# Effort on sail

This article may be expanded with text translated from the corresponding article in French Wikipedia. (June 2010)After translating, `{{Translated|fr|Effort on sail}}` must be added to the talk page to ensure copyright compliance. Translation instructions · Translate via Google |

The purpose of sail is to take wind energy and transmit this energy to boat. Effort of wind is on all surface of sail. We need to calculate :

- location of center of pressure
- intensity of
**effort on sail**

This calculating is very important for the design of ship (stability). Calculation is very complex and much more complex that a wing ^{[1]}. Calculus is part of Fluid mechanics and Aerodynamics.

In the reality the sail is not in-deformable, the wind is not constant, the boat is not in uniform speed, the mast is not infinitely stiff, the air is viscous (losses by friction). The flow of the air varies (turbulent, laminar), the mast perturbs the flow (except when it is profiled). For clarity, these phenomena will not be taken into account ^{[2]}^{[3]}^{[4]}^{[5]}.

## Contents

- 1 Center of pressure of sail
- 2 Effort on sail
- 3 Lift effect on sail
- 3.1 turbulent flow or downwind
- 3.2 laminar flow
- 3.3 influence of effort on sheet on lift performance
- 3.4 Influence of Aspect ratio
- 3.5 Shape of luff of leech, and foot
- 3.6 Contribution of the lift at the progress of the vessel
- 3.7 Evolution of lift coefficient following incidence: polar sail
- 3.8 Sail twist

- 4 Several sails: multidimensional problem resolution
- 5 Annexe

## Center of pressure of sail

The effort on sail could be summarize and could be reduce in on point. This point is named center of pressure.
In first approach, the center of pressure is the geometric center of sail. With the wind, the sail have a ball shape and if shape is stable, then the location of center of pressure is stable. On a sail of topsail and by rear wind, the center goes back up(raises) little towards the top according to the tension of the sheep and the tangon. On a sail of genoa and in the speed(look) of near, the center move to the front of the boat from around 10 to 15 % ^{[6]}.

### Chord of sail

The chord of sail is a straight line between leading edge and training edge. This notion is useful for a good approximation of direction of effort on sail. The effort of sail is quasi perpendicular to chord and located quasi at the maximum camber of sail.

## Effort on sail

At the microscopic level, in a perpetual motion of air parcels moving continuously. However, macroscopically, the air can not move. If the air does not move, it means that each parcel is more or less in the same place (random motion). The parcel of air moves around an imaginary fixed point in space without too much away from this point. If air moves mean that overall the parcel moving in large numbers in the same direction (directed movement). Of course the resulting motion can be a combination of both. The movement of air parcels have two origins : temperature and mechanical influence of wind.

### Role of Atmospheric pressure

Thanks to the energy of the temperature, the parcel are constantly shifting erratically. By moving an air parcel will soon meet one another, and the meeting is a shock. Shock changes the trajectories. The two parcels bounce one on the other. Each off go approximately to its starting point. Again, it meets another parcel for a new shock reducing it back to its point of departure etc, etc.. Overall in distance, it gives impression that parcel do not move (random motion).

More air parcel is at high altitude, less gravity is important. It is therefore less strength for the return to Earth, and shocks are less violent and frequent. Thus more parcel is close to sea level, more collisions are frequent and violent.

When the plot is very close to the sail, the shock occurs between sail and parcel. These collisions generate a considerable force on sail, the force at sea level is about 10 tons per square meter on sail. This force is exerted on a surface, therefore force is a pressure. This pressure is atmospheric pressure. If a wing has two sides, air pressure is on both sides. Finally, two pressures are perfectly balanced, sail does not move.

### Role of wind

Now a part of the movement of parcels is globally ordered (seen far away), parcels move together in the same direction. From a distance the air moves, so call wind.

