Forces on sails 1

Forces on sails

Example of wind force on different sail types on different points of sail. Regattas inCannes, 2006.

Understanding the forces on sails isimportant for the design and operationof the sails and whatever they aremoving, sailboats, ice boats,sailboards, land sailing vehicles orwindmill sail rotors. When air movespast a sail, aerodynamic forcesdevelop. These forces occur along theentire surface of the sails, but can besummed into one net force vector.

Net aerodynamic force may bedecomposed with respect to a boat'scourse over water into componentsacting in six degrees of freedom. Twocomponents with respect to winddirection can also be resolved: drag,which is the component directed down wind, and lift, which is the component normal to the wind and perpendicularto drag. This analysis is important to boat design, operation, balance, stability, seakindliness and seaworthiness.[1]

Briefly, when the sailboat is sailing directly downwind (i.e. the direction the wind blows toward), the aerodynamicforce is almost entirely derived from drag - the wind "pushes" the boat along in the direction of the wind.When the boat is traveling across or into the wind the sails act as airfoils. A lift component is created by redirectingthe wind coming in from the side towards the rear. The wind moves the sail as the sail redirects the air backwards inaccordance with the law of conservation of momentum.[2][3][4]

OverviewThe analysis of the forces on sails takes into account the theoretical location of the propulsive force or centre ofeffort, the direction of the force, and the intensity and distribution of the pressure related surface force and/or lift.The fluid mechanics and aerodynamics airflow calculations for a boat are more complex than for a rigid wingedaircraft. Structural analysis also is involved in modern optimal sail design and manufacture. Aeroelasticity models,combining computational fluid dynamics and structural analysis, are at the frontiers of sail study and design.[5]

However, turbulence and detachment of the boundary layer are not yet fully understood.[6] Computational limitationspersist.[7][8] The theoretical results are corrected by reality. So, wind tunnel scale model and full scale testing of sailsare required for optimum sail design, function and trim.Some complexities of sails:•• The wind is not constant.•• The boat is not traveling in uniform velocity.•• There may be a mast in front of the sail, disturbing the airflow, although this may be mitigated by profiling it.•• A mast is not infinitely stiff.•• The boat profile and position influence the airflow.•• A sail is usually made of thin and deformable fabric.•• The air is viscous, causing losses by friction.• The flow of the air varies from slow to fast and turbulent to laminar.

Forces on sails 2

Though some software algorithms attempt to model these complexities,[9][10][11] the following assumptions make theanalysis much simpler:•• the water more or less flat•• the wind more or less constant•• the sail is set and is not adjusted

Centre of effortThe point of origin of net aerodynamic force on sails is the centre of effort (or also centre of pressure). In a firstapproximate approach, the location of the centre of effort is the geometric centre of the sail. Filled with wind, the sailhas a roughly spherical polygon shape and if the shape is stable, then the location of centre of effort is stable. Theposition of centre of effort will vary with sail plan, sail trim or airfoil profile, boat trim and point of sail.[12]

Direction of force on sailsThe net aerodynamic force on the sail is located quasi at the maximum draught intersecting the camber of the sailand passing through a plane intersecting the centre of effort, normal to the mast, quasi perpendicular to the chord ofthe sail (a straight line between the leading edge (luff) and the trailing edge (leech)).Net aerodynamic force may be decomposed into the three translation directions with respect to a boat's course in aseaway: surge (forward/astern); sway (starboard/port, relevant to leeway); heave (up/down). The force terms oftorque in the three rotation directions, roll (rotation about surge axis, relevant to heeling). pitch (rotation about swayaxis), yaw (rotation about heave axis, relevant to broaching) may be also derived. The scalar values and direction ofthese components may be very dynamic and dependent on many variables on a boat and in a seaway including thepoint of sail.[1]

The net force vector, , is resolved into components in relation to course in a seaway with:• = the driving force directed along the course sailedand• = the heeling force perpendicular to the course and the mast.The heeling force can be resolved as a function of heel angle, , to:

• , the lateral or leeway forceand

• , the vertical or heave force.Net aerodynamic sail force can also be resolved into two components with respect to wind direction: drag, which isthe component directed down wind, and lift, which is the component normal to the freestream wind andperpendicular to drag. The generation of lift and drag, components of , and their contribution to boat motion arediscussed below.

Pressure on the sailFor the purposes of modern sail making and study, pressure distribution measurements are done in wind tunnel andfull scale experiments as well as in computer models.[13][14][15][16][17][18]

According to kinetic theory, at the microscopic level, air pressure is the result of collisions between perpetuallymoving air particles. Their energy, measured by temperature, determines their velocity. In still air, the averageparticle of air randomly moves around an imaginary fixed point in space, colliding with other particles without toomuch average movement away from this point. Wind is the particles moving in large numbers in the same direction.So, air pressure on a sail has two origins: temperature and the mechanical influence of wind.

Forces on sails 3

Pitot tubes[19] and other types of manometers are used in wind tunnel and full scale testing to measure the differencesbetween local static pressures at various points on the sail and atmospheric pressure (static pressure in undisturbedflow). Results are graphed as pressure coefficients (static pressure difference over wind induced dynamic pressure)to obtain windward "pressure" to leeward "suction" distribution curves along the mast/sail's chord.[20][21][22][23][24]

Role of Atmospheric pressureThere are fewer air particles at high altitude. Collisions between slower, colder, particles are less violent and lessfrequent. Thus, there is less pressure. At sea level there are more particles with more energy, resulting in morefrequent and violent collisions, or higher pressure.Close to the sail, collisions occur between sail and air particle. These collisions generate a force on the sail at sealevel of about 10 tonnes-force per square meter of sail (101325 Pa). If the pressures on each side of a sail areperfectly balanced, the sail does not move.

Role of wind

Pressure effect on theparticle level: The particleenters with momentum (1)

and exits with themomentum (2) transmitting

to the sail the amount ofmomentum (3) (Note this isnot an explanation of whythe pressures vary along

the sail.)

A part of the movement of air particles is globally ordered as particles move together inthe same direction, as wind.

Depending on the configuration of the sail, a particle of air near the sail can be in variousstates:•• If the sail is parallel to the wind, the wind has no resistance except to the thickness

and coarseness of the sail's fabric. Air particles pass without being significantlydisturbed (although in reality the sail will flutter).

•• If the sail is perpendicular to the wind (e.g. sail or spinnaker downwind), a particle ofair crashes against the sail. It is almost stopped. Other particles behind stronglyprevent reversing or rebound. The air particle transfers maximum kinetic energy tothe sail. All energy of movement is quasi ordered.

• In the intermediate cases, the air particles are deflected by the sail which results in anet change in the momentum of the air and consequently a force on the sail. Thus, animbalance in air pressure occurs between the two sides of the sail, as explained by lifttheory.

The particles disrupt orderly movement of particles near the collision which in turncascades to other nearby particles. This rebounding will result in a disruption of balanceof air pressure, creating an overpressure on the windward face of the sail and depressionon the leeward side. However, for a small area of the sail,S1, pushed on by the wind, adomino effect of collisions would cancel the disturbance of a surface depression locatedopposite its S2 colleague downwind. Similarly, the same small area downwind S2 wouldproduce a vacuum through a domino effect, negating the overpressure of the initialsurface S1. D'Alembert's paradox implies that if overpressures fill the depressions, generally nothing would happen.However, after thousands of collisions transmitting the original collision, the kinetic energy of the original collisionhas virtually disappeared due to the friction losses of viscosity. At the human level the energy disappears quickly(see boundary layer). This gives the impression that the pressures of the wind and the depressions of the leeward sideare independent, and are not affected by domino effects.

Contrast the collision of the air parcel on a small particle of solid sail fabric, which transfers energy with virtually noloss to sailing air. The sail, being made of a solid material, is not subject to large dissipative processes like a fluid.So, there are two relevant wind phenomena:•• conditions which push the sail (direct wind pressure) and

Forces on sails 4

•• conditions which pull the sail (depression due to wind).

Direction of forceThe push of air particles on the sail set it back. Glancing collisions only push very little. The resulting force is almostperpendicular to the surface of the sail.

Value of forceForces of air on each side of the sail are due to:•• On the windward side, atmospheric pressure, wind pressure, and virtually no depression due to wind.•• On the leeward side, 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 for the entire surface of theprofile in a simple formula valid for airplane wings, rudders, sails or keels (see lift):

with• E = force obtained with maximum wind (see Max Q);• C = Aerodynamic coefficientExplanationBriefly, each parcel of air crashing onto a small surface element of the sail, dS, generates a force dF. The forceexerted on that part of the sail equals the pressure of the air, p, on the sail times the small surface area in a directionopposite to the normal unit vector, n: .[25][26]

The airflow formula is derived from Bernoulli's principle. In steady state, along a pathline, and if heat transfer isneglected, with V, the speed of the wind relative to the sail; (rho), the air density; g, the acceleration of gravity; zthe elevation of a point on the pathline; and p, the pressure at the point:

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

The fluid is considered as incompressible or has little density change. (At Mach = 0.4, the error remains below 2%).At constant speed, consider that the sail which moves in the air at speed or that the air reaches the speed onthe sail are exactly equivalent. Suppose that the air is fixed and the sail moves. Applying the formula to the parcel ofair over the sail and then the same parcel of air before its arrival on the sail:

so

The pressure on the sail, , is the difference between total pressure(stagnation pressure), , and the dynamicpressure, . The total pressure is constant on the pathline. So overall it vanishes when integrating the formuladF over the entire surface of sail, as pressure from one side of the sail is exactly balanced by the total pressure

on the other side of sail, and is therefore eliminated. This leaves the dynamic pressure remaining:

The dynamic pressure is equal to . The dynamic pressure is the volume density of the kinetic

energy of the air parcel .

where .

Forces on sails 5

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

Its maximum is Max Q

With a percentage of the kinetic energy density ranging from 0-100%. The percentage is unknown, it must bedetermined by other means (additional equation or testing).Hence integrating over the whole surface : with

• E = force obtained with maximum wind ;

• C = aerodynamic coefficient. The portion of dynamic pressure (or the energy density) transmitted to the sail.Note that the total surface area, , is equal to the surface of the windward inner surface, , plus the leewardexterior surface, : .with•But for practical reasons of comparisons of airfoils, the surface, , used in the tables is not the total surface of theobject (wing, rudder or sail), but a characteristic surface. The virtual surface intersecting the chord, , is oftenused.The form factors, and relate inner and outer to chord surfaces: , .[27]

So = coefficient of lift found in tables.

As the tables are based on the characteristic surface , it follows that the coefficient in the tables depends ontwo factors:• a percentage factor of transmission of the dynamic pressure (or energy) and • form factors. and .In a slim sail airfoil profile, the surface intersecting the chord is close to the other surfaces, that is to say

.Usually, when it is said that sail is 10m ², it actually means that the surface of the upper surface of the airfoil wing is10m ². The actual surface of the sail including both sides is 20m ², but it is the value of 10m ² which must be used inthe formula for lift tables.Although this calculation is an aid to understanding, the exact calculation of C is complex and uses the fundamentalprinciples of dynamics. The calculation is discussed in the section: The case of multiple sails. In the followingsection, to lighten the notation, will be noted and will be noted .According to the Bernoulli equation, the maximum stress of wind or maximum density of kinetic energy for theentire surface of the sail is:

The full expression of the force is:

with

Forces on sails 6

• F = lift, expressed in Newtons• (rho) = air density ( varies with the temperature and the pressure) ;•• S = typical surface, for sail, it is the sail area in m²• C = aerodynamic coefficient, which is dimensionless. It is the sum of two percentages: the percentage of

recovered energy on the leeward side + the percentage of the recovered energy on the windward side[28]. For thisreason, the aerodynamic coefficient can be greater than 1, depending 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, taking an airfoil form. When the flow of air around the profile is laminar thetelltales of the sail (tufts of yarn or ribbon attached to it) are stable, and the wind induced depression factor becomescrucial. This effect is called lift. Based on studies and theories of sail design:[29]

•• 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.

Breakdown of force: introduction to the concepts of lift and drag

The general form of the force is calculated or measured in an air stream, with speed

as uniform as possible, arriving on the sail. The force is decomposed along three dimensions with respect to winddirection. The viscous air rubs on the airfoil, and creates resistance to movement. More importantly, this viscositydisrupts the air flow around the airfoil. This disturbance causes a considerable force perpendicular to the airfoil.Because the airfoil is not infinite in length, the ends also generate a force in the remaining dimension.

