forward, except for the fact that the keel is doing the very same thing as the sails just in the opposite direction under the water. As the yacht travels through the water, the keel creates a side force that cancels out the side force from the lift of the sail leaving only the forward component of the sail lift force; this enables the boat to travel forward instead of just being blown to the side by the huge lift force. This effect makes sailboats very maneuverable and versatile because they do not have to follow the direction of the wind; instead, a yacht can make forward
Armet - 5 progress in any direction it likes, so long as there is some wind. Yachts cannot sail directly into the wind, but they can sail as close as thirty degrees into the wind,
Lift of sails
Lift of keel
Fig 5: The large lift force perpendicular to the sails is mainly directed at pulling the yacht to the side, but the lift force of the keel counteracts that side force canceling it out and only leaves a driving force to pull the yacht forward through the water. Taken from Larrson and Eliasson's Principals of Yacht Design 2nd Edition
Armet - 6 meaning that if the wind was coming from due north they could sail both north north west and north north east. By alternating between these two directions (Tacking) they can make forward progress directly into the wind. With this amazing force, lift, always comes its counterpart drag. Drag is simply the component of force parallel to the sails or wing instead of perpendicular to them. It generally thought of as wasted work in the case of an airplane wing, it is the force due to friction of air running across the wing, which pulls the airplane back, which is then why the propeller is needed to keep the airplane flying. Yachts also have this lift drag relationship. For example when a yacht is sailing perpendicular to the wind, the lift force can be as much as 50 times the drag force, but when sailing with the wind the lift force is zero and nothing but the air moving with the yachts drags the yacht forward. Although lift and drag are seemingly simple concepts for airplanes (lift always good and drag always bad), the relationship is much more complicated for yachts. For this reason different sails are used when traveling in different directions relative to the wind. Since sailing is a modern competitive sport new sailboat design is done mainly to win races like the America's cup, where teams from all over the world build racing yachts and try to put together the best team to win the competition with the most modern methods Fig 6: An Americas cup team races into the wind crashing through waves. Taken from A.P. America's cup press release photos
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How Physics Determines the Sailing of a Yacht
Let's take an in-depth look at how a yacht sails into the wind during a match race. A match race is where two yachts race a predetermined course upwind around a buoy and then back down-wind back across the start line. Because the yachts cannot sail directly into the wind, they must tack. And to win a race when tacking they try to travel the tightest path as close as they can get to traveling directly into the wind. Sailing into the wind, on the first leg of the race is very windy for both the sailors and the sails because not only does the yacht feel the true wind (of a stationary observer), but also the wind created by the yacht's movement. These two wind velocities add together, causing the apparent wind that sailors and their sails feel. Thus the sails feel a wind of a magnitude greater than that of the actual wind. Due to the addition of the wind the sails feel a very strong wind rushing by the sails creating both a lift and drag. But the lift is much stronger than the drag and thus when added the lift and drag force have a net component forward and that component of the sum of forces drives the yacht forward. Fig 7: Wind is additive meaning that when traveling into the wind the magnitude of the wind and the boat velocity add together to give a very strong wind, but when traveling with the wind, on the deck of a yacht there is nearly no wind because the boat velocity and true wind are "subtractive" Apparent wind
Baw
True wind
Btw Direction of travel
Armet - 8 Unfortunately, the forward driving force comes at the cost of a huge force to the side; worse still that side force is created by the sails that are high above the water. The side force itself is counteracted by the equal and opposite force side of the keel located below the water, which enables the yacht to sail forward rather than be pushed to the side. But because the forces are acting at a distance apart from each other, one up in the sails and the other down below the surface of the water, they create a torque (or moment) that causes the boat to rotate to the side. This is similar to a person sitting on one side of a see-saw. If that person wishes Sail not to sit only on the ground he needs another person to sit on the other side of the force see-saw as to balance the forces. This effect causes the yacht to tilt and if not balanced somehow the yacht would tilt completely over.
How The Moment is Balanced
The moment caused by the counteracting forces of the sail's and the keel is called the heeling moment because
Point of rotation Fig 8:
Keel force
it causes the yacht to heel or tilt to one side. The counter moment that then balances that unbalanced moment comes from the geometric shape of the hull and is called the righting moment because it "rights" this wrong situation of the boat tilting completely over.
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Fig 9: The righting moment created by the distance between the two equal and opposite forces of the gravity and buoyancy separated by a distance GZ counteracts the heeling moment and keeps the yacht from turning over onto its side. Taken from Larrson and Eliasson's Principals of Yacht Design 2nd Edition
Armet - 10 As the hull tilts to one side or the other, the center of buoyancy and the center of gravity come out of alignment and create a couple of equal and opposite forces separated by a distance. This effect is exactly the same as the keel and the sail side force, but where the separation of the keel and sail force tries to turn the yacht to the side the separation of the buoyancy and gravity force turns the yacht upright. These two moments one turning the yacht to the side and the other turning it back upright completely balance each other and keep the yacht from turning over.