Depending on the configuration of the sail, a parcel of air near sail could have different state :

- sail is free, wind have no resistance except thickness of cloth, air parcel passes without being significantly disturbed
- sail is perpendicular to the wind (sail or spinnaker topsail downwind), parcel of air crashes against sail. It is almost stopped. Other parcel behind strongly prevent reversing (rebound). Air parcel send a maximum of energy to sail, quasi all energy of movement ordered.
- In the intermediate cases, the air parcel bounces more or less, they give a portion of its energy. They disrupt orderly movement of those who accompany the collision. These in turn disrupt the orderly movement of other parcel by other collisions, etc, etc.. But rebounding, it will also result, disrupt balance of air pressure, creating an overpressure in wind face of sail and depression under wind face of sail.

There are two phenomena, condition which push sail (wind pressure) and condition which pull sail (depression due to wind).

### Direction of effort

The shock of air parcel on sail set back the sail. Shock moves only push very little on the side. Effort is almost perpendicular to the surface of sail.

### value of effort

The plots of air strike sail on both sides so:

- On the windward side efforts are atmospheric pressure, wind pressure, and virtually no depression due to wind.
- On the leeward side efforts are atmospheric pressure, a bit of depression and almost no wind pressure.

To simplify the manipulation of these forces, the forces are summed into a single force and that the entire surface of the profile (sailing) in a simple formula (valid for airplane wings like a rudder, a sail, an anti Plan -drift) ^{[7]}:

<math>F = C \times E </math> ^{[8]}

with

- E = effort that can give up the wind (see Max Q);

- C = coefficient Aerodynamic

According to the Bernoulli, the maximum stress of wind or density of kinetic energy maximum for the entire surface of the sail:

<math>E = e_c \times S = Max Q \times S = \frac12 \times \rho \times S \times V^2 </math>

The full expression of the force is:

<math> F = \frac12 \times \rho \times S \times C \times V ^ 2 </math>

with

- F = lift, expressed in Newton

- <math> \rho </math> (rho) = density air (<math> \rho </math> varies with the temperature and the pressure) ;

- S = typical surface, for sail, it is the sail area in m²

- C = coefficient Aerodynamic. Aerodynamic coefficient is unit-less, it is the sum of two percentages: the percentage of recovered energy leeward side + percentage of the recovered energy into the wind. For this reason, the coefficient aerodynamics can be greater than 1. It depends on the angle of upwind sailing.
- V = Speed is the speed of the wind relative to the sail (Apparent wind) in m / s.

The sail is deformed by the wind and takes a form named Airfoil. When the flow of air around the profile is Laminar ^{[9]}, the factor against depression in the wind becomes crucial. This effect is then called lift. Studies and theory to draw a sail ^{[10]} that:

- Depression on the upper (leeward side) represents two thirds of the lift,
- The pressure on the lower surface (facing the wind) represents one third of the lift.

## Lift effect on sail

The study effect of lift can compare cases with and without lift ^{[11]}. A typical example is a gaff sail. The sail is rectangular and is approximately vertical. The sail has an area of 10 sqm, with 2.5m of foot by 4m of leech. The apparent wind is 8.3 m / s (about 30 km / h). The boat is supposed to uniform velocity, no wave. It does not heel, does not pitch. The density of air is set at: <math>\rho = 1.2 kg / m^3</math>

### turbulent flow or downwind

The boat is running downwind. The shape of the sail is approximated by a plane perpendicular to the apparent wind.

The depression effect on the sail is second order, and therefore negligible, it remains:

- On the windward side, efforts are atmospheric pressure and wind pressure
- On the leeward side, there remains only the atmospheric pressure

Efforts to atmospheric pressure cancel out. There remains only pressure generated by the wind.

Roughly speaking, shock of parcel on the sail forward all their energy from wind in 90% of the surface of the sail. This means that the Cz or aerodynamic lift coefficient is equal to 0.9.

<math>F = \frac12 \times 1.2 \times 10 \times 0.9 \times 8.3^2 = 372 \ newton</math>

### laminar flow

The boat is Close hauled. The wind has an angle of about 15 degrees with chord of the sail.

