In
addition to the recreational pleasure sailing affords, it involves some interesting physics.
Sailing starts with the force of the wind on the sails. Analyzing that interaction yields some results
not commonly known to non-sailors. It turns out, for example, that downwind is not the fastest direction
for sailing. And there are aerodynamic issues. Sails and keels work by providing "lift" from the
fluid passing around them. So optimizing keel and wing shapes involves wing theory.
The resistance experienced by a moving
sailboat includes the effects of waves, eddies, and turbulence in the water, and of the vortices
produced in air by the sails. To reduce resistance effectively by optimizing hulls, keels, and
sails, one has to understand its various components.
Wind power
Moving air has kinetic energy that can,
through its interaction with the sails, be used to propel a sailboat. Like airplane wings, sails
exploit Bernoulli's principle. An airplane wing is designed to cause the air moving over its top
to move faster than the air moving along its undersurface. That results in lower pressure above
the wing than below it. The pressure difference generates the lift provided by the wing.
There is much discussion
of whether the pressure difference arises entirely from the Bernoulli effect or partly from the
wing's impact and redirection of the air. Classic wing theory attributes all the lift to the Bernoulli
effect and ascribes the difference in wind speeds above and below the wing to the wing's asymmetric
cross-sectional shape, which caused the path on top to be longer. But it's well known that an up–down
symmetrical wing can provide lift simply by moving through the air with an upward tilt, called the
angle of attack. Then, despite the wing's symmetry, the wind still experiences a longer path and
thus greater speed over the top of the wing than under its bottom. A NASA website has an excellent
discussion of the various contributions to lift by an airplane wing.1 It disputes
the conventional simple version of wing theory and emphasizes that lift is produced by the turning
of the fluid flow.
The case is similar for
sailboats. A sail is almost always curved and presented to the wind at an angle of attack. The situation
is shown schematically in figure 1a. The wind moving around the "upper," or downwind, side of the
sail is forced to take the longer path. So the presence of the surrounding moving air makes it move
faster than the air passing along the "lower," or upwind, side of the sail. Measurements confirm
that relative to the air pressure far from the sail, the pressure is higher on the upwind side and
lower on the downwind side.
For downwind sailing,
with the sail oriented perpendicular to the wind direction, the pressure increase on the upwind
side is greater than the pressure decrease on the downwind side. As one turns the boat more and more
into the direction from which the wind is coming, those differences reverse, so that with the wind
perpendicular to the motion of the boat, the pressure decrease on the downwind side is greater than
the pressure increase on the upwind side. For a boat sailing almost directly into the wind, the pressure
decrease on the downwind side is much greater than the increase on the upwind side.
Experimenting with what
can be done, a beginner finds some surprising results. Sailors know well that the fastest point
of sail (the boat's direction of motion with respect to the wind direction) is not directly downwind.
Sailboats move fastest when the boat is moving with the wind coming "abeam" (from the side). That's
easily understood: When a sailboat is moving directly downwind, it can never move faster than the
wind because, at the wind speed, the sails would feel no wind. In fact, a boat going downwind can never
attain the wind speed because there's always some resistance to its motion through the water.
But when the boat is moving
perpendicular to the wind, the boat's speed doesn't decrease the force of the wind on the sails.
One sets the sails at about 45° to the direction of motion—and to the wind. The boat's
equilibrium speed is determined by the roughly constant force of the wind in the sails and the resistance
against the boat's motion through the water. If the resistance can be made small, the velocity can
be large. That's seen most dramatically for sail iceboats, which skate on the ice with very little
resistance. They can glide along at speeds in excess of 150 km/h with the wind abeam at speeds of only
50 km/h! Of course sailboats plowing through the water experience much more resistance. Nonetheless,
some specially constructed sailboats have attained speeds of more than twice the wind speed.
Keels
It was recognized centuries ago that
a sailboat needs something to help it move in the direction in which it's pointed rather than just
drifting downwind. The answer was the keel. Until the development of modern wing theory, it was
thought that one needed a long, deep keel to prevent side-slipping. But now it's understood that
a keel, like a sail, works by providing sideways lift as the water flows around it, as shown in figure
1a. A keel must be symmetrical for the sailboat to move to either side of the wind.
A keel works only if the
motion of the boat is not exactly in the direction in which it's pointed. The boat must be moving somewhat
sideways. In that "crabbing" motion, the keel moves through the water with an angle of attack. Just
as for the sails in the wind, that causes the water on the "high" (more downstream) side of the keel
to move faster and create a lower pressure. Again, the net lift force on the keel is due to the combination
of that decreased pressure on the high side and increased pressure on the other (low) side.