Airfoil diagram showing the relation of drag and lift to angle of attack.Lateral lift is almost never shown because the airfoil is treated as having infinite aspect

ratio and measured values are low.

The breakdown according to threedimensions is:[30][31][32]

.with:• : The axis parallel to the

direction of particles' movement not yet disrupted by the sail, that is,well before the particles arrive onthe sail. Force projected on this axis

is called drag. For convenience,the force has the same equation

.

The aerodynamic coefficient isreplaced by a coefficient, ,adapted to this axis. By nature thisforce is resistive, i.e. the profiletakes energy from the air. In theliterature is also noted ,with D for drag.

• : The axis perpendicular to the direction of movement of particles not yet disrupted by the sail, and perpendicularto the wingspan.[33] Force projected on this axis is lift. For convenience the force has the same

equation , where the aerodynamic coefficient is replaced by a coefficient, , adapted to this axis. The direction of

this force varies with the value of the incidence. In the literature is also with L for lift.

Forces on sails 7

• : The last axis. The force of the last axis is called lateral lift. The lateral lift's equation

. It is zero for an infinitely long airfoil. For a sail, the profile has two ends and

thus the lateral air pressures are balanced perfectly. The airfoil shape is usually straight, long and thin, (bentairfoils such as the gull wing are rare) which creates a low lateral lift compared to the first two axes. In our case ofa sailing boat, usually the side lift is negligible. The airfoil model is then reduced to a simpler two-dimensionalsystem. ( Note, 3D effects are taken into account as they may have some influence. For example, the induced dragis a purely 3D but the modeling is done done in 2D.) The case of a spinnaker is a perfect counter example. Thespinnaker has a low aspect ratio and a high camber draught and it is difficult to determine clearly the axis of lift.The spinnaker generates forces along the three axes, and the vertical force has a great importance for pitching.

Lift's effect on the sailTo study the effect of lift we can compare cases with and without lift.[34] As an approximation of a gaff sail, take asail that is rectangular and approximately vertical, with an area of 10 m² - 2.5 m of foot by 4 m of leech. Theapparent wind is 8.3 m/s (about 30 km/h). The boat is presumed to have uniform velocity, no heel and no pitch andthere are no waves. The density of air is set at: ρ = 1.2 kg/m³.

Sailing in stalled flowThe 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. The remaining pressures are:•• on the windward side atmospheric pressure and wind pressure•• on the leeward side only the atmospheric pressureForces of atmospheric pressure cancel out. There remains only pressure generated by the wind.Roughly speaking, collisions of particles on the sail forward all their energy from wind to 90% of the surface of thesail. This means that the Cz or aerodynamic lift coefficient is equal to 0.9.

Sailing in attached flowThe boat is close hauled, with the sail set at, for example, 15° relative to the apparent wind. The camber of the sailcreates a lift. In other words, the effect of depression on the leeward side comes into play. As air pressure forcescancel out, significant resulting forces are:•• on the windward side wind pressure•• on the leeward side wind depressionThe only unknown to be determined is the drag coefficient. In a well trimmed sail the curve profile is close tooptimal airfoil shape NACA 0012.[35][36] A less well trimmed sail, perhaps of older technology, will have greaterdraft with more camber. The coefficient of aerodynamic lift will be higher but the sail will be less efficient with alower lift/drag ratio (L/D). The sail profile may be similar to NACA 0015, NACA 0018.[37]

For a given profile, there are tables which give the lift coefficient (Cz), which depends on several variables:• Incidence angle of apparent wind to sail profile,• The slope of lift of the sail, which depends on its Aspect ratio,• The surface roughness and Reynolds number, which affect the flow of fluid (laminar, turbulent).The coefficient is determined for a stable and uniform fluid, and a profile of infinite extension.

The Reynolds number is:

Forces on sails 8

with• - fluid velocity or apparent wind [m/s]• - characteristic length - since this is a rectangular sail the length at any height will do, e.g. the foot of the sail

[m]• - fluid kinematic viscosity: [m/s]• - air density [kg/m³]• - air dynamic viscosity [Pa] or Poiseuille [pl]so for this sail about With an incidence angle of 15° and a Reynolds number of one million a NACA0012 profile reached a Cz of 1.5 (asopposed to 0.9 for 90° incidence).

The lift has increased by 50%. The force on the sheets and rig also increases by 50% for the same apparent wind.

Contribution of lift to the progress of the vessel

Detailed diagram outlining the boatvelocity vectors (V), wind (W) and

apparent wind (A) for a sailing boat.

When running downwind the direction of the apparent wind is equal to that ofthe true wind and most of the sail force contributes to the advancement of theship. There is no sail lift, so the boat can not go faster than the wind, andpropulsive force decreases gradually. When the ship approaches the speed ofthe true wind, the apparent wind speed and the force drop to zero.

In the cases with lift, the sail has an angle of incidence with the apparentwind. The apparent wind also forms an angle with the true wind. Similarly,wind creates an angle to the direction taken by the ship. Forces on the sail donot contribute fully to the advancement of ship. With a ship pointing closehauled, an example scenario is:•• 40° angle between apparent wind and ship's course, β.•• 20° incidence angle between apparent wind and sail chord, α.•• 10° leeway, λ.(See diagram under section Lift/Drag. Upwind sail cut and trim forillustration and definition of relevant angles on upwind sailing.)

The lift force vector, perpendicular to the apparent wind, does not participate fully in the progress of the vessel. Itforms an angle of 40° to the course sailed. The propulsive force vector is more than 76% of the total value. Theremaining 36%[38] is perpendicular to the vessel, and generates leeway angle and heeling moment.For the same sail with the same apparent wind speed, lift coefficient is 1.5 close hauled and 1 downwind. The vectorof force towards advancement of the vessel remains 15% above cases without lift.

Forces on sails 9

Velocity vectors boat, wind and apparent wind at different points of sail. For the same boat, apparent wind speed close hauled is muchhigher than downwind. Consequently close hauled speed is well over 15% of sailing downwind

The more the boat accelerates the more the apparent wind increases. So the force on the sail increases. At each speedincrease apparent wind direction moves. So, re-trimming the sail is needed for optimum effect (maximum lift). Themore the boat accelerates, the smaller the angle of the apparent wind to the direction of the ship. So sail force angleis less oriented towards the course of the boat, requiring bearing down a bit to gain maximum power sailingconditions. The ship can go faster than the true wind. The ship to wind angle can be quite small. Consequently thepoint of sail may approach the dead zone requiring the boat to back away from the wind.

Influence of apparent windWhen a ship is moving, its velocity creates a relative wind in the horizontal plane. (Ignoring for now the verticalelement to the velocity across the sail.) The sum of the true wind and the relative wind is called the apparent wind. Ifthe ship moves upwind, the two winds are cumulative, and the apparent wind is more important than the actual wind.Downwind, the effect is reversed, winds are subtracted, the apparent wind is lower than the true wind. Thecombination of these two winds can, in some cases, increase the performance of a sail boat.

Forces on sails 10

Apparent wind with a boat at fixed speed 7 knots red curve. Apparent wind with a boat speed fixed at half of apparent wind:wind in yellow; boat speed in blue.

The graph above depicts the relation of apparent wind speed of the boat for the following two cases. The true wind is14 knots.In the first case (red curve), the boat is at a constant 7 knots speed, say its maximum speed.In the second case, the ship has not yet reached its speed limit. For simplicity assume the performance oreffectiveness of the sail boat, the true wind direction and sails are constant. So the boat is able to go to half theapparent wind speed. Of course in reality this efficiency depends on the parameters mentioned above. Thisapproximation is realistic enough for pleasure boats. Racing yachts come to exceed the speed of true wind.Hydrofoils may reach twice the true wind speed. In yellow is shown the apparent wind speed and blue vessel speed.At a fixed speed of ship (red curve), the apparent wind increases gradually. The apparent wind exceeds the boatspeed around all the points and exceeds true wind from broad reach to close-hauled. Close-hauled the apparent windwill be doubled. In a headwind the apparent wind is 50% higher than the true wind.In the second case (yellow line) the apparent wind increases slightly and then increases rapidly upwind. In aheadwind, the apparent wind will be doubled. But even with a low efficiency of the sail boat, the gain of wind isstronger than in the first case. Sailing upwind would be as fast as the true wind. This explains why sail boats performoptimally upwind. Of course speeds on the graph are not reached because the boat can not exceed close-hauledwinding up in irons in the dead zone. This advantage can be reduced even further, with poor sail boat performanceand trim.Thus, two phenomena are cumulative:•• the apparent wind speed upwind is much higher than downwind;•• for the same apparent wind, the sail pointed upwind provides an additional 15% force compared to the tailwind

case.

Forces on sails 11

Influence of rigging tension on lift performanceTrimming a sail involves two parameters:• Angle of attack, or incidence. i.e. the angle of apparent wind to sail chord to create maximum lift, or a maximum

L/D. This angle varies with sail height, which is aerodynamic twist.• Airfoil profile which is composed of: 1. Camber of the sail as defined as the ratio of maximum draft depth to

chord length and 2. Draft position.Sail set and shape is generally flexible.[39] When the sail is operating in lift, if a sail is not properly inflated andstretched, there are wrinkles on the sail. These folds form a break in the profile. The air does not slip along the sail.The air streams come off the airfoil profile. Areas of recirculation or turbulent separation bubbles appear. Theseareas considerably diminish the performance of the sail. The assumption of a non wrinkled profile will simplify sailanalysis.A sail may be rigid where the canopy is composed of non stretchy fiber. Tightening a flat piece of such cloth inflatedby the wind results in folds at the attachment points. To avoid wrinkles, the sail could be tightened harder. Thetension can be considerable to eliminate all wrinkles. So, in the case of a taut rigid sail, the inflated shape is static,hollow and with its draft position immobile.The more elastic sail deforms slightly to its locations of high stress on the material, thereby eliminating wrinkles.The sail is no longer flat. Consequently, the sail can take several forms. By varying the tension of the sail, it is moreor less empty. It is possible to vary the shape of the sail without folds. The potential sail shapes are intrinsicallylinked to the cut of the sail. So in the elastic case, there is a family of possible forms and draft depths and positionsthe sail may take.Sailmakers try to build rigidity into sails for a predictable working shape with a degree of advantageous resiliencedepending on the sail's type, application and range: racing, cruising, high, moderate or variable wind, etc.The airfoil profile of the sail changes depending on the sail trim. At a given incidence, the sail can take differentforms. The shape depends on the rigging tensions such as on clew corner of the sail, the tack with Cunninghamadjustment, the backstay, the outhaul, the halyards or the boom vang (kicking strap). These elements help determinethe shape of the sail. More exactly, they can decide position of maximum draft along the camber of the sail.[40]

Each profile represents an appropriate value of Cz (lift). The position of the draft along the chord with the most lift isabout 40% of the foot from luff. The leeward side of a sail is close to the NACA series 0012 (NACA 0015, NACA0018, etc.) within the possibilities of trimming.The position of the draft is not independent of the camber setting. These parameters are linked by the shape of sail.Modifying the camber modifies the position of the draft.

Camber

The curves of propulsive component of lift and heel versus the angle of attack vary with the camber of the sail, thatis to say, the biggest draft depth relative to the chord of the sail. A sail with high camber has a higher aerodynamiccoefficient and, potentially, a greater propulsive force. Though the heeling coefficient varies with draft depth in thesame direction. So finding the optimal camber will be a compromise between achieving a large propulsive force andan acceptable list.[41] · [42]

Forces on sails 12

Propulsive and heel aerodynamic coefficients and sail camber depth.

Note that with a small camber (1 / 20), performance degrades significantly. The propulsion coefficient plateausaround a ceiling of 1.0.

Forces on sails 13

Draft position

The curves of propulsive lift and heel as a function of the angle of attack also depend on the position of the draft'sproximity to the luff.[43] · .[44]

Propulsive and Heel aerodynamic coefficients, point of sail for varied masted mainsail draftpositions. After Larsson and Eliasson wind tunnel data. Note a more forward position would likely

suit the jib.