Because the setting of the sail at 15 ° relative to the apparent wind, the camber of the sail creates a lift. In other words, the effect of depression on the leeward side is not neglected. As air pressure forces cancel out, efforts remain are:

- On the windward side, wind pressure,
- On the leeward side, wind depression.

The only unknown is the drag coefficient to be estimated. Curve takes a good adjustment of sail is close to upper shape
NACA 0012^{[12]}^{[13]}. A sail less well adjusted or older technology (old rig), will be more hollow, more camber. The coefficient of aerodynamic lift will be higher but the sail will be less efficient (lower finesse). The profiles would be more suitable profiles as NACA 0015, NACA 0018 ^{[14]}.

For a given profile, there are tables which giving the lift coefficient of the profile. The lift coefficient (Cz) depends on several variables:

- Incidence (angle: apparent wind / Profile)
- The lift hill of the sail, which depends on its extension,
- The surface roughness and Reynolds number, which affect the flow of fluid (laminar, turbulent).

The coefficient is determined for a fluid stable and uniform, and a profile of infinite extension.

The Reynolds number is: <math> \mathrm{Re} = {{\rho {\bold \mathrm U} L} \over {\mu}} = {{{\bold \mathrm U} L} \over {\nu}}</math>

with

- <math> U </math> - fluid velocity or apparent wind [m / s]
- <math> L </math>- characteristic length or foot of the sail [m]
- <math> \nu </math> - kinematic viscosity fluid: <math> \nu = \eta / \rho </math> [m / s]
- <math> \rho </math> - density air [kg / m³]
- <math> \mu </math> - dynamic viscosity air [Pa] or Poiseuille [pl]

so for this sail about <math> \mathrm{Re} = 10^6</math>

Under an incidence of 15 ° and a Reynolds number to one million, reached a NACA0012 Profile Cz 1.5 instead of 0.9 or 1-90 ° incidence.

<math>F = \frac12 \times 1.2 \times 10 \times 1.5 \times 8.3^2 = 620 \ Newton</math>

The lift has increased by 50%. This also corresponds on sheet an increase of 50% effort for the same apparent wind.^{[15]}^{[16]}

### influence of effort on sheet on lift performance

Set a sail is the setting of two parameters:

- Addressing the impact, ie adjust the angle "apparent wind / sail" is at maximum lift, or a maximum of finesse (ratio lift / drag). This angle varies in height (aerodynamic twist)
- Set the profile (camber) of the sail.

Sailing is a generally flexible ^{[17]}. The profile of the sail changes depending on the settings of the sail. At a given incidence, the sail can take different forms. The shapes depends on the stress on clew corner of sail thanks to main sheep. Other stress: at the point of tack, the Cunningham, the backstay. These elements help determine a possible shape of sailing. More exactly they can decide position of maximum camber on sail.^{[18]}

Each profile represents an appropriate value of Cz. The position of the trough along the chord with the most lift is about 40% foot from luff. Leeward side of sail is close to the NACA series 0012 (NACA 0015, NACA 0018, etc. within the possibilities of tunning).

### Influence of Aspect ratio

A sail is not infinitely long. So there have ends, in our case of mainsail :

- Boom
- Horn.

When the sail propels the ship, leeward side is depressed, windward side is under pressure. At border of sail, depression is in contact with pressure. Naturally, compressed air molecules (many and frequent shocks) will rush into the area depression (low impact and less frequent). Consequence is that the area was depressed more air molecules than expected so the depression is lower (more pressure than expected). Similarly, the area is under pressure from air molecules than expected so the pressure is less strong. Propulsive effect is less.

The distance between the downwind and upwind ends of the veil is very low ^{[19]}, a pressure zone closer to a depression zone, the transfer movement of molecules from one side of the sail to another is very violent. This creates significant turbulence. On a sail Bermuda, foot and leech are two areas where this phenomenon exists. The drag of leech is included in drag "usual lift curves, profile is considered as infinite (ie no border). But foot drag is to be calculated separately. This loss of efficiency of the sail the border is called Lift-induced drag.

