Language of velocity and force

To understand forces and velocities, discussed here, one must understand what is meant by a "vector" and a "scalar." Velocity (V), denoted as boldface in this article, is an example of a vector, because it implies both direction and speed. The corresponding speed (V ), denoted as italics in this article is a scalar value. Likewise, a force vector, F, denotes direction and strength, whereas its corresponding scalar (F ) denotes strength alone. Graphically, each vector is represented with an arrow that shows direction and a length that shows speed or strength. Vectors of consistent units (e.g. V in m/s or F in N) may be added and subtracted, graphically, by positioning tips and tails of the arrows, representing the input variables and drawing the resulting derived vector.

Components of force: lift vs. drag and driving vs. lateral force

Lift on a sail (L), acting as an airfoil, occurs in a direction perpendicular to the incident airstream (the apparent wind velocity, V_A, for the head sail) and is a result of pressure differences between the windward and leeward surfaces and depends on angle of attack, sail shape, air density, and speed of the apparent wind. Pressure differences result from the normal force per unit area on the sail from the air passing around it. The lift force results from the average pressure on the windward surface of the sail being higher than the average pressure on the leeward side.^[1] These pressure differences arise in conjunction with the curved air flow. As air follows a curved path along the windward side of a sail, there is a pressure gradient perpendicular to the flow direction with lower pressure on the outside of the curve and higher pressure on the inside. To generate lift, a sail must present an "angle of attack" (α) between the chord line of the sail and the apparent wind velocity (V_A). Angle of attack is a function of both the craft's point of sail and how the sail is adjusted with respect to the apparent wind.^[2]

As the lift generated by a sail increases, so does lift-induced drag, which together with parasitic drag constitutes total drag, (D). This occurs when the angle of attack increases with sail trim or change of course to cause the lift coefficient to increase up to the point of aerodynamic stall, so does the lift-induced drag coefficient. At the onset of stall, lift is abruptly decreased, as is lift-induced drag, but viscous pressure drag, a component of parasitic drag, increases due to the formation of separated flow on the surface of the sail. Sails with the apparent wind behind them (especially going downwind) operate in a stalled condition.^[3]

Lift and drag are components of the total aerodynamic force on sail (F_T). Since the forces on the sail are resisted by forces in the water (for a boat) or on the traveled surface (for an ice boat or land sailing craft), their corresponding forces can also be decomposed from total aerodynamic force into driving force (F_R) and lateral force (F_LAT). Driving force overcomes resistance to forward motion. Lateral force is met by lateral resistance from a keel, blade or wheel, but also creates a heeling force.

• 
  Decomposition of forces (in horizontal cross-section) acting on a sail, generating lift.
  F_T is the Total Force acting on the sail for the Apparent Wind (V_A), shown. This resolves into forces felt by the sail, Lift (L) and Drag (D), with vectors shown in red and angle of attack noted as α.
• 
  Wind forces acting on a sailboat sail (L and D) and being transmitted to the boat (F_R—propelling the boat forward—and F_LAT—pushing the boat sideways), while close-hauled, are both components of total aerodynamic force (F_T).

Effect of points of sail on forces

Apparent wind (V_A) is the air velocity acting upon the leading edge of the most forward sail or as experienced by instrumentation or crew on a moving sailing craft. It is the vector sum of true wind velocity and the apparent wind component resulting from boat velocity (V_A = -V_B + V_T). In nautical terminology, wind speeds are normally expressed in knots and wind angles in degrees. The craft's point of sail affects its velocity (V_B) for a given true wind velocity (V_T). Conventional sailing craft cannot derive power from the wind in a "no-go" zone that is approximately 40° to 50° away from the true wind, depending on the craft. Likewise, the directly downwind speed of all conventional sailing craft is limited to the true wind speed.^[4]

• 
  Apparent wind and forces on a sailboat.
  As the boat sails further from the wind, the apparent wind becomes smaller and the lateral component becomes less; boat speed is highest on the beam reach.
• 
  Apparent wind on an iceboat.
  As the iceboat sails further from the wind, the apparent wind increases slightly and the boat speed is highest on the broad reach. The sail is sheeted in for all three points of sail.^[5]

A sailboat's speed through the water is limited by the resistance that results from hull drag in the water. Sail boats on foils are much less limited. Ice boats typically have the least resistance to forward motion of any sailing craft. Craft with the higher forward resistance achieve lower forward velocities for a given wind velocity than ice boats, which can travel at speeds several multiples of the true wind speed.^[5] Consequently, a sailboat experiences a wider range of apparent wind angles than does an ice boat, whose speed is typically great enough to have the apparent wind coming from a few degrees to one side of its course, necessitating sailing with the sail sheeted in for most points of sail. On conventional sail boats, the sails are set to create lift for those points of sail where it's possible to align the leading edge of the sail with the apparent wind.^[4]