Influence of Aspect ratio and Sail Planform on Induced DragSails are not infinitely long. They have ends. For the mainsail:•• Boom• Head.When the sail propels the ship, the leeward side is depressed and the windward side is under pressure. At the borderof the sail, depression is in contact with pressure. Naturally, compressed air molecules, with many and frequentcollisions, will rush into the area of depression, with lower impact and less frequent collisions. The consequence isthat the area which was depressed has relatively more air molecules than expected. So, the depression is lower (morepressure than expected). Similarly, the area under higher pressure from air molecules has fewer collisions thanbefore. So, the pressure is less. The propulsive effect is less.The distance between the downwind and upwind sides of the sail is very low,[45] a pressure zone closer to adepression zone. The transfer of molecules from one side of the sail to another is very violent. This createssignificant turbulence. On the end of a wing this is manifest as wingtip vortex. On a Bermuda sail, foot and leech aretwo areas where this phenomenon exists. The drag of leech is included in drag in the usual lift curves. The sail airfoilprofile is considered as infinite (i.e. no ends). But foot drag is calculated separately. This loss of efficiency of the sailat the foot is called Lift-induced drag.

Forces on sails 14

Influence on coefficients

Aspect ratio influences on driving and side forces at different points of sail. From windtunnel data of C A Marchaj.

Lift-induced drag is directly related tothe narrowness of the extremities dueto premature stall over the heavilyloaded short chord profile. The longeris the narrow head, the higher isinduced drag. Conversely, the sail canbe reefed, i.e. reduce surface of the sailwithout reducing the length of thehead. This means that value of thelift-induced drag will be substantiallythe same. For a given length of head,the more sail area, the lower is the ratiolift-induced drag on lift. The moreelongated the sail, the less lift-induceddrag alters value of the lift coefficient.

The curved shape of the mast and the battens tomaintain the curved profile of the leech are

clearly visible in this picture of a windsurfingsail.

Lift-induced drag on the sail also depends on aspect ratio, λ. Theequation is defined:[46]

with• b the length of luff• S the surface area of the sail.Lift-induced drag is:

with• Cz : Lift coefficient of airfoil• (pi) : 3.1416• λ : Aspect ratio (wing) (dimensionless)• e : Oswald efficiency number (less than 1) which depends on the

distribution of lift over the sail span. "e" could be equal to 1 for an"ideal" distribution of lift (elliptical). Elliptically shaped ends helpreduce induced drag. In practice "e" is the order of 0.75 to 0.85.Only a three-dimensional model and tests can determine the value of "e".

Optimal distribution for maximum reduction of lift-induced drag is elliptical in shape.[47][48] Accordingly, the luff will be elliptical. So, the mast is not straight as on a classic boat, rather designed with the closest possible form to an

Forces on sails 15

ellipse. An elliptically configured mast is possible with modern materials. This is very pronounced on surfboards. Onmodern sailboats the mast is curved thanks to shrouds and backstays. Similarly, the leech will be elliptical[49]. Thisprofile is not natural for a flexible sail. So, mainsails have battens to maintain this roach curve.An ideal lift-induced drag distribution creates an elliptical sail. But current sails are rather a half-ellipse, as if thesecond half part of the ellipse was completely immersed in the sea. This is logical because, as wind speed is nil at thesea level (0 m), the sea is equivalent to a mirror from an aerodynamic point of view.[50] So only half an ellipse in airis necessary.

Influence on efforts

Formula are :

Then :

This result is important(cf. Lift/Drag ratio and Power paragraph). This result show that the induced drag (force, notcoefficient) regardless of whatever AR ( ) is. In sailing, lift is limited by max rightning moment quite often,making it practical to use that formula to see induced drag doesn't depend on AR ( ), but ( ) instead doessince sail area does, if AR ( ) is changed while keeping span the same. This concept is used a lot by airplanedesigners also.

Influence of the height of the foot relative to sea levelThe gap between the edge of the sail and the sea surface has a significant influence on performance of a Bermudantype sail. In effect it creates an additional trailing edge vortex. The vortex would be nonexistent if the border were incontact with the sea. This vortex consumes extra energy and thus modifies the coefficients of lift and drag. The holeis not completely empty, as the sail is partially filled by the freeboard and superstructure of any sailboat.For a height between the edge of the sail and the deck of the sailboat of 6% of the length of the mast, changes are:•• a 20% increase in the drag coefficient• a 10% loss in the lift coefficient.[51]

The crab claw sail may partially circumvent this problem by harnessing the delta-wing's vortex lift.

Shape of luff, leech, and footA sail hauled up has a three-dimensional shape. This form is chosen by the sailmaker. The 3D shape is different forthe hauled up form compared to when empty of wind. This must be taken 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 a Bermuda sail andheavily distorted in the case of a spinnaker. The shape of edges empty is different from shape of edges once the sailis hauled up. Convex empty can go to straight edge when the sail is hauled up.Edges can be:•• convex•• concave

Forces on sails 16

•• straightWhen the convex shape is not natural (except for a free edge in a spinnaker), the sail is equipped with battens tomaintain this pronounced convex shape. Except for the spinnaker with a balloon shape, the variation of edge emptycompared to straight line remains low, a few centimeters.Once hauled up, an elliptical sail would be ideal. But as the sail is not rigid:•• You need a mast, which for reasons of technical feasibility, needs to be quite straight.•• Flexibility of the sail can bring other problems, which are better to fix at the expense of an ideal convex elliptic

shape.

Leech

On a Bermudan type sail the oval is the ideal (convex), but a concave shaped leech improves the twist at the top ofthe sail and prevents overpowering the top of the sail in the gusts, thereby improving the boat's stability. The concaveleech makes sailing more tolerant and more neutral. A convex shape is an easy way to increase the sail area (roach).Marchaj[52] discusses crescent shaped foils like a raked wing tip device as seen on various fish fins, Brazilianjungada sails, crab claw sails, and America's cup boat Stars and Stripes to reduce lift induced drag.

Luff

Once hauled up, the edge must be parallel to the forestay or mast. Masts and spars are very often, except inwindsurfing, jangada boats, and proas, straight. So, a straight luff is usually needed.But the draft of the sail is normally closer to luff than foot. So to facilitate the implementation of draft of the sailwhen hauled up, the empty form of luff is convex.[53] This convexity is called the luff curve. Sometimes rigging iscomplex and the mast is not straight.[54] In this case, the shape of luff empty can be convex at bottom and concave atthe top.

Foot

Foot form has little importance, particularly on sails with a loose foot or free edge. Its shape is more motivated byaesthetic reasons. Often it is convex empty to be straight once hauled up. When the border is attached to a spar orboom a convex shape is preferred to facilitate formation of draft of the sail. On retractable booms, the shape of theedge of the border is chosen based on technical constraints associated with the reel than consideration ofaerodynamics. A winglet[55] as used on airplanes to minimise lift induced drag is so far not practically seen on sails.

Forces on sails 17

Relationship of lift coefficient to angle of incidence: polar diagram

Polar curves showing the relationship between lift and drag for sails of aspect ratios 6, 3,1 and 1/3 over varied incidence angles.

The lift coefficient of the sail varieswith angle of incidence. Thecoefficient is often divided into twocomponents:

•• The component perpendicular to theapparent wind is called lift;

•• The component parallel to theapparent wind is called drag.

Each incidence angle corresponds witha single lift-drag pair. Sailmakersprovide a relation of drag and lift in apolar graph.Behaviour of the sail due toincidence[56] (angle: apparent wind /sail) is:

• The sail is free, equivalent to havingno sail, lift and drag from the sailare null;[57]

• Sailing is perpendicular to the wind,the movement is turbulent.[58] Thisis the case of no lift and maximumdrag;

•• These are the intermediate cases:•• sail free to maximum lift: the flow is attached, i.e. there is an airfoil. There are no eddies (dead zones) created

on the sail. It is noted in the case of a good well trimmed sail, 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. Air becomes gradually lifted or taken off. This creates an area on leeward side, a dead zone wheredepressions form on the sail. At typical angle, dead zone has invaded the leeward side.

•• the dead zone to maximum drag: Dead zone has invaded the whole face on the leeward side, only on thewindward side is there an effect. Air in these high angles, is somewhat deviated from its trajectory. Airparticles are just crashing on all surfaces of windward side. Force is almost constant, so the polar sailingdescribes 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 thezone of lift, i.e. increase force of lift and angle of incidence. The task of a knowledgeable sailmaker is to decrease thesize of the dead zone at high angles of incidence, i.e. in the control of the boundary layer.[59]

Forces on sails 18

Influence of altitude: aerodynamic twist and sail twist

The yacht The Bluenose in 1921. The photographshows the twist on sails properly trimmed,

particularly on the mainsail.

The rapid increase of the wind speed with altitude will increase bothapparent wind speed and its angle of incidence to course sailed,(β).[60][61] When using sails with lift, the sail must be twisted to have aconsistent angle of incidence of sail with apparent wind, (α), along theleading edge (luff). This results in the lower sail chords being atsmaller angles to the course sailed, (β - α), ( see decomposition offorces diagram below) than the upper chords to compensate for thesmaller β angle closer to the deck.

The air moves primarily in slices parallel to the ground or sea. Whileair density can be regarded as constant for our calculations of force,this is not the case for wind speed distribution. Wind speed willincrease with altitude. At the sea surface, the difference of speedbetween air particles and water is zero. The wind speed increasesstrongly in the first ten meters.[62][63][64][65] KW Ruggles gives agenerally accepted formula for the relation of the wind speed withaltitude:

[66][67][68]

With data collected by Rod Carr[69] the parameters are:•• k = 0.42,•• z altitude in meters;•• z0 is an altitude that reflects the state of the sea, i.e. 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;

• = 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 theboom for a Bermuda mainsail. The more the boom is pulled down, the less twist. For the foresail, depending on therig, twist is controlled by adjusting jib leech tension through sheet tension adjustments of: sheet angle with sheetblock track (fair lead) position, jib halyard tension, jib Cunningham tension, or forestay tension.[70][71]

Influence of the roughness of the sailAs on a hull or wing, roughness plays a role on the performance of the sail. Small humps and hollows may have astabilizing effect or facilitate stalls as when switching from laminar to turbulent flow. They also influence frictionlosses.This area is the subject of research in real and wind tunnel conditions. It is currently not not simulated numerically. Itappears that at high Reynolds number, well chosen roughness prolongs the laminar mode incidence a few degreesmore.[72][73]

Forces on sails 19

Influence of the Reynolds numberThe Reynolds number is a measure of the ratio of inertial forces to viscous forces in moving fluids. It also indicatesdegrees of laminar or turbulent flow. Laminar flow occurs at low Reynolds numbers, where viscous forces aredominant, and is characterised by smooth, constant fluid motion. Turbulent flow occurs at high Reynolds numbersand is dominated by inertial forces.The stronger the wind, the more the air particles tend to continue moving in a straight line, so are less likely to stickto the wing, making the transition to turbulent mode nearer. The higher the Reynolds number the better theperformance of the sail (within other optimal parameters.)[74]

The lift force formula is practical and easy to use. The aerodynamic lift coefficient,

C, depends on wind speed, V, and surface characteristics. The lift coefficient depends on Reynolds number as shownin the tables and polar diagrams. The Reynolds number is defined by . The Reynolds number depends

on wind speed, U, and length, L, travelled by the air (characteristic chord length) and kinematic viscosity, . Butthe influence of the Reynolds number is second order relative to other factors. The performance of the sail changesvery little for a variation of the Reynolds number. The influence of very low Reynolds number is included within thetables (or chart) by plotting the lift coefficient (or drag) for several values of the Reynolds number (usually threevalues).Increasing the incidence or the maximum lift coefficient by good choice of the Reynolds number is very interestingbut secondary. The Reynolds number depends only on three parameters: speed, viscosity and length:Viscosity is a physical constant, it is not an input variable for optimisation.Wind speed is a variable of optimisation. It is obvious that we look for the highest possible wind speed on the sail forsailing maximum force much more than for reasons of Reynolds number. This parameter has already beenoptimised.The sail is inherently inelastic and of fixed size. So, the characteristic length is fixed for a given sail. Lengthoptimisation is the responsibility of the naval architect, except for sail changes by the sailor. Performance tuning ofthe sails by varying the characteristic length of the Reynolds number is masked by the optimisation of otherparameters, such as looking for better sailing performance by adjusting the weight of the sails. The weight of the sailis an important point for the balance of the ship. Just a little more weight in the higher part of the sail may create amajor change affecting the balance of the ship. Or, for high winds, the sail fabric must resist tearing, so be heavy.The sailor is looking for a set of sails adapted to each range of wind speeds for reasons of weight more than forreasons of Reynolds number: jib, storm sail, main sail, spinnaker, light genoa, heavy genoa, etc. Each wind speed hasits sail. Higher winds tend to force small characteristic lengths. The choice of the shape of the sails and therefore thecharacteristic length is guided by other criteria more important than the Reynolds number. The price of a sail is veryhigh and therefore, limits the number of sails.The coefficients of lift and drag, including the influence of the Reynolds number, are calculated by solving theequations of physics governing the flow of air over a wing using computed simulation models. The results found arewell correlated with reality, less than 3% error.[75]

Forces on sails 20

Lift/Drag ratio and Power

Example of sailboat racing upwind, making the crew hike out to decrease the heel.