Lift-induced drag is directly related to the length of the extremities. More horn is long more induced drag is high. Conversely sail can reefing, i.e. surface of the sail without reducing the length of the horn changes. This means that value of the lift-induced drag will be substantially the same. For a given length of horn, more sail is bigger, lower is ratio lift-induced drag on lift. More sail is stretched, more lift-induced drag alter slightly value of lift coefficient.

Lift-induced drag depends only on aspect ratio. The extension is defined ^{[20]} :

- <math>\lambda = {b^2 \over S}</math>

with

*B*is the length of luff*S*on the surface of the sail.

Lift-induced drag is:

- <math>Ci = {{Cz^2} \over {\pi \times \lambda \times e}} </math>

with

- Cz : Lift coefficient of airfoil
- <math>\pi</math> (pi) : 3.1416
- λ : Aspect ratio (wing) (dimensionless)
- e : Oswald efficiency number (less than 1) which depends on the distribution of lift in scale. "e" could be equal to 1 for a distribution of lift "ideal" (elliptical). An elliptical shape of the ends to reduce induced drag better. In practice "e" is the order of 0.75 to 0.85. Only a three-dimensional model and tests to determine the value of "e".

Optimal distribution of maximum reducing lift-induced drag is elliptical in shape ^{[21]}^{[22]}. Accordingly, luff will be elliptical, so the mast is not straight as on classic boat, but the mast is design with the closest possible form of ellipse. Elliptical configuration mast is possible with modern materials. It is very pronounced on surfboards. On modern sailboats, mast is curved thanks to Shroud (sailing). Similarly leech will be elliptical. This profile is not natural for a flexible sail, its reason sial uses batten for maintain leech for this curvature.

An ideal lift-induced drag distribution creates a elliptical sail, but current shape of the sails is rather a half-ellipse, as if second half part of ellipse was completely immersed in the sea. It is logical, because as wind speed is nul at the sea level (0 m), sea is equivalent to "mirror" for an aerodynamic point of view. So only half of an ellipse in air is necessary.

### Shape of luff of leech, and foot

A sail hauled up have a three-dimensional shape. This form is form chosen by the sailmaker. Now 3D shape is different between hauled up form and empty (no wind, see for example when sail is deployed at sailmaker). We must take this into account when cutting the sail.

The general shape of a sail is a deformed polygon. The polygon is slightly distorted in the case of Bermuda sail, heavily distorted in the case of spinnaker. Shape of edges empty is different from shape of edges once the sail is hauled up. Convex empty can go to straight edge when sail is hauled up.

Edge can be:

- convex
- concave
- right

When the convex shape is not natural (except for a free edge, a spinnaker), sail is equipped with batten to maintain this form when the convex shape is pronounced. Except for the spinnaker with a balloon-shaped, variation of edge empty compared to straight line remains low, a few centimeters.

Once hauled up, sail elliptical would be ideal. But as sail is not rigid:

- You need a mast for reasons of technical feasibility is quite right.
- Flexibility of the sail can bring other problems, it is better to fix at the expense of ideal elliptic shape (convex).

#### Leech

The oval is the ideal (convex), but a fall concave vacuum improves the twist at the top of the sail and prevents the collapse of "padding" in the gusts, thereby improving its stability. The fall concave make sailing more tolerant and more neutral. A convex shape is also an easy way to increase the sail area (roach).

#### Luff

Once hauled up, edge must be parallel to forestay or mast. Similarly when the veil is horn. Masts and spars are very often (except surfing) rights, right luff is priori in form to use.

But hollow of the sail is normally closer to luff than foot. So to facilitate the implementation of hollow of sail hauled up, the empty form of luff is convex^{[23]}. Convexity is called the luff luff round. When rigging is complex, shape of the mast is not straight ^{[24]}. In this case we must take into account, and the shape of luff empty can be convex at bottom and concave to top.

#### Foot

Foot form has little importance, particularly on sails with free edge. Its shape is more motivated by aesthetic reasons. Often convex empty to be right once hauled up. When the border is attached to a spar or boom a convex shape is preferred to facilitate formation of hollow of sail. By conception on retractable booms, the chosen of shape of foot edge is more based on technical constraints than aerodynamics consideration.