For a sailboat, point of sail affects lateral force significantly. The higher the boat points to the wind under sail, the stronger the lateral force, which requires resistance from a keel or other underwater foils, including daggerboard, centerboard, skeg and rudder. Lateral force also induces heeling in a sailboat, which requires resistance by weight of ballast from the crew or the boat itself and by the shape of the boat, especially with a catamaran. As the boat points off the wind, lateral force and the forces required to resist it become less important.^[6] On ice boats, lateral forces are countered by the lateral resistance of the blades on ice and their distance apart, which generally prevents heeling.^[7]

Force components on sails

Windsurfers employ lift vertical to the water to reduce drag on the board by leaning the sail towards the wind.

The approximate locus of net aerodynamic force on a craft with a single sail is the centre of effort (CE ) at the geometric centre of the sail. Filled with wind, the sail has a roughly spherical polygon shape and if the shape is stable, then the location of centre of effort is stable. On sailing craft with multiple sails, the position of centre of effort varies with the sail plan. Sail trim or airfoil profile, boat trim and point of sail also affect CE. ^{[6] [9]} On a given sail, the net aerodynamic force on the sail is located approximately at the maximum draught intersecting the camber of the sail and passing through a plane intersecting the centre of effort, normal to the leading edge (luff), roughly perpendicular to the chord of the sail (a straight line between the leading edge (luff) and the trailing edge (leech)). Net aerodynamic force with respect to the air stream is usually considered in reference to the direction of the apparent wind (V_A) over the surface plane (ocean, land or ice) and is decomposed into lift (L), perpendicular with V_A, and drag (D), in line with V_A. For windsurfers, lift component vertical to the surface plane is important, because in strong winds windsurfer sails are leaned into the wind to create a vertical lifting component ( F_VERT) that reduces drag on the board (hull) through the water.^[10] Note that F_VERT acts downwards for boats heeling away from the wind, but is negligible under normal conditions.

The three dimensional vector relationship for net aerodynamic force with respect to apparent wind (V_A) is:^[8]

${\displaystyle \mathbf {F_{T}} =\mathbf {L} +\mathbf {D} +\mathbf {F_{VERT}} }$

Likewise, net aerodynamic force may be decomposed into the three translational directions with respect to a boat's course over the surface: surge (forward/astern), sway (starboard/port—relevant to leeway), and heave (up/down). The scalar values and direction of these components can be dynamic, depending on wind and waves (for a boat).^[6] In this case, F_T is considered in reference to the direction of the boat's course and is decomposed into driving force (F_R), in line with the boat's course, and lateral force (F_LAT), perpendicular with the boat's course. Again for windsurfers, the lift component vertical to the surface plane ( F_VERT) is important.

The three dimensional vector relationship for net aerodynamic force with respect to the course over the surface is:^[8]

${\displaystyle \mathbf {F_{T}} =\mathbf {F_{R}} +\mathbf {F_{LAT}} +\mathbf {F_{VERT}} }$

The values of driving force (F_R ) and lateral force (F_LAT ) with apparent wind angle (α), assuming no heeling, relate to the values of lift (L ) and drag (D ), as follows:^[8]

${\displaystyle F_{R}=L\cdot \sin(\alpha )-D\cdot \cos(\alpha )}$

${\displaystyle \ F_{LAT}=L\cdot \cos(\alpha )+D\cdot \sin(\alpha )}$

Reactive forces on sailing craft

Reactive forces on sailing craft include forward resistance—sailboat's hydrodynamic resistance (R_l), an ice boat's sliding resistance or a land sailing craft's rolling resistance in the direction of travel—which are to be minimized in order to increase speed, and lateral force, perpendicular to the direction of travel, which is to be made sufficiently strong to minimize sideways motion and to guide the craft on course.