Polar curves of lift versus drag initiallyhave a high slope. This is very wellexplained by the theory of thinprofiles. The initial constant drag andlift slope becomes more horizontal, asmaximum lift is approached. Then athigher angles of incidence a dead zoneappears, reducing the effectiveness ofthe sail. The goal of the sailor is to setthe sail in the incidence angle wherethe pressure is maximum. Consideringthe proper tuning for a Bermuda-riggedboat, it is rare to set a sail withtheoretical optimum L/D. The apparentwind is not constant for two reasons:wind and sea. The wind itself is notconstant, or even simply variant. Thereare swings in the wind, there are gustsof wind and wind shifts. Evenassuming constant wind, the boat canbe raised with the swell or wave, the top of the sail finding faster winds, or in the troughs there is less wind. Up ordown a wave the boat pitches, that is to say, the top of the sail is propelled forward and back constantly changing theapparent wind speed, relative to the sail. The apparent wind changes all the time and very quickly. It is oftenimpossible to adapt to sea conditions with correspondeningly fast adjustments of the sails. Therefore, it is impossibleto be at the theoretical optimum. This is not necessarily a disadvantage as the "pumping" phenomenon of abruptchanges in incidence has been shown to increase lift beyond the steady flow situation.[76] Nevertheless, setting to themaximum optimum may prove quickly disastrous for a small change in wind. It is best to find an optimum settingmore tolerant to changing conditions of apparent wind, state of equipment and weather.

The important parameter influencing the type of sail trim is the shape of the hull. The hull shape is elongated toprovide a minimum of resistance to progress. We need to consider effects of wind on direction of the hull tilt:forward (pitch) or heel (roll). Downwind, sailing thrust is oriented in the direction of travel so will result in a forwardpitch. Maximizing sail area may be important as the heeling force is minimal. The situation changes if part of theforce is perpendicular to the vessel. For the same force as sailing downwind, the force perpendicular to the vesselmay result in a substantial heel. Under heavy list, the top of the sail does not take advantage of stronger winds ataltitude, where the wind can give maximum energy to sail and boat.[64][77][78][79][80]

The heel phenomenon is much more sensitive than sail induced pitching. Accordingly, to minimize the list, the typeof setting will be different close to the wind versus downwind: Close hauled, the setting is for L/D. When sailingdownwind, the setting is for power.

Forces on sails 21

Performance limitations of a sailA sail can recover energy of the wind. Once the particles have passed their energy to the sail, they must give way tonew particles that will in turn give energy to the sail. As the old particles transmitting energy to the sail evacuate,these particles have retained a certain energy in order to escape. The remaining energy of the particle is notnegligible. If the old particles evacuate too soon to make way for new particles, these particles carry with them a lotof energy. They then hand the sail less energy. So there is little energy per unit time, or power, transmitted to the sail.Conversely, if the old particles evacuate too slowly they certainly convey a lot of energy to sail but they prevent thenew transmission of power. So there is little power transmitted to the sail. There is a balance between incomingparticle speed and exit velocity, giving maximum power to the sail. This limit is called the Betz limit :

with

with : fluid density (1.23 kg / m³ in air at 20 °C)

S: surface wind "cut" by the sail m²: speed incident (upstream) of the fluid in m / s, ie the apparent wind speed.

So the sail can not recover more than 60% of the energy in the wind. The rest being used to evacuate the air parcelsoff the surface of the sail. Note that the surface of the Betz limit is not the surface of the sail but the surface wind"cut" through the sail[81].

The formula for the force on the sail is

whereis a characteristic surface in the case of sail on the surface of the chord.is the aerodynamic coefficient.

represents the percentage of energy recovered over the upper(outer) surface multiplied by the upper(outer)surface area plus the percentage of energy recovered from the lower surface multiplied by the surface area of thelower(inner) surface. By definition for a sail, the fabric is thin, so the upper surface area is identical to the lowersurface area. Considering the sail as inelastic, the sail airfoil is relatively thin. The camber of the sail can be veryimportant in lift mode lest the airflow comes off the airfoil and thus decreases the performance of the sail. Even for ahighly deformed spinnaker, the spinnaker must be set to catch maximum wind. The upper surface area or the lowersurface area are approximately equal to the surface area through the chord. The surface area of the sail isapproximated to the surface area through the chord . So the drag coefficient has upper limit 2.On the other hand, the apparent wind is related to the true wind from the formula:

with , the angle between true wind and the direction of movement of the boat in radian.The apparent wind depends on the true wind and boat speed. The true wind speed is independent of the boat. Theboat can take any apparent wind speed. So if the sailor increases the apparent wind with the true wind fixed, the boatspeed increases, with some practical limits.Research is intended to improve the speed of boats. But improvements are limited by the laws of physics. With allthe advanced technology available, the aerodynamic coefficient has a theoretical limit, which limits the recoverableforce at constant speed. Recovered energy from the wind intercepted by the canopy is limited to 60%. The only wayfor the sailor to go faster is to increase the energy recovered per unit time (or power ) by increasing the surface windintercepted by the canopy. Without going into calculations, the faster the boat moves, the more the surface area

Forces on sails 22

intercepted increases, the vessel has more energy per unit of time, it goes even faster. If the boat is faster, the areaintercepted is even greater. It gets even more energy. It goes even faster than before. The boat then enters a virtuouscycle. If the apparent wind increased indefinitely. with no heeling problem and hull resistance, the boat wouldaccelerate indefinitely. The other possibility is to increase the area of the sails. But, the sailor can not increase thesurface of the sails indefinitely. Increasing the surface area of sail, the responsibility of the naval architect, is limitedby the strength of materials.

Lift/drag. Upwind sail cut and trim

Decomposition of force on sailing upwind: Apparent wind (W) atincidence, (α) and angle to course sailed (β). Aerodynamic force

(A). Lift (C), perpendicular to flow. Drag (B), parallel to flow. C1is portion of lift propelling the boat and C2 the portion causingheeling and leeway (λ). Drag (B) will also contribute to heeling,

leeway and reduce propulsion.

In the example of upwind sailing, the apparent wind,with incidence, α, to the sail chord, is at an angle, β, tothe course sailed. This means that:•• a (small) part of the drag slows the boat.• the other part of the drag of the sail is involved in the

vessel's heel and leeway, λ.•• much of the lift of the sail contributes to the

advancement of the vessel,• the other part of the lift of the sail is involved in the

vessel's heel and leeway.[82]

Sailing high to the wind generates a perpendicularheeling force. Naval architects plan optimum heel togive maximum forward drive. Technical means used tocounter the list include ballast, hydrofoils and counterballasted keels. Heel can be almost completely offset bythe counter- heel technology such as boom / swing keel,type of hydrofoil, etc. These technologies are costly inmoney, weight, complexity and speed of change ofcontrol, so they are reserved for elite competition. Innormal cases, the heel remains as extra ballast begins todecrease forward drive. The architect must find acompromise between the amount of resources used toreduce the heel and the heel remaining reasonable. Thenaval architect often sets the optimal heel between 10°and 20° for monohulls.[83] As a result, the sailor muststick as much as possible to the best heel chosen by thearchitect. Less heel may mean that the boat is notallowing maximum sail performance. More heel meansthat the head of the sail drops, thus reducing pressure, in which case the sailing profile is not the best.

The sailor desires optimum heel, a heel giving optimum perpendicular force for the best resulting driving force andto minimise the ratio of perpendicular force to driving force.This ratio depends on the point of sail, incidence, the drag and lift for a given profile.As the lift is the main contributor to the force that drives the boat, and drag usually the main contributor toperpendicular heeling and leeway forces, it is desirable to maximise the L/D.The point of sail depends on the course chosen by the sailor. The point of sail is a fixed parameter, not a variable ofoptimisation. But each apparent wind angle relative to the axis of the ship has a different optimum settings.

Forces on sails 23

The sail trimmer will first select the trim profile giving maximum lift. Each profile corresponds to a different polardiagram. A sail is generally flexible, the sailor changes the trim through:[84]

•• the position of the draft of the sail by adjusting the elements acting on the tension of the fabric of the sail•• adjusting twist of the sail using the leech tension lines such as boom vang or jib sheet angle adjustment.Many polar curves exist for the possible sail twists and draft positions. The goal is choosing the optimal one.The twist will be set for constant incidence angle along the luff for maximum sail performance, remembering thatapparent wind strength and angle varies with altitude.The best L/D is usually obtained when the draft is as far forward as possible. The more forward the draft, the greaterthe angle of incidence over the luff area. There comes an attack angle when the air streams do not stick to the sail,creating a dead zone of turbulence which reduces the efficiency of the sail. This inefficient zone is located just afterthe luff on the windward side. The tell tales in this area become unstable.[85] The flatter the stretched fabric over thesail is, the less the draft. The yacht has several elements acting on the tension of the fabric of the sail:•• Cunningham tension,•• the tack,•• the head point,•• the clew of the sail.•• the backstay,•• shrouds. They act indirectly.These elements can interact. For example, backstay tension also affects the tension of the head point and thereforethe shape of the luff. Both high clew sheet tension tightening the foot and a tighter backstay cause slackening of theleech.For a flexible sail, the camber of the sail and position of the draft are linked. This is a result of their dependence onshape of the cut of the sail. The camber is a major factor for maximising lift. It is the naval architect or sail makerthat sets the cut of the sail for the draft-camber relationship. The thickness of the airfoil profile corresponds to thethickness of sail fabric. Variations in thickness of a sail are negligible compared to the dimensions of the sail. Sailthickness is not a variable to optimise. Contrast mast thickness and profile which are much more important[86].For the naval architect the sail-shape offering a large L/D is one with a large aspect ratio. (see previous polardiagram) This explains why modern boats use the Bermudan rig.Sail drag has three influences:•• induced drag (see influence of aspect ratio on the lift). As the profile is not of infinite length, the ends of the sail,

foot and head, equalise the depression of the leeward surface with the pressure of the upwind surface. Thisdissipated pressure balance becomes the induced drag.

•• friction drag, related to boundary layer laminar turbulent flow and roughness of fabric•• form drag, related to choice of the airfoil profile, camber, draft position, and mast profile•• (parasitic drag is related to parts extraneous to the sail, but may influence rigging, boat and sail design)Prandtl's lift theory applied to thin profile is less complex than the resolution of Navier-Stokes equations, but clearlyexplains the aspect ratio's effect on induced drag. It shows that the principal factor influencing L/D is induced drag.This theory is very close to reality for a low-impact thin profile.[87] Smaller secondary terms include the form dragand friction drag. This theory shows that the factor with main influence is aspect ratio.[88] The architect chooses thebest aspect ratio for best sailing, confirming the choice of Bermudan rig. The sailor's choice of sail trim affects thefactors of secondary importance.ExplanationThin profile theory[89] is applied to a 3D profile. A conventional infinitely extended 2D profile of the thin profiletheory is truncated. At each end section of the truncated profile there is a vortex that swirls over its whole periphery.The theory then gives the drag:[90]

Forces on sails 24

with• Cz : Lift coefficient of profile• : the circle circumference to diameter ratio, pi or 3.1416

• λ : Aspect ratio (dimensionless) with b the length of the luff and S the surface area of the sail.