### Contribution of the lift at the progress of the vessel

In case without lift, apparent wind direction is identical to the wind. If ship direction is identical to wind, all sailing effort contributes to the advancement of ship. Without lift of sail, ship can go faster than the wind, and propulsive force decreases gradually as ship approaches speed wind and effort down to zero.

In case with lift, sail has an impact with apparent wind. Apparent wind also forms an angle with the wind. Similarly, wind creates an angle to the direction taken by ship. Effort of sail does not contribute fully to the advancement of ship. With a ship in close hauled, hypothesis are ^{[25]}:

- Apparent wind angle / course of the ship 40° degrees
- Sailing with 20° incidence.

Lift does not participate fully in progress of vessel, it forms an angle of 40° is the propulsive force is more than 76% of its value. The remaining 36% ^{[26]} is perpendicular to the vessel, and this effort generates the drift boat.

If the same sail with the same apparent wind speed, lift coefficient is 1.5 close hauled and 1 downwind. Part of effort involved for advancement of vessel remains above 15% cases without lift. Another advantage, more the boat accelerates more apparent wind increases, effort of sail increases. At each speed increase apparent wind direction moves, it must tunning the sail to be the optimum effect (maximum lift). More ship accelerates, more the angle "apparent wind / direction of ship" is closer, so sail thrust is less oriented towards course of ship, forcing a shift to be back in maximum thrust sailing conditions. The ship can go faster than the wind. The angle "ship and wind" can be quite small, consequently the ship may be close hauled to reaching. The ship goes back to the wind.

Another consequence, for same boat apparent wind speed close hauled is much higher than downwind. Consequently the gain of close hauled well over 15% of sailing downwind.

### Evolution of lift coefficient following incidence: polar sail

The lift coefficient of the sail varies with angle of incidence. Coefficient is often divided into two components:

- The component perpendicular to the apparent wind is called lift;
- The component parallel to the apparent wind is called drag.

For each incidence angle matches a single pair of lift-drag. Sailmakers provides an evolution of the drag and lift in a graph called a polar sailing.

Behavior of the sail due to incidence ^{[27]} (angle: apparent wind / sail) is:

- The sail is free, equivalent to have no sail, it does lift and drag null;
^{[28]} - Sailing is perpendicular to the wind, the movement is turbulent
^{[29]}. This is the case no lift and maximum drag; - It is the intermediate cases:
- sail free to maximum lift: the flow is attached, i.e. the wind glue airfoil. There are no eddies (dead zones) created on the sail. It is noted in case of a good sail (well regulated), maximum lift is greater than maximum drag;
- maximum lift to maximum dead zone: the wind does not stick properly to profile of the sail. Flow is less stable it becomes gradually lifted or taken off. This creates an area on leeward side, a dead zone which depress perform of sail. At typical angle, dead zone has invaded the whole face on leeward side.
- the dead zone to maximum drag: Dead zone has invaded whole face on leeward side, only windward side have an effect. Air in these high incidence, is somewhat deviated from its trajectory, Air parcel are just crashing on all surface of windward side. Effort is almost constant, so the polar sailing describes an arc of a circle.

As the lift is more effective than drag to contribute to the advancement of ship, sail makers trying to increase the zone of lift, i.e. increase effort of lift and angle of incidence. Any knowledge of a sailmaker is to decrease size of dead zone at height incidence, i.e. in the control of the boundary layer ^{[30]}.

### Sail twist

The air moves primarily slices parallel to the ground, in our case sea. If air density can be regarded as constant for our calculations of effort, this is not the case of wind speed distribution, it will be different according to altitude. As at the sea surface, the difference of speed between air parcel and plots of water is zero, the speed of wind varies strongly in the first ten meters ^{[31]}^{[32]}^{[33]}. This rapid increase of the wind speed with altitude, consequently, will also vary apparent wind. It follows that intensity and speed of wind vary widely when altitude is between 0 and 20 meters ^{[34]}. In the case of using sails with lift, sail must be twisted to have a good incidence with apparent wind along the leading edge (luff) ^{[35]}.