Fig 10: As the yacht tilts (heels) to the side the separation between the buoyancy and gravity force grows up to 60 degrees of tilt and this separation of two forces counteracts the separation of the sail and keel forces. Taken from Larrson and Eliasson's Principals of Yacht Design 2nd Edition
GZ is the distance between the "centers" of mass and buoyancy and is completely analogous to the distance between the "centers" of the keel and the
Armet - 11 sails, but whereas the sails and keel are always at a constant distance apart, one can see from figure 10 that the GZ distance changes with respect to how much the yacht is tilted to the side. So depending on the magnitude of the side force created by wind, the yacht will find equilibrium in the righting and heeling moments at a given angle usually between 0-60 degrees of tilt. The more side wind, the more the tilt and the less side wind the less the tilt. When yachts travel directly with the wind the side force is zero so there is no heeling moment and thus the boat has no heeling angle and is thus perfectly flat in the water. The steepness of the first section of the GZ curve (0 to 60 degrees of tilt) really determents the sailing stability of a yacht. The steeper the curve the better because that means that one's yacht can counter a larger side force without having to tilt so much. At a given angle of tilt the yacht with the steeper GZ curve will sail faster because its side force and thus forward driving force are larger than a yacht with a less steep GZ curve.
Fig 11, 12: By changing the shape of the keel from the short fat one to the longer slender one, more of the mass of the yacht could be placed deeper under water and that change effects the GZ curve by raising it and making it steeper. The steeper GZ curve enables the side force and thus speed of a yacht to increase. Taken from Armet and Lerch's Velocity Prediction Program Modifications to the DS Match racer
Armet - 12 One way to make a GZ curve steeper is by distributing more of the weight of the yacht lower, thus moving the center of gravity lower, which in turn makes the yacht more stable. As seen by figure 11 and 12 by simply changing the shape of the keel can increase the steepness of the GZ curve and make the yacht much more stable. This newer more slender keel gives better performance, but it requires stronger materials, and the yacht can no longer sail in such shallow water.
Keel Lift
Now the forces on a yacht are pretty well explained except for one final point. The keel creates a large side force that counteracts the sails large side force but how is this force produced? This force is another "wing" type force, but instead of being the sails in the air or an airplane in the air, it is the force due to water rushing around the keel. Because water is much denser than air the keel can be much smaller and still give an equitant force to that of the sails. The keel creates lift force, but is a symmetric wing so it works equally well to each side. A good way to think of this force is to remember back to your childhood where when driving on the freeway sometimes you would put your hand out side of the window and tilt it slightly upwards and your whole arm would float upwards then if you tilted your hand downwards your arm would be forced down. The keel is being tilted much like your hand in fast moving water flow and that creates a huge force, but instead of pulling the yacht up or down it is a side force that keeps the yacht on course so it is not blown to the side by the wind.
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Fig 13: As the ke differences identi the force that cou and Eliasson's Pr
Fig 14: The lift and drag of a keel add together and counteract the sail side force. Taken from Larrson and Eliasson's Principals of Yacht Design 2nd Edition
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A honed design
So with years of continual improvement and the natural advantage that the lift force is many times greater than the drag force, yachts and other sailing craft can even go faster than the wind that they are traveling in. This happens because a yacht's sails do not feel the true wind, but instead the sum of both the true wind and the velocity of the craft's own movement. Some modern designs like ice boats can reach speeds of 60 MPH and typically go 2-4 times the speed of the true wind2 and yachts can break 1.5 times the wind speed and go almost up to 50knots with certain designs that drastically decrease the water and wave friction.
Fig 15: Because of the incredibly low friction of skates on ice and the fact that a sail feels not only the true wind, but also the wind created by the movement of the craft, this ice boat can typically go 2 to 3 times the speed of the wind and up to 60 MPH. Taken from the International DN Ice Yacht Racing Association
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Goodwin, Paul and Bowen, Ted. International DN Ice Yacht Racing Association http://www.sailingsource.com/ice/ Accessed 3/14/03
Armet - 15 A way to drastically decrease the water and wave friction of a yacht is to no longer keep the yacht hull in the water but to lift it out with small water wings called hydrofoils. These hydrofoils are just like airplane wings but are beneath the surface of the water and because of the advantage that water is many times as dense as air they can lift the entire yacht out of the water at relatively slow speeds. Ice boats and hydrofoils do not rely on any physics that is different than that of the yachts described earlier, but instead they reach their extreme speeds, just by reducing friction in the case of the ice boat by sailing on ice skates and in the case of a hydrofoil, by lifting the hull out of the water. These speeds of upwards to 60 mph are truly amazing though because they do not rely on oil or motors, but simply the free and plentiful power of the wind. Fig 16: Hydrofoils lift the hull out of the water to drastically reduce the friction created by the hull in the water. Taken from Brandon Mercer's article in Tech Live "Sailboats That Fly" Jan 16 2002
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Works Cited
Fig 5, 9, 10,13,14 Larrson, L and Eliasson, R. Principals of Yacht Design 2nd Edition. London: Adlard Coles Nautical, 2000. Figure 12, 13 Armet, David, Lerch Bengerman. Velocity Prediction Program. Modifications to the DS Match racer. Danish Technical University, 2002. Figure 1, 2, 7 Associated Press, Americas cup photos, 2001 Figure 3, 4 Hand. Mr. Hand's 8th Grade Science Site, Accessed 2/15/03. http://www.mansfieldct.org/schools/mms/staff/hand/ Figure 15 Goodwin, Paul and Bowen, Ted. International DN Ice Yacht Racing Association. Accessed 3/14/03. http://www.sailingsource.com/ice/ Figure 16 Mercer, Brandon. "Sailboats That Fly; In race to break world speed record, designers lift boats out of the water." Tech Live Jan 16 2002. Accessed 3/14/03 http://www.techtv.com/news/sportstech/story/0,24195,3368430,00.html