In figure 1b, the keel lift
thus generated points almost in the opposite direction from the lift provided by the sails. The
two vectors can be resolved into components along and perpendicular to the boat's direction of
motion. For a sailboat moving in equilibrium—that is, at constant speed in a fixed direction—the
transverse lift components from sail and keel cancel each other. The component of the driving force
from the sails in the direction of motion is the force that is actually moving the boat forward. For
equilibrium motion, that force is balanced by the opposing component of the keel lift plus the total
resistive force.
Wing theory, developed
over the past 100 years for flight, indicates that the most efficient wing is long and narrow. Vortices
produced at the wing tip cost energy. A long, narrow wing maximizes the ratio of lift to vortex dissipation,
thus providing the best performance for a given wing surface area. That also applies to sailboat
sails and keels.
It is now recognized that
the most efficient keels are narrow from front to back and deep. Such a keel can have much less surface
area than the old long keels. Less area means less resistance. Most modern racing sailboats, such
as those used in the America's Cup races, have deep, narrow keels that are very efficient at providing
the lift necessary to prevent side-slipping. Of course, such keels are a problem for recreational
sailors in shallow waters.
Resistive forces
A sailboat experiences several kinds
of resistance. The first is simply the resistance of the hull moving through water. As the boat moves,
it shears the water. Water molecules adhere to the hull's surface. So there must be a shear—that
is, a velocity gradient—between the adhering molecular layer at rest with respect to the
hull and the bulk of water farther away. The shear means that van der Waals couplings between water
molecules are being broken. That costs energy and creates the resistive force, which becomes stronger
as the boat's speed increases. The energy dissipation also increases with the total area of wetted
surface.
Although the effect is
called frictional resistance, it's important to realize that the resistive force in water is basically
different from the frictional force between solid surfaces rubbed together. To reduce ordinary
friction, one can polish or lubricate the sliding surfaces. That makes surface bumps smaller,
and it substitutes the shearing of fluid lubricant molecules for shearing of the more tightly bound
molecules on the solid surfaces.
For a boat moving through
water, however, polishing the hull doesn't eliminate the shearing of the molecules of water, which
is already a fluid. The resistive force cannot be reduced significantly except by reducing the
wetted surface. It does help to have a smooth surface, but that's primarily to reduce turbulence.
The generation of turbulence
is a general phenomenon in the flow of fluids. At sufficiently low speeds, fluid flow is laminar.
At higher speeds, turbulence begins. Its onset has to do with the shearing of the molecules in the
fluid. When the shearing reaches a critical rate, the fluid can no longer respond with a continuous
dynamic equilibrium in the flow, and the result is turbulence. Its onset is quantified in terms
of the Reynolds number
R = (Lυ)/(µ/ρ), (1)
where υ is the velocity
of the flowing fluid, µ
is its viscosity, ρ
is its density, and L is the relevant length scale of the system. Rearranging factors in
equation 1, one can think of R as the ratio of inertial forces (ρυ)
to viscous forces (µ/L).
In the late 19th century, English engineer Osborne Reynolds found that, with surprising universality,
turbulence begins when that dimensionless parameter exceeds about a million.
For a boat of length L
moving through water at velocity υ to see when turbulence begins in the flow along the hull,
R is about 106Lυ (in SI units). A typical speed for a sailboat
is 5 knots (2.4 m/s). At that speed, then, one should expect turbulence for any boat longer than half
a meter. (Used worldwide as a measure of boat speed, a knot is one nautical mile per hour. A nautical
mile is one arcminute of latitude, or 1.85 km.)
Because turbulence dissipates
energy, it increases the resistance to motion through the water. With turbulence, a sailboat's
resistance is typically four or five times greater than it is when the flow along the hull is laminar.
A rough surface will cause turbulence to be greater and begin sooner. That's the main reason to have
a smooth hull surface.
Turbulence also occurs
in the air flowing along the surface of the sail. Water is a thousand times denser than air and 50 times
more viscous. So for the air–sail system one gets
R = 7 × 104Lυ. (2)
For a typical wind speed
of 5 m/s, then, one gets turbulence if the sail is wider than about 3 meters. When turbulence forms
in the air flow along the sail, the desired pressure difference between the two sides of the sail—its
lift—is diminished.