Forward resistance comprises the types of drag that impede a sailboat's speed through water (or an ice boat's speed over the surface) include components of parasitic drag, consisting primarily of form drag, which arises because of the shape of the hull, and skin friction, which arises from the friction of the water (for boats) or air (for ice boats and land sailing craft) against the "skin" of the hull that is moving through it. Displacement vessels are also subject to wave resistance from the energy that goes into displacing water into waves and that is limited by hull speed, which is a function of waterline length, Wheeled vehicles' forward speed is subject to rolling friction and ice boats are subject to kinetic or sliding friction. Parasitic drag in water or air increases with the square of speed (V_B² or V_A², respectively);^{[11] [12]} rolling friction increases linearly with velocity;^[13] whereas kinetic friction is normally a constant,^[14] but on ice may become reduced with speed as it transitions to lubricated friction with melting.^[5]

Ways to reduce wave-making resistance used on sailing vessels include reduced displacement—through planing or (as with a windsurfer) offsetting vessel weight with a lifting sail—and fine entry, as with catamarans, where a narrow hull minimizes the water displaced into a bow wave.^[15] Sailing hydrofoils also substantially reduce forward friction with an underwater foil that lifts the vessel free of the water.^[16]

• 
  International Moth class sailboat on foils.
• 
  DN class ice boat.
• 
  Land sailing craft.

Sailing craft with low forward resistance can achieve high velocities with respect to the wind velocity:^[17]

• High-performance catamarans, including the Extreme 40 catamaran and International C-class catamaran can sail at speeds up to twice the speed of the wind.^[18][19]
• Sailing hydrofoils achieve boat speeds up to twice the speed of the wind, as did the AC72 catamarans used for the 2013 America's Cup.^[20]
• Ice boats can sail up to five times the speed of the wind.^[21][22]

Lateral force is a reaction supplied by the underwater shape of a sailboat, the blades of an ice boat and the wheels of a land sailing craft. Sailboats rely on keels, centerboards, and other underwater foils, including rudders, that provide lift in the lateral direction, to provide hydrodynamic lateral force (P_LAT) to offset the lateral force component acting on the sail (F_LAT) and minimize leeway.^[8] Such foils provide hydrodynamic lift and, for keels, ballast to offset heeling. They incorporate a wide variety of design considerations.^[23]

Rotational forces on sailing craft

The forces on sails that contribute to torque and cause rotation with respect to the boat's longitudinal (fore and aft), horizontal (abeam) and vertical (aloft) rotational axes result in: roll (e.g. heeling). pitch (e.g. pitch-poling), and yaw (e.g. broaching). Heeling, which results from the lateral force component (F_LAT), is the most significant rotational effect of total aerodynamic force (F_T).^[8] In stasis, heeling moment from the wind and righting moment from the boat's heel force (F_H ) and its opposing hydrodynamic lift force on hull (F_l ), separated by a distance (h = "heeling arm"), versus its hydrostatic displacement weight (W ) and its opposing buoyancy force (Δ), separated by a distance (b = "righting arm") are in balance:^[8]

${\displaystyle h\times F_{H}=b\times \Delta =h\times F_{l}=b\times W}$
(heeling arm × heeling force = righting arm × buoyancy force = heeling arm × hydrodynamic lift force on hull = righting arm × displacement weight)

Sails come in a wide variety of configurations that are designed to match the capabilities of the sailing craft to be powered by them. They are designed to stay within the limitations of a craft's stability and power requirements, which are functions of hull (for boats) or chassis (for land craft) design. Sails derive power from wind that varies in time and with height above the surface. In order to do so, they are designed to adjust to the wind force for various points of sail. Both their design and method for control include means to match their lift and drag capabilities to the available apparent wind, by changing surface area, angle of attack, and curvature.

Wind variation with elevation

Wind gust increasing heeling moment on the right-hand J/22-class boat.

Wind speed increases with height above the surface; at the same time, wind speed may vary over short periods of time as gusts. These considerations may be described empirically.

Measurements show that wind speed, (V (h ) ) varies, according to a power law with height (h ) above a non-zero measurement height datum (h₀ —e.g. at the height of the foot of a sail), using a reference wind speed measured at the datum height (V (h₀ ) ), as follows: ^[24][25]

${\displaystyle V(h)=V(h_{0})\left({\frac {h}{h_{0}}}\right)^{p}}$

Where the power law exponent (p) has values that have been empirically determined to range from 0.11 over the ocean to 0.31 over the land.

This means that a V (3 m) = 5-m/s (≈10-knot) wind at 3 m above the water would be approximately V (15 m) = 6 m/s (≈12 knots) at 15 m above the water. In hurricane-force winds with V (3 m) = 40-m/s (≈78 knots) the speed at 15 m would be V (15 m) = 49 m/s (≈95 knots) with p = 0.128.^[26] This suggests that sails that reach higher above the surface can be subject to stronger wind forces that move the centre of effort (CE ) higher above the surface and increase the heeling moment.