•• e : Oswald efficiency numberThe theory only models the induced drag. The form and friction drag are neglected.[91] To set the order ofmagnitude:•• Cz ranges from about 0 to 1.5, a value of 1 is taken•• e is between 0 and 1 for a sail is at about 0.8•• λ aspect ratio.For an Edel 2 sloop, the mainsail is 10 m2 with a foot of 2.5 m is λ = 1.6 for an Edel 2 is 0.2.But the sail's reflection in the sea must be taken into account. If we neglect the distance between the sea and the foot,the sail's surface area and length doubles. So is 0.1.In reality, lies between these two values. This value varies depending on the state of the sea, that is dependingon the quality of the mirror.The calculation of the Oswald efficiency number is based on integral calculus. It is not addressed in the literaturebecause it is calculated indirectly. The formulas are not all written with the Oswald efficiency number. There is alsoanother notation :

Thin profile theory applied in 3D gives the formulas for the induced drag.[92] Note the case of a large cambered sailsuch as a genoa. The recalculation based on this camber is no longer negligible. Also note that can be calculated notusing a Fourier series, but via the calculus.[93]

The lift/drag ratio is directly derived from the formula of the induced drag:

The theory then gives for the lift:[94][95]

with

• λ: Aspect ratio (dimensionless) with b is the length of the luff, S the surface area of the sail.

•• Cz: lift coefficient calculated by the theory with an infinite extension profile.It should be noted that in the case of a sail wind speeds are far removed from Mach, it follows that the correctionfactor of the Mach number is approximated to 1.The theory then gives for the infinite extension lift profile:

with α incident angle between the chord of the sail and the apparent wind. α0 3D coefficient is actually slightlydifferent from the 2D calculation.hence

Forces on sails 25

The naval architect fixes e and λ, while the sailor sets α and α0. The sailor does not have a choice of total factor α0,i.e. She can not choose the full form of the profile. The set of possible profiles is limited. Indeed, the sail maker setsa cut of the sail therefore defines a set of profiles that can be made possible depending on sail trim. To illustrate, thesail may be cut to give a profile NACA0009 but if the sail is not taut it can become NACA0012 NACA0015NACA0018 or in between, though the general shape remains the same. The relationship between draft position andcamber, camber / chord, etc. are fixed. The choice of the general form is the responsibility of the sail maker or navalarchitect.A high L/D is in the range of possible profiles of the sail set to a maximum draft, forward on the sail, with the sailtaut.[96] Trimming to advance the draft position is desirable. But that does not mean it can be placed anywhere. Thechoice of the profile by the sail maker could for example result in maximum draft positioned at 30% or 50% ofchord. In the first case flexibility of draft position will be between 30% and 100%, while in the second case between50% and 100%.Form drag and friction drag, have a secondary but significant impact for competition sailing.[97] The surfaceperpendicular to the wind is a factor, the greater the depth of the draft (camber) the smaller the L/D due to increaseddrag. Similarly increasing twist increases drag.[98] A flatter sail is preferable to a balloon shape. This implies that thegeneral shape of a sail is set properly by a tense sail. But the sail should not be too tight as too much flatnessdecreases the lift.[99]

These calculations are approximations of reality. They are still relatively simple. They avoid the very heavy 3Dcalculations (see Forces on sails# Several sails : multi-dimensional problem resolution). They are handy for sizing arig or to model its behaviour in full sail.A higher L/D means less drag, for the same heeling force. The maximum L/D will be preferred. So among theremaining profiles that give maximum lift, the sailor selects the profile with maximum L/D (draft forward on thesail). Now that the profile of the sail is set, it remains to find the point of the polar diagram of the profile giving themaximum forward force to the vessel, that is to say the choice of the angle of incidence.On a triangular sail the zone of maximum lift coefficient (0.9 to 1.5) has two characteristic points (see Marchaj'spolar diagram above, and[14]):•• Point 1: the maximum L/D (0 to 5° incidence, the correct zone)•• Point 2: maximum lift (15° incidence on the polar diagram).As total aero and hydrodynamic drag slows the boat, it is necessary that the portion of the lift that moves the boat isgreater than the contribution of the total drag:

and hence with:• incident angle between the chord of the sail and the apparent wind,• angle between apparent wind and boat's course made good (including leeway).• Lift/Drag of sail at α• propulsive force• total of aerodynamic and hydrodynamic drag• liftThis means that we should not increase incidence beyond the point of the polar curve or decrease the tangent at thispoint less than the tangent of β. Hence between the maximum L/D noted point 1 (end of correct zone) and a L/D oftan(β) noted point 2. The evolution of the propelling force is as follows: at 0° incidence to point 1 both forward force

Forces on sails 26

and heel increase linearly. Point 1 to the optimum forward force still leaves the polar curve flattening, which meanthat the drag slowed the boat's progress more than lift adds. But overall as the heel increased, the sail has a lowerapparent wind. The top of the sail is no longer at the altitude of fast winds. From the optimum point to point 2,forward force decreases until it becomes zero, the ship stands up. Optimal adjustment of incidence is between point 1and point 2. The optimum point depends on two factors:•• changes in the L/D•• changes in the heel.The sailor will find a compromise between these two factors between points 1 and 2. The optimum operating point isclose to point 1 and close-hauled, where the heel is dominant factor. Since it is difficult to heel on a broad reach, theoptimum will be closer to point 2.Note that the L/D is determined through the polar of the sail. The polar is determined regardless of the apparent windspeed, yet the heel is involved in setting speed (wind in the sail), so the L/D of the polar of the sail does not dependon the heel.

Oops! Heel is too much for the smooth running of the yacht.

The position of the draft is the dominantfactor in the search for the optimum. All theknowledge of ocean racing is to advance thedraft forward. With a setting of "too much",the sail answers. The optimum trim isalways on the verge of dropping out. The jibluff lift and main leech lift are so veryimportant. At this optimum the main leechand jib luff tell tales are horizontal andparallel to the surface of thesail.[100][101][102]

The purpose of trimming the boat is to havemaximum propulsive force (Fp). A simpleway might be to set a giant sail, except theboat will capsize due to Fc, the capsizeforce. The ratio Fp/Fc is an importantconsideration.In summary, at points of sail where lift acts, the L/D is determined by the height of the sail, sail fabric and cut, butespecially good sail trim. Close-hauled, there can be variations in the L/D of 100% comparing one sailing crew toanother. In the race, boats are often close in performance (the role of racing rating rules). The dominant factor for thespeed of the boat is the crew. The L/D is not a secondary concept.[103][104][105]

A sail boat can drift, this leeway creates lift from the submerged form, the force used to counteract the force pushedperpendicular to the sail. So in other words, minimising the heel also amounts to minimising the leeway of the ship.Minimising leeway gives better upwind performance. The L/D of a yacht enhances its ability to go upwind.Similarly, the concept of balancing L/D, is in various forms:•• trimming the boat for optimal sailing upwind,•• Fp / Fc, the inverse of capsizing tendency,•• design capacity of the boat to go upwind,•• L/D of the sail, or slope of the polar.

Forces on sails 27

Power. Downwind sail cut and trim

Decomposition of the aerodynamic sail force toforward, and lateral components at different

points of sail.

Downwind sailing forces tend to pitch the boat forward. Heeling (rollingaxis) forces are less important (in theoretical steady state conditionsonly). The apparent wind is at an acutely aft angle to the axis of the ship.The chord of the sail is roughly square to the axis of the ship. So:

•• much of the sail's drag contributes to the advancement of the ship•• the other part of the sail drag is involved in the vessel's heel•• much of the lift is involved in slowing the vessel,• the other part of the lift is involved in the vessel's heel.[14]

The optimum setting depends on the apparent wind angle relative to thecourse. The sail profile is chosen for maximum drag. The heel is not abig factor reducing boat speed. The L/D is not a factor in applying theright profile. The overriding factor is to get the sail profile to give themaximum forward drive based on drag or "power". To maximize thepower or maximize propulsive effort are equivalent.

ExplanationPower in physics is defined for the sail:

Where:

( In N ) is the force vector on the sail.

(In m / s ), boat velocity vector over the ground with . With boat speed over the ground,is the instantaneous power (in W ).

At constant boat velocity, forward force is exactly balanced by hull drag:

;

Hence,

.Force projected on the course:

The hull acts as an underwater profile. The hull lift is perpendicular to the advancement of the ship. So, it does notaffect propulsion. The hull's driving resistance is due to its drag. For simplicity, this formula is chosen:

Forces on sails 28

.

With , the water density, C, the hull drag coefficient, , the hull wetted surface area and , the boatspeed over the ground.The following factors depend somewhat on the speed of the boat:

Hence,

Hence,

Therefore, optimising power amounts to optimising the propulsive effort.

with With:•• α Incidence, angle between the chord of the sail and the apparent wind,•• β angle between apparent wind and the course (including drift).Hence, the complete formula is:

To vary the power the sailor sets only three factors:•• L/D•• incidence•• lift.This means that in some cases we must take into account the lift in optimising the speed of the boat. The lift equationis with C the lift coefficient, S the surface area of the sail, and V is the

apparent wind speed. The speed is not identical along the luff. A weighted average, determined through testing, isused. The lift depends on several factors including primarily the apparent wind speed, averaged over the entire sail,which depends heavily on the vessel's list.The stability of a sail boat

Physical power can not increase indefinitely. The boat will capsize if the heel or the pitch is too large. The power islimited by the capacity of the ship to withstand the heel, i.e. the righting moment.The righting moment is the work needed to keep the boat upright.[106][107] This prevents the boat from capsizing.Hull shape, ballast and keel all contribute to the righting moment. This moment exactly balances the momentgenerated by the force of the wind in the sails when the ship is at constant speed. This constant speed simplificationwill avoid a too complex formula:

with•• G centre of gravity of the boat•• E centre of effort of sail•• righting moment of the boat, this time including the effects of ballast, keel of the hull and so on.

Forces on sails 29

These moments are decomposed along the axes of the ship: roll, pitch, yaw. Each axis has its limits (Fc low to highdownwind heel), namely:

Maximum moment for heeling:

Maximum moment for pitch: .Of course, it is not possible to capsize at the yaw axis. The pitch axis will not have the same righting moment as theheel axis. A hull is made to offer the least resistance to movement. So, the limit along the axis of pitch isconsiderably more important than along the heel:

To set the order of magnitude of a monohull, . Although this value is lower for amultihull.A sail boat is usually composed of several sails. But for the calculation, a single equivalent sail will approximatemultiple sails. This sail will have its own polar plot. The results for this sail are applicable to multiple sails as thesails influence each other. The correct trim is different from the equivalent multiple sail trim. This difference can bedetermined via computation or experience of the sailor on his yacht. (See case of several sails: multi-dimensionalresolution of the problem).The force in the sails changes direction and intensity depending on the point of sail. The force is divided into sail liftand drag. is the lever arm of the pitch and is the lever arm of the heel acting on the centre of effort ofthe sail in relation to the centre of gravity.Hence,

with course to apparent wind angle.[108]

These moments should be expressed in conjunction with the means against heel. Analysis of hull heel counteringmeasures is beyond the scope of this article. Nevertheless, the results of calculation show that and evolve respectively along the angle of heel and pitch. The value of moments rise linearly to pass through a maximumand then decrease. The metacentre approach is generally used:

with

• heel or pitch angle• distance to the centre of gravity metacenter• heel or pitch constant[109] · .[110]

The relevant moments are: for the sail, the heeling moment and for the hull, the righting moment.The literature often uses a simplified equation for calculating the heeling moment or sail.

.Lift and drag are of the form:

The wind speed is not constant with altitude. Speed depends on the heel. Different simplified formulas are used:

•which leads to:

with

Forces on sails 30

average pressure on the sail.

is called the heeling arm. (Moment divided by the displacement of the vessel. Not to beconfused with wind heeling arm.)

angle of heel or pitch.surface area of the sails.constant.

n a coefficient to be determined.[111]

• or [112] will give another formula,• or other formula.[113]

The higher the heel, the greater the boat approaches its safety limit. Heel is not desirable because it reduces theforces for the good performance of the boat. The two phenomena, speed and safety, act in the same direction. Takinginto account the effect of wind speed in the rest of the explanation does nothing. It would reinforce the results foundwithout bringing anything new. In the following explanation angles (pitch heel) will be low, therefore neglected.