KW Ruggles gives a generally accepted formula for the evolution of the wind speed with altitude:

<math>U = \frac {\mu'} {k} \ ln ( \frac {z + z0} {z0}) </math>^{[36]}^{[37]}^{[38]}

With data collected by Rod Carr ^{[39]} the parameters are:

- k = 0.42,
- z altitude in meters;
- z0 is an altitude that reflects the state of the sea, ie the wave height and speed:
- 0.01 for 0-1 Beaufort;
- 0.5 2-3 Beaufort
- 5.0 to 4 Beaufort;
- 20 5-6 Beaufort;

- <math>\mu' </math>= 0335 related to viscosity of air;
- U m / s.

In practice, the twist must be adjusted to optimize the performance of the sail. The primary means of control is the boom for a sail Bermuda. More boom will be pulled down, less twist will be important ^{[40]}.

## Several sails: multidimensional problem resolution

A boat is rarely rigged with one sail. Previous method for estimating the thrust of each sail is no longer valid but it remains a good approximation.

Sails are often close to each other. They influence each other ^{[41]}. In the case of a sloop-rigged sailboat, sail of bow (Genoa) changes air flow entering on mainsail. The genoa could blanket mainsail, as mainsail can prevent flow of air from Genoa to "get out".

Condition of a stable fluid constant and uniform, necessary for tables which gives lift coefficient is no longer respected.

Cumulative effect of several sails on a boat can be positive or negative. It is well known that the same total surface sail, two sails are properly set more effective than a single set correctly. Two sails can increase the thrust sailing 20% ^{[42]}. Only a two-dimensional model explains the phenomenon.

## Annexe

### See also

- Sail
- Sailing
- Sailcloth
- Points of sail
- Sail-plan
- Rigging
- Wing
- Sail twist
- Spar
- Stays (nautical)
- Sheet (sailing)

### Notes

- ↑ sail are not rigid
- ↑ Software include partially this phenomena or parameter
- ↑ http://www.mh-aerotools.de/airfoils/javafoil.htm
- ↑ http://www.mecaflux.com/voile.htm
- ↑ For example, see
*Xfoil*and*AVL*programmed by Mark Drela - ↑ http://www.adeps.be/pdf/Theorie2005.pdf
- ↑ see Lift (force)
- ↑
This section requires expansion. due to Bernoulli's principle, in steady state, along a pathline, and if Heat transfer are neglected, airflow formula is :

- <math> \frac{V^2}{2.g} + z +\frac{p}{\rho.g} = \mathrm{constante}</math>

As the elevation changes are small and are negligible compared to other terms, then :

- <math> \frac{V^2}{2.g} +\frac{p}{\rho.g} = \mathrm{constante}</math>

The fluid is considered as incomprehensible or have little density change (Mach = 0.4, error remains below 2%). At constant speed, considered that the sail which moves in the air at speed <math>V0 </math> or is it that the air reaches the speed <math> V0 </math> on the sail is exactly equivalent. Suppose that the air is fixed and is sailing on the move. Applying the formula to the parcel of air over the sail and then the same parcel of air before his arrival on the sail

- <math>\frac{p0}{\rho.g} = \frac{V^2}{2.g} + \frac{p}{\rho.g}</math> so <math>p0 = \frac{\rho . V^2}{2} + p</math>

Either pressure on the sail <math> p </math> is the difference in static pressure <math> p0 </math> is the infinite and the dynamic pressure <math> q </math>. The static pressure <math> p0 </math> is constant regardless of pathline. So overall it vanishes when integrating the formula dF over the entire surface of sail, as pressure <math> p0</math> from one side of the sail is exactly balanced by the pressure <math> p0 </math> on the other side of sail, and is therefore eliminated. It is the dynamic pressure is <math> \ p = q </math>

The dynamic pressure is equal to <math> q = \frac12 \times \rho V ^ 2 </math>. The dynamic pressure is the volume density of the kinetic energy of the air parcel <math> q = \frac12 \times \rho V ^ 2 = dE </math>.

where <math> \vec dF \cdot \vec n = dE \, dS </math>.

in this formula is dE is unknown but bounded. Indeed V is between 0 and V0 as if the speed exceeds V0, so that means the surplus energy which come from another energy than the Bernoulli principle have been neglected, ie sailing generate aerodynamic phenomena not negligible ever seen in reality (shock wave ...).