Additionally, apparent wind direction moves aft with height above water, which may necessitate a corresponding twist in the shape of the sail to achieve attached flow with height.^[27]

Wind variation with time

Hsu gives a simple formula for a gust factor (G ) for winds as a function of the exponent (p ), above, where G is the ratio of the wind gust speed to baseline wind speed at a given height: ^[28]

${\displaystyle G=1+2p}$

So, for a given windspeed and Hsu's recommended value of p = 0.126, one can expect G = 1.5 (a 10-knot wind might gust up to 15 knots). This, combined with changes in wind direction suggest the degree to which a sailing craft must adjust to wind gusts on a given course.

Lift predominant (attached flow)

Sail angles of attack and resulting (idealized) flow patterns for attached flow, maximum lift, and stalled for a hypothetical sail. The stagnation streamlines (red) delineate air passing to the leeward side (top) from that passing to the windward (bottom) side of the sail.

Sails allow progress of a sailing craft to windward, thanks to their ability to generate lift (and the craft's ability to resist the lateral forces that result). Each sail configuration has a characteristic coefficient of lift and attendant coefficient of drag, which can be determined experimentally and calculated theoretically. Sailing craft orient their sails with a favorable angle of attack between the entry point of the sail and the apparent wind as their course changes. The ability to generate lift is limited by sailing too close to the wind when no effective angle of attack is available to generate lift (luffing) and sailing sufficiently off the wind that the sail cannot be oriented at a favorable angle of attack (running downwind). Instead, past a critical angle of attack, the sail stalls and promotes flow separation.

Drag predominant (separated flow)

When sailing craft are on a course where the angle of attack between the sail and the apparent wind (α ) exceeds the point of maximum lift on the C_L–C_D polar diagram, separation of flow occurs.^[34] The separation becomes more pronounced until at α = 90° lift becomes small and drag predominates. In addition to the sails used upwind, spinnakers provide area and curvature appropriate for sailing with separated flow on downwind points of sail.^[35]

• 
  Broad Reach: With apparent wind behind the sail (α = 45°), the sail has stalled and lift has diminished.
• 
  Running before the wind: With apparent wind directly behind the sail (α = 90°), drag forces dominate.

Again, in these diagrams the direction of travel changes with respect to the apparent wind (V_A), which is constant for the sake of illustration, but would in reality vary with point of sail for a constant true wind. In the left-hand diagram (broad reach), the boat is on a point of sail, where the sail can no longer be aligned into the apparent wind to create an optimum angle of attack. Instead, the sail is in a stalled condition, creating about 80% of the lift as in the upwind examples and drag has doubled. Total aerodynamic force (F_T) has moved away from the maximum lift value. In the right-hand diagram (running before the wind), lift is one-fifth of the upwind cases (for the same strength apparent wind) and drag has almost quadrupled.^[33]

• 
  Spinnaker set for a broad reach, generating both lift with separated flow and drag.
• 
  Spinnaker cross-section trimmed for a broad reach showing transition from boundary layer to separated flow where vortex shedding commences.
• 
  Symmetric spinnaker while running downwind, primarily generating drag.
• 
  Symmetric spinnaker cross-section with following apparent wind, showing vortex shedding.

Downwind polar diagram to determine velocity made good at various wind speeds for a hypothetical displacement sailboat and sail plan.

A velocity prediction program can translate sail performance and hull characteristics into a polar diagram, depicting boat speed for various windspeeds at each point of sail. Displacement sailboats exhibit a change in what course has the best velocity made good (VMG), depending on windspeed. For the example given, the sailboat achieves best downwind VMG for windspeed of 10 knots and less at a course about 150° off the wind. For higher windspeed the optimum downwind VMG occurs at more than 170° off the wind. This "downwind cliff" (abrupt change in optimum downwind course) results from the change of balance in drag forces on the hull with speed.^[35]

Sail interactions

Sailboats often have a jib that overlaps the mainsail—called a genoa. Arvel Gentry demonstrated in his series of articles published in "Best of sail trim" published in 1977 (and later reported and republished in summary in 1981) that the genoa and the mainsail interact in a symbiotic manner, owing to the circulation of air between them slowing down in the gap between the two sails (contrary to traditional explanations), which prevents separation of flow along the mainsail. The presence of a jib causes the stagnation line on the mainsail to move forward, which reduces the suction velocities on the main and reduces the potential for boundary layer separation and stalling. This allows higher angles of attack. Likewise, the presence of the mainsail causes the stagnation line on the jib to be shifted aft and allows the boat to point closer to the wind, owing to higher leeward velocities of the air over both sails. ^[33][36]