Another important resistive
force comes from vortex generation at the bottom of the keel and at the top of the sails. When the air
or water moves around the longer-path side of the sail or keel, its speed increases and therefore
its pressure falls. As the air or water moves along the sail or keel, it will respond to the resulting
pressure difference by trying to migrate from the high-pressure side to the low-pressure side.
Figure 2 sketches that effect for a keel. What actually happens, as shown in the figure's side view,
is that the flow angles a bit up on one side and down on the other. When those flows meet at the back of
the sail or keel, the difference in their arrival angles has a twisting effect on the fluid flow that
can cause a vortex to come off the top of the sail or the bottom of the keel.
The effect is well known
for airplane wings. Called induced drag, vortex formation costs energy. Figure 3 shows vortices
generated at the tops of sails by racing sailboats moving through a fog. A long keel will generate
very large vortices. By making the keel short and deep, one can increase the ratio of lift to energy
dissipated by vortices. The same is accomplished—especially for sailboats racing upwind—by
having tall, narrow sails. It's also why gliders have long, narrow wings.
Because it's often impractical
to have a short, deep keel or a narrow, long wing, one can install a vane at the tip to reduce the flow
from the high-pressure to the low-pressure side. On planes they're called winglets, and on keels
they're simply called wings. A modern recreational or cruising sailboat will have a keel that's
a compromise between the old-fashioned long keels and the modern deep, narrow keels—with
a wing at the bottom rear end to reduce induced drag. Such keel wings were first used by the victorious
sailboat Australia II in the 1983 America's Cup race. Modern wing theory also suggests
that to minimize induced drag, keels and sails should have elliptic or tapered trailing edges.2
Such shaped edges are now common.
A sailboat also has a resistance
component due simply to its deflection of water sideways as it advances. That's called form resistance,
and it obviously depends on hull geometry. It's easy to see that narrow hulls provide less resistance
than do wider hulls. Any boat will always be a compromise between providing low form resistance
and providing passenger and cargo space. Seeking to minimize form resistance for a given hull volume,
shipbuilders have tried many basic hull shapes over the centuries. Even Isaac Newton weighed in
on the question. He concluded that the best hull shape is an ellipsoid of revolution with a truncated
cone at the bow.
Extensive computer modeling
and tank testing have resulted in a modern hull design that widens slowly back from the bow and then
remains fairly wide near the stern. Even with a wide stern, designers try to provide enough taper
toward the back to allow smooth flow there. That taper is often accomplished by having the stern
rise smoothly from the water rather than by narrowing the beam. If the flow from the stern is not smooth,
large eddies will form and contribute to resistance.
As a boat moves through water, it creates
a bow wave that moves with the speed of the boat. Water waves are dispersive; long waves propagate
faster than short ones. Therefore the length of the full wave generated by the bow is determined
by the boat's speed. As a boat starts to move slowly through the water, one sees at first a number of
wave crests and troughs moving down the side of the hull. As the boat speeds up, the wavelength gets
longer and one sees fewer waves down the side. Eventually at some speed, the wave will be long enough
so that there's just one wave down the side of the boat, with its crest at the bow, a trough in the middle,
and another crest at the stern (see figure 4). That's called the hull speed.
If the boat speed increases
further, the wavelength increases so that the second crest moves back behind the boat and the stern
begins to descend into the trough. At that point, the boat is literally sailing uphill and the resistance
increases dramatically. That's called wave resistance. Of course, if one has a powerboat with
a large engine and a flat-bottomed hull, one can "gun" the engine and cause the boat to jump up on the
bow wave and start to plane on the water's surface. Most sailboats don't have either the power or
the hull geometry to plane. So they're ultimately limited by wave resistance.
The wave-resistance limit
also applies to all other so-called displacement boats: freighters, tankers, tugs, and most naval
vessels bigger than PT boats—that is, any boat that can't rise to plane on the surface. The
functional dependence of water-wave speed υ on wavelength λ
is well known. From the limiting case for deep-water waves for the solution of the two-dimensional
Laplace wave equation,3 or from a simple derivation due originally to Lord Rayleigh,4
one gets υ = √gλ/2π,
where g is the acceleration of gravity. In the form commonly used by sailors in the US,
υ = 1.34
√λ, (3)
where the λ
is in feet and υ is in knots.
If one equates the wavelength
to the waterline length of a boat, equation 3 gives the boat's hull speed. For a sailboat with a waterline
length of 20 feet (6 m), the hull speed is 6 knots. For a large cruising sailboat with a waterline of
40 feet (12 m), it's about 8 knots. And for a 300-foot-long naval vessel, it's 23 knots. In practice,
it's very difficult to make a displacement boat go faster than about 1.5 times its hull speed.