From these curves are determined and . These are the maximum moments and not forpractical use because if exceeded, then the boat is in danger. In addition, the angles are too high and the sail does nottake advantage of fast winds at altitude. For these reasons, the linear region of the curve is used where there arelower angle limits for optimum heel and pitch.Upwind lift propels the boat and drag slows it. Downwind the reverse is true. The sail tunings are to optimizerespectively the driving force. At the transition between these two at a transverse wind, the setting mode thenswitches to search for maximum drag at maximum lift. The transition also corresponds to a flow on the sail ofturbulent (search for drag) to laminar (search for lift).To consider low heel (i.e. a GM approach), if the yacht is well designed, then it is neither tender nor stiff. So, thesailing centre is approximately above the centre of gravity on the same vertical:

More is still somewhat varying in first approximation:•• The sailing centre is near the geometric centre of the sails•• The most common case is the boat without ballast over a ton. So, the major part of the weight is fixed, the centre

of gravity moves slightly.

Downwind

Downwind, the leading edge is the leech and the trailing edge is the luff. The situation is reversed close hauled. Thedownwind sail profile works in reverse. As downwind drag moves the boat, and lift slows it, so you have amaximum drag, a sail that blocks the wind set to a high incidence, therefore, a mode of turbulent air on the sail.

Consider the simple case of downwind . Hence,

Or downwind sailing works exclusively in drag. The lift is zero then:

This simplification suggests that the heel is not a problem. But, other phenomena in non steady state conditions likeoscillations set up from spinnaker vortex shedding and rudder induced roll may cause "death roll"broaching.[114][115] Too much power could also cause pitchpoling. Downwind, the boat can not go faster than thewind. The more the boat approaches the true wind speed the lower the apparent wind pushing the sail. The goal istherefore to safely raise a maximum sail area to move the boat as close as possible to the true wind speed.

Forces on sails 31

As soon as the wind is not exactly aft, the apparent wind effect appears. As the apparent wind increases, so, also theforce on the sail. If the sails were hoisted for a calculated safe limit charging downwind, this limit is exceeded.Setting smaller sails moderates this apparent wind effect. The boat is always at a lower speed than the true wind.Speed limits of the hull are affected by high wind shifts (from breeze to storm). In such conditions, the sailor is morereasonable to temper speed in a search for maximum boat security. The loss of speed compared to the case ofmaximum sail area will be minimal because the downwind boat can not go faster than the wind.

From downwind to a broad reach

The lift slows the ship, so, minimize it. The caution profile is reversed. The polar plot of the sail giving maximumdrag and minimum lift is for an incidence of 90°.[116] So, keep the incidence perpendicular to the apparent wind. Inthis case the lift is zero. Then the constraints are:

Furthermore, the lift is no longer parallel to the course. The perpendicular part causes the heel.With the apparent wind, the boat accelerates. The boat's heel is compensated by the leeway. Leeway increases theresistance efforts of the hull. From downwind to a broad reach, the boat is more rapid, taking advantage of theincreasing apparent wind effect. Approaching a beam reach, the resistance of the hull takes over. The boat slowsdown a bit.[117]

Like , here .

There is a tipping point or constraint design changes from the stress of pitching to the stress of heel:So,

The angle is close to downwind. For a low ratio , it is still 165 degrees.

So, the constraint is:

Downwind, the transition zone

Downwind, if the boat keeps the same profile of drag at a broad reach, sailing well adjusted bearing capacity is zero.The propulsive effort follows the formula:

So the more the boat approaches the beam reach, the more propulsive drag decreases to zero. So, the choice comes torunning downwind or to switching to lift on the beam.Similarly in reverse, if the boat keeps the same profile as for beam reaching, as the incidence is not too close to zero,the more the sail keeps the profile, and propels the boat in lift mode, the more the point of sail moves from beam tobroad reach, the more the incidence decreases, and the propulsive effort is reduced. So, is it better to switch to drag,positioning the wing to cut the path of maximum wind like on a broad reach? This particular point differs betweensailboats and sets of sails available. For example, using a multihull with better anti heel properties than a monohull,the tipping point will be different.

Beam Reach

Near a beam reach, the lift is reversed. It contributes to the advancement of the ship, and drag slows the boat. Thepropulsive effort follows the formula:

Forces on sails 32

As the angle is not perfectly in the ideal case would be to find a point on the polar of the sail with no dragand maximum lift. Unfortunately, unlike running downwind with drag and maximum lift is zero, the theory of thinprofiles shows that drag exists. The choice of the correct incidence of the sail is going to depend on the L/D of thesail. For a sail working in lift .The constraints are:

As , the constraint is:

if sailing through work liftOn beam reach, we must use lift as much as possible. So the choice of the profile turns to a sail which can be themost hollow.[118] But as the sail is more hollow, flow becomes more turbulent. We need to find the limit, because inturbulent flow the lift breaks down.The profile selected, then the right incidence is selected. The good incidence is the point of the polar plot with thehigher lift (an incidence of 20°, which varies according to the sails).The heel does not pose a problem yet because the heel component contains only drag, which is quite low. The drag isnot as small as possible because we selected a profile with a maximum lift with a big draft, therefore generating a lotof drag for a working lift profile.As the optimum incidence is 20°, it is possible to adjust the sail lift mode corresponding to the lower points of sail

. The limit is . At this point of sail incidence is zero. The propulsive effort is zero. The sail isno longer inflated by the wind.It is not uncommon to see a sail set in drag. This situation, as evidenced by the formulas, give a heavy list. The sail isset incorrectly. The propulsive effort is provided by the drag , with closeto zero. The propulsive effort is low. Almost all of the drag makes the boat heel.Close reach

At points of sail closer to the wind, the sail works to lift. As the heel becomes more important, we must limit the heelby improving the L/D (see L/D this Wikipedia page). It should be sailing with a less and less draft.The constraints are:

As , here .

There is a tipping point where design constraint changes from the pitching factor to the heel factor:

Hence,

with

the Inverse function of the tangent .in radians.

However, , in practice then .

Forces on sails 33

The tipping point is near the beam reach, even with a low ratio . So, soon after the beam reach,

the sail, set to the greatest lift (deep draft), creates an excessive heel. The setting needs gradual adjustment to obtainthe highest possible L/D (sail more flat).The constraint is:

Close hauled

The formulas are the same as above. As against the position of the polar of the sail changes, the incidence decreases.Indeed, the angle with the wind becomes weaker and weaker, until it becomes so weak that the sail is not inflated.So, there is no profile. The sail flaps in the wind.Analysis of results

Constraints following the gaits are:

downwindbeam reach

and

on a broad reach

upwind.So, each will look at a different limit. Or,

on a broad reachdownwind

beam reach

upwind and partly reachingTherefore,

on a broad reach

beam reachdownwind

upwindThe point of sail is not an optimization variable but the incoming data. It depends on the chosen course. The point ofsail varies from near or close to downwind of about 30° to 180°.The L/D of a sail using lift is similar to a thin profile. The theory of thin profile gives the formula of L/D. But thenaval architect or sail maker to increase the performance of the boat will try to have a L/D as high as possible (seeL/D and sail trim). This setting is by design quite high. Factor is

quite independent of the forward drive and varies little in point of sail. But, sufficiently varied so that, as the formulashows, optimizing the performance also depends on the optimization of L/D. This optimization will give settingsdifferent from the pure power settings, called L/D.So, the constraints are directly related to the forward drive. But the physical power is directly related to the forwarddrive . So, the stability is directly related to the power.

Finally, we return to the same concept as the physical power. The harder it is to capsize the boat, the more the boat supports a large area of canvas for the propulsive force, the more the boat is moving fast, so the more powerful . It is well understood that the explanation is a guide to load. If the reader wishes, it is possible to obtain more complete

Forces on sails 34

formulas operating less severely than the approximations made in this guide. (See the scientific literature from theUniversity of Southampton)[119] · [120] · [121]

On the other hand, the apparent wind is the vector sum of the true wind minus the boat speed. The mathematicsshows that:

The formula clearly shows the potential gain following the apparent wind speed of the boat; potential gain that theyacht will build between close hauled and beam reach. The propulsive effort is determined by:

and the wind speed . Without going into details,

the effort of the sail is balanced by gravity, hydrodynamic forces of the hull and buoyancy. So, by including allformulas, it is possible to determine the maximum speed of a sailboat with good accuracy according to its set of sailsand point of sail. The software that performs this calculation are named the Velocity prediction program. The resultsindicate that the ship is the fastest in about beam reach (wide at close reach), that is to say the area where the lift ofthe sails is felt, and the apparent wind gained without compensation efforts for heel (drift leading to a strongresistance of the hull) too high.The common thread: to maximize the speed of the boat. But according to course conditions, other choices arepossible, leading to another type of setting. Settings will differ in a gale, or for a sailboat one is sailing against.In reality, downwind must be respected at best an incidence of 90° with the apparent wind, simple enough for amainsail with boom but, for headsail, it is more difficult, even with outriggers. But, this condition is not met then thelift is not zero. Performance is then lower than expected.This approach is for the sail. It helps to understand the settings and optimizations to be made. For the naval architectthe same process is conducted using the formulas including the hull .[122] For more completeness, this optimizationcan include, rather than static (constant speed of the boat and wind), dynamic variables such as sea swell and windgusts.A moment is a force multiplied by its lever arm. So, for the hull the arm length should be as high as possible and forthe sail as low as possible. The sailor has little control over the length of the lever arm. The bulk of the optimizationwork will be done by a naval architect for the hull (keel ballast) and sailmaker for the sail. This optimization is ofcourse not independent. It is linked to other elements. It is limited for example by looking for winds aloft givingmaximum propulsive effort. The end result will be one between all the constraints:•• Upwind, L/D is the major factor for the sail maker.•• With a tailwind, minimising the lever arm of the sail's centre of effort is the major factor.The polar "power" plots have a higher maximum propulsive effort compared to polar "L/D" plots. The polar plotgiving the maximum drag is a draft located behind the sail. Unlike the optimum setting for close hauled, there is nosudden drop in pressure if the trough is set a little too far. The setting of the sail is wider, more tolerant .[85]

The power of the sail depends almost solely on the part of the sail force contributing to the advancement of the ship(along the axis of vessel speed or course made good). The power is treated as part of the sail force contributing to theadvancement of the boat. The power is determined by the polar plot of the sail. The polar plot is independent of theapparent wind speed. Nor, in steady state theory as opposed to dynamic reality,[115] does the heel on the sailintervene with the speed setting. So the heel is not taken into account in the polar plot (same for the L/D of a polarplot). The profile of maximum power is not the profile of maximum L/D, where a setting of "power" creates toomuch heel, a fairly standard error.[123]

Forces on sails 35

Several sails: multidimensional problem resolutionThe previous method for estimating the thrust of each sail is not valid for boats with multiple sails, but it remains agood approximation.Sails close to each other influence each other. A two-dimensional model explains the phenomenon.[124] In the case ofa sloop-rigged sailboat, the foresail changes air flow entering onto the mainsail. The conditions of a stable fluid,constant and uniform, necessary for tables which give lift coefficient, are not respected with multiple sails. Thecumulative effect of several sails on a boat can be positive or negative. It is well known that for the same totalsurface sail, two sails properly set are more effective than a single sail set correctly. Two sails can increase thesailing thrust 20% compared a single sail of same area.[55][125]

Notes and references[1] Marchaj, C. A. (2003). Sail performance : techniques to maximise sail power (Rev. ed. ed.). London: Adlard Coles Nautical. pp. Part 1 ch 5

p20 fig 16 "Seakindliness and Seaworthiness". Part 2 Ch. 4 "The effects of Aerodynamic Forces" p76 fig 58. ISBN 978-0-7136-6407-2.[2] "When air flows over and under an aerofoil inclined at a small angle to it's direction, the air is turned from its course. Now, when a body is

moving at a uniform speed in a straight line, it requires a force to alter either its direction or speed. Therefore, the sails exert a force on thewind and, since action and reaction are equal and opposite, the wind exerts a force on the sails." Sailing Aerodynamics New Revised Edition1962 by John Morwood Adlard Coles Limited page 17

[3] Gilbert, Lester. "Momentum Theory of Lift" (http:/ / www. onemetre. net/ design/ downwash/ Momentum/ Momentum. htm). . Retrieved 20June 2011. "errata should read F=mw/unit time"

[4] "The physics of sailing" (http:/ / www. phys. unsw. edu. au/ ~jw/ sailing. html). . Retrieved 21 June 2011.[5] Fossati, Fabio; translated by Martyn Drayton (2009). "10.3 The frontiers of numerical methods: aeroelastic investigation".