Its maximum is Max Q <math> = \frac{\rho . V0^2}{2}</math>

<math> dE = c \times MAXQ </math>

with <math> \, c </math> a percentage of the kinetic energy density ranging from 0-100%. The percentage <math> \, c </math> is unknown, it must be determined by other means (additional equation or testing).

hence integrating over the whole surface : <math> F = C \times E </math>

with

- E = effort that can give a maximum wind <math> = Max Q \times S =\frac12 \times \rho V0^2 \times S</math>;

- C = coefficient aerodynamic end integration;

It is noted for practical reasons for comparing the surface profile S is not the total surface of the object (or sail) but a typical surface. The surface of the chord is often used as a rudder or wing. The surface of the rope is very close to the surface of the underside of the profile. It follows that the coefficient C depends on two factors:

- The percentage of transmission dynamic pressure (or energy)
- Form factor. In a thin profile such as sailing, rudder surface S is taken similar to the half surface of the object. By abuse of language, when a person indicates that a sail is 10m ², it means made in the surface of the upper surface of the sail is 10m ². The real surface of the sail (upper + lower surfaces) is 20m ², but it is the value of 10m ² which is used in the formula <math> F = C \times E </math>.

Although this calculation is an aid to understanding, the exact calculation of C is complex and the fundamental principle of dynamics. The calculation is discussed in the section: The case of multiple sails.

- ↑ Telltales of sails are stable
- ↑ http://www.deltavoiles.com/technique/notions_aero.htm
- ↑ http://www.francelaser.org/lettre/mf-jvp/jvp1.htm
- ↑ http://www.lmm.jussieu.fr/~lagree/TEXTES/RAPPORTS/rapportsX/voilesNorvezPernot.pdf
- ↑ indeed if airfoil is symmetrical and sail shape not symmetrical
- ↑ http://www.ae.metu.edu.tr/tuncer/ae443/docs/NACA-All-Re.pdf
- ↑ Efforts are transmitted to half mast, the other half point tack. Is an effort on sheet to half the effort of the sail.
- ↑ In practice, the sail-maker double this efforts for design of sail. This allows to include gust of wind.
- ↑ not the case for hydroptere, wind surf...
- ↑ book partially scanned
*Bien naviguer et mieux connaître son voilier*by Gilles Barbanson,Jean Besson sheet 72-73 - ↑ distance is thickness of sailcloth adding Spar (boom or horn)
- ↑ http://www.dedale-planeur.org/horten/Horten%20critique%20par%20Deszo.pdf
- ↑ http://air-et-terre.info/aerodyn_theorique/ligne_portante_3D.pdf
- ↑ http://j.haertig.free.fr/aerodyn_theorique/ligne_portante_3D.pdf
- ↑ http://chazard.org/emmanuel/cours-de-catamaran-reglage-de-la-grand-voile-gv
- ↑ http://www.finn-france.fr/TECHNIQUE%20VOILE/michaud1.pdf
- ↑ drag and drift are not taken into account
- ↑ It is enough to push the sail on the major axis (Fprin) of the vessel and its perpendicular (Fper) F is the thrust of all sail Fprin = F * cos 40° = 76% * F = F * sin Fper 40° = 36% * F
- ↑ http://membres.multimania.fr/tpevoile/faero.htm
- ↑ If the sail is loose, the sail faseille, thus providing some resistance. The sailing ship is slightly back, in this case there is a slight drag. It is also noted that under its conditions the mast, the rigging, superstructure and topsides will provide much more effort aerodynamic that sail itself.
- ↑ telltale are in-stable
- ↑ http://www.onera.fr/mecao/aerodynamique/phototheque/video/naca12.htm
- ↑ http://hal.archives-ouvertes.fr/docs/00/16/72/71/PDF/B104.pdf
- ↑ http://www.ignazioviola.com/ignazio_maria_viola/Publications_files/Viola_EACWE2005.pdf page
- ↑ http://heikki.org/publications/ModernYachtLePelleyHansen.PDF
- ↑ http://techniques.avancees.free.fr/tipe/techniquesAvanceesGeneral.pdf
- ↑ http://denismerlin.blogspot.com/2009/12/comment-wikipedia-manipule-lhistoire.html
- ↑ http://syr.stanford.edu/JWEIA557.pdf sheet 2
- ↑ http://airsea.ucsd.edu/papers/MELVILLE%20WK%20-%20JOURNAL%20OF%20PHYSICAL%20OCEANOGRAPHY%207%20-%201977.pdf formula is given in introduction
- ↑ http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD734670&Location=U2&doc=GetTRDoc.pdf
- ↑ http://www.onemetre.net/Design/Gradient/Gradient.htm
- ↑ http://media.wiley.com/product_data/excerpt/0X/04705165/047051650X.pdf
- ↑ http://www.arvelgentry.com/techs/The%20Aerodynamics%20of%20Sail%20Interaction.pdf
- ↑ http://jestec.taylors.edu.my/Issue%201%20Vol%201%20June%2006/p89-98.pdf draw sheet 94 for example