Combining all the components
of resistance for a sailboat moving at close to its hull speed, one finds that the frictional resistance
contributes about a third of the total, and the wave resistance another third. Form resistance
accounts for about 10%, as does the induced drag from vortex generation at the bottom of the keel.
The assorted remaining contributions, including eddy formation behind the boat and aerial vortex
generation by the sails, provide the remaining 10 to 15%. Of course the fractional contributions
vary with boat speed, wave conditions, and the direction of motion relative to the wind.
Predicting speed
One can exploit the physics of sailing
to calculate boat speeds for a given sailboat for different wind speeds and points of sail. Such
calculations are usually performed iteratively by computer programs that start from two basic
vector equations to be solved simultaneously:
Fdrive = Fresistance and Mheel = Mrighting. (4)
Here Fdrive
is the total driving force in the direction of motion provided by the wind in the sails, and Fresistance
is the sum of all the resistive forces. The torques Mheel and Mrighting
are the heeling and righting moments caused by the wind in the sails and the weight of the hull and
keel.
The force of the wind on
the sail is calculated as a lifting force perpendicular to the apparent wind direction and a drag
force in the direction of the apparent wind. (The apparent wind is the wind as perceived by an observer
aboard the moving vessel.) These lift and drag forces are then resolved into components along and
perpendicular to the direction of motion. The net force in the direction of motion is then Fdrive,
and the net force perpendicular to the boat's motion is what produces the heeling moment. The two
equations in (4) must be solved simultaneously because the angle of heel affects the total driving
force.
Following Bernoulli's
principle, one takes the force of the wind in the sails to be proportional to the total sail area times
the square of the apparent wind speed. The actual forces are then obtained with empirical lift and
drag coefficients, given as functions of sail geometry and angle of attack. Frictional resistance
is proportional to the hull's wetted surface area and increases as the square of the boat's speed.
All the various contributions to total resistance involve empirical coefficients. Wave and form
resistance are expressed as functions of the hull's "prismatic coefficient," which is an inverse
measure of the tapered slimness of its ends.
There are simple and complex
speed-prediction computer programs. Some that have been refined over decades for racing applications
are kept private and closely guarded. Figure 5 shows the results of calculations I performed for
a 30-foot (10-m) cruising sailboat using a publicly available program.5 The figure
shows the calculated boat speed as a function of wind speed and point of sail. The predicted boat
speeds are greatest when one is sailing about 90° away from the wind direction. Sailors call
that beam reaching. It yields a boat speed of about half the wind speed.
Such calculations are
confirmed experimentally, with a degree of accuracy that depends on the sophistication of the
model and on how much the program has been tuned for a specific kind of sailboat. Broadly speaking,
a sailboat is faster if it is longer and narrower, with bigger sails and a smaller wetted surface.
Such general rules can, of course, yield a boat that's longer than one wants, or tips over too easily,
or has too little room inside.
So every design feature
is a compromise between competing needs. For sailing downwind, one wants fairly square sails,
which are best at catching the wind. But for sailing upwind, taller, narrower sails are best, because
they maximize the ratio of lift to energy lost by generating vortices. The most efficient keel is
deep and narrow, to maximize lift with minimal surface area. But a deep keel is problematic in shallow
waters. Shorter keels with wings or bulbs at the bottom usually represent the best compromise for
overall sailing.
What's the highest speed
a sailboat can reach? The trick is to reduce resistance. An iceboat can outrun the wind because it
has so little resistance. For a sailboat, the resistance comes primarily from having to plow through
the water. The best way to reduce that resistance is to move less and less of the boat through the water.
One answer is hydrofoils. They are vanes placed below the hull that raise it out of the water as the
boat speeds up.
Sailboats with hydrofoils
have reached speeds of more than 40 knots when the wind speed was barely half that. One such craft
is shown in figure 6. These vessels are not usually practical for cruising and other normal recreational
activities. They're sometimes dismissed as low-flying aircraft. A more practical alternative
is the catamaran—a double-hulled sailboat. Catamarans are being developed to provide
relatively stable, fast sailing. Although they are more expensive than traditional single-hull
sailboats for a given amount of living space, catamarans are becoming increasingly popular.
Bryon
Anderson is an experimental nuclear physicist and chairman of the physics department at Kent
State University in Kent, Ohio. He is also an avocational sailor who lectures and writes about the
intersection between physics and sailing.
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