Aero-hydrodynamics and the performance of sailing yachts : the science behind sailing yachts and their design. Camden, Maine: InternationalMarine /McGraw-Hill. p. 307. ISBN 978-0-07-162910-2.

[6] "Millenium Prize Navier Stokes equation" (http:/ / www. claymath. org/ millennium/ Navier-Stokes_Equations/ ). .[7] "Pressure PIV and Open Cavity Shear Layer Flow" (http:/ / www. me. jhu. edu/ lefd/ PPIV/ index. html). Johns Hopkins U. Laboratory for

Experimental Fluid Dynamics. . Retrieved 22 October 2011.[8] http:/ / knol. google. com/ k/ d-alembert-s-paradox# J. Hoffman, C. Johnson. D'Alembert's Paradox new resolution. So they say, but not all

their peers![9] JavaFoil (http:/ / www. mh-aerotools. de/ airfoils/ javafoil. htm)[10] Logiciel Calcul Voile Bateau Aile Portance (http:/ / www. mecaflux. com/ voile. htm)[11] For example, see XFOIL and AVL programmed by Mark Drela[12] http:/ / www. adeps. be/ pdf/ Theorie2005. pdf[13] Marchaj, Czeslaw A. Sail Performance, Techniques to Maximize Sail Power, Revised Edition. London: Adlard Coles Nautical, 2003. Part 2

Aerodynamics of sails, Chapter 2 "How and Why an Aerodynamic Force is Produced", page 49 "Pressure differences - the right way toexplain sail forces"

[14] http:/ / www. finot. com/ ecrits/ Damien%20Lafforgue/ article_voiles_english. html Damien Laforge Sails: from experimental to numerical[15] Fossati, Fabio; translated by Martyn Drayton (2009). Aero-hydrodynamics and the performance of sailing yachts : the science behind sailing

yachts and their design. Camden, Maine: International Marine /McGraw-Hill. pp. ch 8.12 Wind tunnel tests; ch 10.2 numerical methods.ISBN 978-0-07-162910-2.

[16] http:/ / knol. google. com/ k/ why-it-is-possible-to-sail# J Hoffman, C Johnson. 2009. "...lift and drag of a wing in three-dimensional realityresults from a three-dimensional instability mechanism at separation generating turbulent streamwise vorticity."

[17] http:/ / appliedfluidtech. com Applied Fluid Tech, Maryland USA[18] http:/ / www. wb-sails. fi/ news/ index. html WB-Sails Finland[19] "The Engineering toolbox. Pitot tubes" (http:/ / www. engineeringtoolbox. com/ pitot-tubes-d_612. html). . Retrieved 25 October 2011.[20][20] Marchaj p 57 Part 2 Ch 3 Distribution of pressures over sails figs 39 and 41[21][21] Fossati 8.12.2 p229 Test apparatus and measurement set-up[22] Crook, A. "An experimental investigation of high aspect-ratio rectangular sails" (http:/ / ctr. stanford. edu/ ResBriefs02/ crook2. pdf) (PDF).

see Figure 2. Center for Turbulence Research Annual Research Briefs. . Retrieved 22 October 2011.[23] "An explanation of sail flow analysis" (http:/ / syr. stanford. edu/ SAILFLOW. HTM). . Retrieved 22 October 2011.[24] Viola, Ignazio; Pilate, J, Flay, R. (2011). "UPWIND SAIL AERODYNAMICS: A PRESSURE DISTRIBUTION DATABASE FOR THE

VALIDATION OF NUMERICAL CODES" (http:/ / www. ignazioviola. com/ ignazio_maria_viola/ home_files/ Viola_IJSCT2011. pdf)(PDF). Intl J Small Craft Tech, 2011 153 (Part B1). . Retrieved 22 October 2011.

[25][25] Roussel, J. [perso.ensc-rennes.fr/jimmy.roussel/apprendre/fluides_parfaits.pdf "MÉCANIQUE DES FLUIDES"].perso.ensc-rennes.fr/jimmy.roussel/apprendre/fluides_parfaits.pdf. Retrieved 23 October 2011.

Forces on sails 36

[26][26] [fr.wikiversity.org/wiki/Intégrales_en_physique/Intégrales_multiples "Intégrales en physique : Intégrales multiples"]. Wikiversite.fr.wikiversity.org/wiki/Intégrales_en_physique/Intégrales_multiples. Retrieved 23 October 2011.

[27] "Chapter: 05. Aerodynamic Characteristics" (http:/ / www. lissys. demon. co. uk/ pug/ c05. html). Piano software. . Retrieved 26 October2011.

[28][28] coeficient of shape are neglectrd, because these are to close to 1. In general a sail have a ridiculous thickness in front of other length.[29] Eliasson, Lars Larsson & Rolf E. (2007). Principles of yacht design (3rd ed. ed.). Camden, Me: International Marine. pp. Ch 7 Sail and Rig

Design pp 142, 143 Fig 7.1. ISBN 978-0-07-148769-6.[30] (French) texte very educational in french (http:/ / www. aerodrome-ecuvillens. ch/ pilote guide/ aerodynamique. pdf)[31] (French)http:/ / www. onera. fr/ conferences/ mesures-aerodynamique/ cours-aerodynamique-mesures-efforts. pdf[32] (French)http:/ / docinsa. insa-lyon. fr/ polycop/ download. php?id=157288& id2=1[33] The reference for wingspan is often based on the line of all the first quarter of the chord. This first quarter chord is chosen because this is at

the aerodynamic center where the pitching moment, M, does not vary with angle of attack (see

Airfoil and Aerodynamic center). The line is often a straight line.[34] Vent réel - vent apparent - forces aéro et hdrodynamiques (http:/ / www. francelaser. org/ lettre/ mf-jvp/ jvp1. htm)[35] http:/ / www. lmm. jussieu. fr/ ~lagree/ TEXTES/ RAPPORTS/ rapportsX/ voilesNorvezPernot. pdf[36][36] indeed if the airfoil is symmetrical and sail shape not symmetrical[37] hnjb324.tmp (http:/ / www. ae. metu. edu. tr/ tuncer/ ae443/ docs/ NACA-All-Re. pdf)[38][38] It pushes the sail on the major axis, Fprin, of the vessel and its perpendicular, Fper. F is the forward thrust of the sail. Fprin = F * cos 40° =

76% * F. Fper = F * sin 40° = 36% * F[39][39] not the case for hydroptere, wind surf...[40] book partially scanned Bien naviguer et mieux connaître son voilier by Gilles Barbanson,Jean Besson sheet 72-73 (http:/ / books. google. fr/

books?id=3xUwXPoJ1loC& pg=PA150& lpg=PA150& dq=voilier+ vitesse+ longueur+ de+ flottaison& source=bl& ots=LshK-LIcxN&sig=wo3DmLfFdrdZpI-Qr_6ikYJ9RV0& hl=fr& ei=rNbKS5uhBcGBOI3T0O0F& sa=X& oi=book_result& ct=result& resnum=10&ved=0CCwQ6AEwCQ#v=onepage& q=voilier vitesse longueur de flottaison& f=false)

[41][41] Principles of yacht design, by Lars Larsson et Rolf E Eliasson ISBN 0-7136-5181-4 or 9 780713 651812 page 140 figure 7.9 and 7.10[42] figure 5 (http:/ / www. hiswasymposium. com/ assets/ files/ pdf/ previous/ 17th - 2002/ 17th -3- Optimization of yard sectional shap and

configurati. pdf)[43][43] Principles of yacht design, by Lars Larsson and Rolf E Eliasson ISBN 0-7136-5181-4 or 9 780713 651812 page 140 figure 7.11[44] figure 5 (http:/ / www. iawe. org/ Proceedings/ 5EACWE/ 142. pdf)[45] distance is thickness of sailcloth adding spar (boom or horn)[46] http:/ / www. dedale-planeur. org/ horten/ Horten%20critique%20par%20Deszo. pdf[47] http:/ / air-et-terre. info/ aerodyn_theorique/ ligne_portante_3D. pdf[48] http:/ / j. haertig. free. fr/ aerodyn_theorique/ ligne_portante_3D. pdf[49][49] This ideal elliptical shape is result of calculus for a stable and uniform flow of wind, as win is not uniform (see :Influence of altitude:

aerodynamic twist and sail twist), ideal shape must be mitigate.[50] in french see page 51 (http:/ / hal. archives-ouvertes. fr/ docs/ 00/ 45/ 37/ 91/ PDF/ these_Roncin. pdf) in this thesis, the autor explaind that

du to proximity of deck, deck can be used as mirror surface instead of sea level.[51] Larsson, Lars; Eliasson, Rolf E. (1999). Principles of yacht design (2nd ed. ed.). London: Adlard Coles Nautical. pp. 139 figure 7.8.

ISBN 978-0-7136-5181-2.[52] Marchaj, C. A. (2003). Sail performance : techniques to maximise sail power (Rev. ed. ed.). London: Adlard Coles Nautical. pp. 208–211.

ISBN 978-0-7136-6407-2.[53] www.emmanuel.chazard.org (http:/ / chazard. org/ emmanuel/ cours-de-catamaran-reglage-de-la-grand-voile-gv)[54] Microsoft PowerPoint - analyse des forces.ppt (http:/ / www. finn-france. fr/ TECHNIQUE VOILE/ michaud1. pdf)[55] http:/ / jestec. taylors. edu. my/ Issue%201%20Vol%201%20June%2006/ p89-98. pdf Al_Atabi, M. The Aerodynamics of wing tip sails.

Journal of Engineering Science and Technology.Vol. 1, No. 1 (2006) 89-98. Multiple sails figure on page 94 of article[56] Etude de la force aérodynamique (http:/ / membres. multimania. fr/ tpevoile/ faero. htm)[57][57] If the sail is loose, the sail shakes, thus providing some resistance. The sailing ship is slightly back, in this case there is a slight drag. It is

also noted that under these conditions the mast, the rigging, superstructure and topsides will provide much more aerodynamic force than thesail itself.

[58][58] telltales are unstable[59] Naca 12 (http:/ / www. onera. fr/ mecao/ aerodynamique/ phototheque/ video/ naca12. htm)[60] figure 5 (http:/ / syr. stanford. edu/ JWEIA557. pdf)[61] Section 2.2 Apparent wind-true wind (http:/ / www. finot. com/ ecrits/ Damien Lafforgue/ article_voiles_english. html#Titre_2)[62] http:/ / hal. archives-ouvertes. fr/ docs/ 00/ 16/ 72/ 71/ PDF/ B104. pdf[63] http:/ / www. ignazioviola. com/ ignazio_maria_viola/ publications_files/ Viola_EACWE2005. pdf Zasso A, Fossati F, Viola I. Twisted

flow wind tunnel design for yacht aerodynamic studies. EACWE4 — The Fourth European & African Conference on Wind Engineering J. N´prstek & C. Fischer (eds); ITAM AS CR, Prague, 11–15 July 2005, Paper #153

[64] http:/ / heikki. org/ publications/ ModernYachtLePelleyHansen. PDF[65] http:/ / techniques. avancees. free. fr/ tipe/ techniquesAvanceesGeneral. pdf

Forces on sails 37

[66] sheet 2 (http:/ / syr. stanford. edu/ JWEIA557. pdf)[67] formula is given in introduction (http:/ / airsea. ucsd. edu/ papers/ MELVILLE WK - JOURNAL OF PHYSICAL OCEANOGRAPHY 7 -

1977. pdf)[68] http:/ / www. dtic. mil/ cgi-bin/ GetTRDoc?AD=AD734670& Location=U2& doc=GetTRDoc. pdf[69] Wind Gradient (http:/ / www. onemetre. net/ Design/ Gradient/ Gradient. htm)[70] http:/ / media. wiley. com/ product_data/ excerpt/ 0X/ 04705165/ 047051650X. pdf[71] Sail Shape (http:/ / www. sailingusa. info/ sail_shape. htm)[72] http:/ / www. usna. edu/ naoe/ people/ SCHULTZ%20PAPERS/ Miklosovic,%20Schultz%20& %20Esquivel%20JoA%202004. pdf[73] http:/ / www. usna. edu/ naoe/ people/ SCHULTZ%20PAPERS/ Schultz%20JFE%202002. pdf[74] figure 6 page 23 (http:/ / aerade. cranfield. ac. uk/ ara/ arc/ rm/ 3726. pdf)[75] http:/ / bbaa6. mecc. polimi. it/ uploads/ validati/ TR02. pdf Viola, I.M., Fossati, F. Downwind sails aerodynamic analysis. BBAA VI

International Colloquium on: Bluff Bodies Aerodynamics & Applications. Milano, Italy, July, 20-24 2008.[76] Marchaj, C. A. (2003). Sail performance : techniques to maximise sail power (Rev. ed. ed.). London: Adlard Coles Nautical. pp. 343–350.