### External links

- PDF Rapport NACA n° 824
- PDF Rapport NACA n° 1218
- PDF Rapport NACA n° 1217
- (English) listing of interesting article on Sailboat-technology.com
- (English) website of Knol.google.com on tubulancy and boundary layer

### Bibliography

- Royce, Patrick M. (1993).
*Royce's Sailing Illustrated: The Sailors Bible Since '56*. Prostar. ISBN 978-0911284089.

- Mulville, Frank (1991).
*Single-handed Sailing*. Seafarer Books. ISBN 978-0850364101.

- Marchaj, C.A. (1985).
*Sailing Theory and Practice, Revised edition*. Putnam. ISBN 978-0396084280.

- Bethwaite, Frank (first edition in 1993; next in 1996, last in 2007).
*High Performance Sailing*. Waterline (1993), Thomas Reed Publications (1996, 1998, et 2001), and Adlard Coles Nautical (2003 and 2007). ISBN 978 0 7136 6704 2.

- Curry, Manfred (1930).
*L'aérodynamique de la voile et l'art de gagner les régates*. Etienne Chiron, Ed. nouv. with new document (1 juillet 1991). ISBN 978-2702700273.

- (French) Bertrand, Chéret (June 2010).
*Les Voiles. Comprendre, régler, optimiser*. Gallimard. ISBN 978-2742407675.

- (French) Leonhard Euler
*Théorie complète de la construction et de la manoeuvre des vaisseaux*imprimé chez Claude-antoine Jombert at Paris in 1773, scan available at [1]

- (Latin) Leonhard Euler
*Scientia navalis*full title is*Scientia navalis seu tractatus de construendis ac dirigendis navibus Pars prior complectens theoriam universam de situ ac motu corporum aquae innatantium. Auctore Leonhardo Euler prof. honorario academiae imper. scient. et directore acad. reg. scient. Borussicae. Instar supplementi ad tom. I. novorum commentar. acad. scient. imper. Petropoli typis academiae scientiarum MDCCXLIX.*scan available atà [2]

- Articles to be expanded from July 2010
- Articles with invalid date parameter in template
- All articles to be expanded
- Articles to be expanded from June 2010
- Articles needing translation from French Wikipedia
- Pages with broken file links
- PDFlink without a parameter
- Engineering disciplines
- Naval architecture
- Sailing vessels and rigging
- Marine propulsion
- Sailboat anatomy