The two sails cause an overall larger displacement of air perpendicular to the direction of flow when compared to one sail. They act to form a larger wing, or airfoil, around which the wind must pass. The total length around the outside has also increased and the difference in air speed between windward and leeward sides of the two sails is greater, resulting in more lift. The jib experiences a greater increase in lift with the two sail combination.^[37]

Sail terminology

Sails are classified as "triangular sails", "quadrilateral fore-and-aft sails" (gaff-rigged, etc.), and "square sails".^[38] The top of a triangular sail, the head, is raised by a halyard, The forward lower corner of the sail, the tack, is shackled to a fixed point on the boat in a manner to allow pivoting about that point—either on a mast, e.g. for a mainsail, or on the deck, e.g. for a jib or staysail. The trailing lower corner, the clew, is positioned with an outhaul on a boom or directly with a sheet, absent a boom. Symmetrical sails have two clews, which may be adjusted forward or back.^[38]

The windward edge of a sail is called the luff, the trailing edge, the leach, and the bottom edge the foot. On symmetrical sails, either vertical edge may be presented to windward and, therefore, there are two leaches. On sails attached to a mast and boom, these edges may be curved, when laid on a flat surface, to promote both horizontal and vertical curvature in the cross-section of the sail, once attached. The use of battens allows a sail have an arc of material on the leech, beyond a line drawn from the head to the clew, called the roach.^[38]

Lift variables

As with aircraft wings, the two dominant factors affecting sail efficiency are its planform—primarily sail width versus sail height, expressed as an aspect ratio—and cross-sectional curvature or draft.

Drag variables

Spinnakers have traditionally been optimized to mobilize drag as a more important propulsive component than lift. As sailing craft are able to achieve higher speeds, whether on water, ice or land, the velocity made good (VMG) at a given course off the wind occurs at apparent wind angles that are increasingly further forward with speed. This suggests that the optimum VMG for a given course may be in a regime where a spinnaker may be providing significant lift.^[44] Traditional displacement sailboats may at times have optimum VMG courses close to downwind; for these the dominant force on sails is from drag.^[43] According to Kimball, C_D ≈ 4/3 for most sails with the apparent wind angle astern, so drag force on a downwind sail becomes substantially a function of area and wind speed, approximated as follows:^[5]

${\displaystyle D\approx {\tfrac {2}{3}}\rho {V_{A}}^{2}A}$

Measurement of pressure on the sail

Modern sail design and manufacture employs wind tunnel studies, full-scale experiments, and computer models as a basis for efficiently harnessing forces on sails.^[6]

Instruments for measuring air pressure effects in wind tunnel studies of sails include pitot tubes, which measure air speed and manometers, which measure static pressures and atmospheric pressure (static pressure in undisturbed flow). Researchers plot pressure across the windward and leeward sides of test sails along the chord and calculate pressure coefficients (static pressure difference over wind-induced dynamic pressure).^{[6][8][45][46]}

Research results describe airflow around the sail and in the boundary layer.^[6] Wilkinson, modelling the boundary layer in two dimensions, described nine regions around the sail:^[47]

1. Upper mast attached airflow.
2. Upper separation bubble.
3. Upper reattachment region.
4. Upper aerofoil attached flow region.
5. Trailing edge separation region.
6. Lower mast attached flow region.
7. Lower separation bubble.
8. Lower reattachment region.
9. Lower aerofoil attached flow region.

Analysis

Sail design differs from wing design in several respects, especially since on a sail air flow varies with wind and boat motion and sails are usually deformable airfoils, sometimes with a mast for a leading edge. Often simplifying assumptions are employed when making design calculations, including: a flat travel surface—water, ice or land, constant wind velocity and unchanging sail adjustment.^[47]

The analysis of the forces on sails takes into account the aerodynamic surface force, its centre of effort on a sail, its direction, and its variable distribution over the sail. Modern analysis employs fluid mechanics and aerodynamics airflow calculations for sail design and manufacture, using aeroelasticity models, which combine computational fluid dynamics and structural analysis.^[8] Secondary effects pertaining to turbulence and separation of the boundary layer are secondary factors.^[47] Computational limitations persist.^[48] Theoretical results require empirical confirmation with wind tunnel tests on scale models and full-scale testing of sails. Velocity prediction programs combine elements of hydrodynamic forces (mainly drag) and aerodynamic forces (lift and drag) to predict sailboat performance at various windspeed for all points of sail^[49]