ISBN 978-0-7136-6407-2.[77] voir figure3b (http:/ / www. gidb. itu. edu. tr/ staff/ insel/ Publications/ Cesme. PDF)[78] see the figures (http:/ / hal. archives-ouvertes. fr/ docs/ 00/ 45/ 37/ 30/ PDF/ articlecorrige. pdf)[79] FTE: Heel For Speed | Sailing World (http:/ / www. sailingworld. com/ experts/ heel-for-speed)[80] Sailing World (http:/ / www. sailingworld. com/ experts/ how-heel-affects-speed-and-handling)[81] A true wind approach will be more rigorous than used a surface cut.[82] Les voiles (http:/ / www. finot. com/ ecrits/ Damien Lafforgue/ article_voiles_english. html)[83] Bob Sterne How to Sail Fast (http:/ / c_r_y_a. tripod. com/ Sterne How to. htm#4 Non Optimum)[84] voiles (http:/ / modelismepassion. ibelgique. com/ voiles. htm)[85] Les réglages de voile - Réglage de grand voile, réglage de génois, réglage de spi (http:/ / www. bretagne-atlantic-yachting. eu/ peda/

reglages_de_voile. html)[86] Viscous Computational Fluid Dynamics as a Relevant Decision-Making Tool for Mast-Sail Aerodynamics (http:/ / vincent. chapin. free. fr/

publications/ MarineTechnology_VGC_Jan2005. pdf)[87] Aerospaceweb.org | Ask Us - Drag Coefficient & Lifting Line Theory (http:/ / www. aerospaceweb. org/ question/ aerodynamics/ q0184.

shtml)[88] http:/ / www. ltas-mct. ulg. ac. be/ who/ stainier/ docs/ aeroelasticite. pdf page 28[89] calcul avec la méthode des lignes portantes avec les deux vortex d'extrémité de profil (http:/ / j. haertig. free. fr/ aerodyn_theorique/

ligne_portante_3D. pdf)[90] Induced Drag Coefficient (http:/ / www. grc. nasa. gov/ WWW/ K-12/ airplane/ induced. html)[91] The Drag Coefficient (http:/ / www. grc. nasa. gov/ WWW/ K-12/ airplane/ dragco. html)[92] http:/ / s6. aeromech. usyd. edu. au/ aero/ liftline/ liftline. pdf[93] voir (3.2.1) page 38 (http:/ / www. engbrasil. eng. br/ index_arquivos/ art52. pdf)[94] aerodynamic lift (http:/ / ocw. mit. edu/ courses/ aeronautics-and-astronautics/ 16-100-aerodynamics-fall-2005/ study-materials/ 1liftdrag.

pdf)[95] Lift (http:/ / www. aoe. vt. edu/ ~neu/ aoe5104/ 23 - LiftingLineTheory. pdf)[96] figures 27 and 29 (http:/ / oa. upm. es/ 2203/ 2/ INVE_MEM_2008_53562. pdf)[97] Principles of yacht design, by Lars Larsson and Rolf E Eliasson ISBN 0-7136-5181-4 or 9 780713 651812 page 151 figure 7.20 This figure

shows well the different types of drag[98] figure 26 (http:/ / oa. upm. es/ 2203/ 2/ INVE_MEM_2008_53562. pdf)[99] figure 17 (http:/ / www. futureship. net/ downloads/ KrebberHochkirchHPYD06. pdf)[100] Voile-habitable : Réglage et conduite au portant sous spi (http:/ / voilehabitable. org/ cms/ tiki-index. php?page_ref_id=116&

PHPSESSID=10939f64b633cbc77fdf4b7710a44e5f)[101] WB-Sails Ltd (http:/ / www. wb-sails. fi/ news/ 95_11_Tellingtales/ Tellingtales. html)[102] tuning @ sailtheory.com (http:/ / www. sailtheory. com/ tuning. html)[103] MD / Voile & Mer (http:/ / marc. donneger. free. fr/ Voile& Mer/ propulse. htm)[104] http:/ / arxiv. org/ ftp/ arxiv/ papers/ 1002/ 1002. 1226. pdf[105] Capacité de porter de la toile (http:/ / www. finot. com/ ecrits/ vitessecoq/ chap4/ chap4. htm)[106] Estimating Stability (http:/ / www. johnsboatstuff. com/ Articles/ estimati. htm)[107] Stability and Trim for Ships, Boats, Yachts and Barges – Part I (http:/ / www. hawaii-marine. com/ templates/ stability_article. htm)[108] page 42 equation 47 breakdown identical with other notation (http:/ / www. orc. org/ rules/ ORC VPP Documentation 2009. pdf)[109] http:/ / www. towage-salvage. com/ files/ stab014. pdf[110] 1 3 Dynamic Stability Ppt Presentation (http:/ / www. authorstream. com/ Presentation/

Garrick-25629-1-3-Dynamic-Stability-Objectives-Heeling-Moments-Moment-Curve-Static-Righting-CGC-JARVIS-November-15-197-as-Entertainment-ppt-powerpoint/)

[111] Heeling arm definition (http:/ / www. formsys. com/ extras/ FDS/ webhelp/ hydromax/ heeling_arm_definition. htm)[112] http:/ / www. gidb. itu. edu. tr/ staff/ insel/ Publications/ Cesme. PDF

Forces on sails 38

[113] PII: 0169-5983(94)00027-1 (http:/ / 202. 114. 89. 60/ resource/ pdf/ 1044. pdf)[114] (French)http:/ / chazard. org/ emmanuel/ cours-de-catamaran-couples-de-rotation-dessalage-heel-enfournement[115] Marchaj, C. A. (2003). "Part 2 Ch 7 Sailing Downwind (Rolling)". Sail performance : techniques to maximise sail power (Rev. ed. ed.).

London: Adlard Coles Nautical. pp. 351–360. ISBN 978-0-7136-6407-2.[116] Figure 19 on page 34 and Figure 17 and Figure 20 an incidence value of 90° is slightly wrong. This is due to the fact that for a flexible sail

the sailor can not place the entire surface of the sail perpendicular to the wind (to cut the wind on the full sail). For a jib, maximum drag is at160° and the incidence of lift is zero. For a mainsail, max 170°. For a headsail genaker type or large genoa, 180° max. (http:/ / www. orc. org/rules/ ORC VPP Documentation 2009. pdf)

[117] Grain de Sel : Navigation à la voile (http:/ / www. grain-de-sel. org/ technique/ voile/ coursvoile. htm)[118] see page 31 (http:/ / digilander. libero. it/ eapisa/ download/ conferenza 2g. ppt)[119] example on ORC class (http:/ / www. wumtia. soton. ac. uk/ papers/ HISWA2008ARC. pdf)[120] The Development of Stability Standards for UK Sailing Vessels, B. Deakin (http:/ / www. wumtia. soton. ac. uk/ papers/ RINA1990BD.

pdf)PDF[121] http:/ / www. orc. org/ rules/ ORC%20VPP%20Documentation%202009. pdf[122] http:/ / www. wumtia. soton. ac. uk/ papers/ FAST2005WHM2BD. pdf[123] Si nous parlions assiette (http:/ / sinousparlionsassiette. blogspot. com/ )[124] The Aerodynamics of sail interaction (http:/ / www. arvelgentry. com/ techs/ The Aerodynamics of Sail Interaction. pdf)[125] Richards, Peter; Lasher, William (20-24 July 2008). "WIND TUNNEL AND CFD MODELLING OF PRESSURES ON DOWNWIND

SAILS" (http:/ / bbaa6. mecc. polimi. it/ uploads/ validati/ TR08. pdf). BBAA VI International Colloquium on: Bluff Bodies Aerodynamics &Applications. . Retrieved 2 June 2012.

Bibliography• Royce, Patrick M. (1993). Royce's Sailing Illustrated: The Sailors Bible Since '56. Prostar.

ISBN 978-0-911284-08-9.• Mulville, Frank (1991). Single-handed Sailing. Seafarer Books. ISBN 978-0-85036-410-1.• Marchaj, C.A. (1985). Sailing Theory and Practice, Revised edition. Putnam. ISBN 978-0-396-08428-0.• Marchaj, C. A. (2003). Sail performance : techniques to maximise sail power (Rev. ed. ed.). London: Adlard

Coles Nautical. ISBN 978-0-7136-6407-2.• 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.

• Eliasson, Lars Larsson & Rolf E. (2007). Principles of yacht design (3rd ed. ed.). Camden, Me: InternationalMarine. ISBN 978-0-07-148769-6.

• Fossati, Fabio; translated by Martyn Drayton (2009). Aero-hydrodynamics and the performance of sailing yachts :the science behind sailing yachts and their design. Camden, Maine: International Marine /McGraw-Hill.ISBN 978-0-07-162910-2.

• (French) 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 978-2-7027-0027-3.

• (French) Bertrand, Chéret (June 2010). Les Voiles. Comprendre, régler, optimiser. Gallimard.ISBN 978-2-7424-0767-5.

• (French) Leonhard Euler Théorie complète de la construction et de la manoeuvre des vaisseaux (http:/ / books.google. fr/ books?id=0qwWAAAAQAAJ& pg=PA166& lpg=PA166& dq=centre+ vélique& source=bl&ots=lm1-n-v_7o& sig=UB58BXVE7ruNiDy9HpnhmWTVbpc& hl=fr& ei=HI7FS5raCYKROJ6fwb4P& sa=X&oi=book_result& ct=result& resnum=10& ved=0CBwQ6AEwCTgo#v=onepage& q=centre vélique& f=false)printed by Claude-antoine Jombert at Paris in 1773

• (Latin) Leonhard Euler E110 Scientia navalis (http:/ / www. math. dartmouth. edu/ ~euler/ pages/ E110. html) 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.

Forces on sails 39

scient. imper. Petropoli typis academiae scientiarum MDCCXLIX.

External links• Rapport NACA n° 824 (http:/ / www. sailboat-technology. com/ links/ naca-report-824. pdf)PDF (18.6 MB)• Rapport NACA n° 1218 (http:/ / www. sailboat-technology. com/ links/ naca-tm-1218. pdf)PDF (4.01 MB)• Rapport NACA n° 1217 (http:/ / www. sailboat-technology. com/ links/ naca-tm-1217. pdf)PDF (4.57 MB)• listing of interesting article on Sailboat-technology.com (http:/ / www. sailboat-technology. com/ links/

online_articles. php)• website of Knol.google.com on turbulence and boundary layer (http:/ / knol. google. com/ k/

the-spell-of-prandtl-s-laminar-boundary-layer#)• Sails: from experimental to numerical Damien Laforgue (http:/ / www. finot. com/ ecrits/ Damien Lafforgue/

article_voiles_english. html)

Article Sources and Contributors 40

Article Sources and ContributorsForces on sails  Source: http://en.wikipedia.org/w/index.php?oldid=500110488  Contributors: Akerans, Alan Liefting, Arjayay, Bcebul, Chris the speller, CommonsDelinker, DaemonicKangaroo, DirkvdM, Dolphin51, Erwan1972, Eumolpo, Favonian, ForemastJack, GoingBatty, John of Reading, Khazar, Kudpung, Life of Riley, LittleWink, Mandarax, Marktaff, Mild BillHiccup, Mr swordfish, Nigelj, Nuttyrave, Tabletop, Thumperward, 5 anonymous edits

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