An **airfoil** (American English) or **aerofoil** (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine.

- Overview
- Airfoil terminology
- Thin airfoil theory
- Derivation of thin airfoil theory
- See also
- References
- Notes
- Citations
- Sources
- Further reading
- External links

A solid body moving through a fluid produces an aerodynamic force. The component of this force perpendicular to the relative freestream velocity is called lift. The component parallel to the relative freestream velocity is called drag. An airfoil is a streamlined shape that is capable of generating significantly more lift than drag.^{ [1] } Airfoils designed for use at different speeds differ in their geometry: those for subsonic flight generally have a rounded leading edge, while those designed for supersonic flight tend to be slimmer with a sharp leading edge. All have a sharp trailing edge. Foils of similar function designed with water as the working fluid are called hydrofoils.

The lift on an airfoil is primarily the result of its angle of attack. When oriented at a suitable angle, the airfoil deflects the oncoming air (for fixed-wing aircraft, a downward force), resulting in a force on the airfoil in the direction opposite to the deflection. This force is known as aerodynamic force and can be resolved into two components: lift and drag. Most foil shapes require a positive angle of attack to generate lift, but cambered airfoils can generate lift at zero angle of attack. This "turning" of the air in the vicinity of the airfoil creates curved streamlines, resulting in lower pressure on one side and higher pressure on the other. This pressure difference is accompanied by a velocity difference, via Bernoulli's principle, so the resulting flowfield about the airfoil has a higher average velocity on the upper surface than on the lower surface. In some situations (e.g. inviscid potential flow) the lift force can be related directly to the average top/bottom velocity difference without computing the pressure by using the concept of circulation and the Kutta–Joukowski theorem.^{ [lower-alpha 1] }^{ [2] }^{ [3] }^{ [lower-alpha 2] }

A fixed-wing aircraft's wings, horizontal, and vertical stabilizers are built with airfoil-shaped cross sections, as are helicopter rotor blades. Airfoils are also found in propellers, fans, compressors and turbines. Sails are also airfoils, and the underwater surfaces of sailboats, such as the centerboard and keel, are similar in cross-section and operate on the same principles as airfoils. Swimming and flying creatures and even many plants and sessile organisms employ airfoils/hydrofoils: common examples being bird wings, the bodies of fish, and the shape of sand dollars. An airfoil-shaped wing can create downforce on an automobile or other motor vehicle, improving traction.

When the wind is obstructed by an object such as a flat plate, a building, or the deck of a bridge, the object will experience drag and also an aerodynamic force perpendicular to the wind. This does not mean the object qualifies as an airfoil. Airfoils are highly-efficient lifting shapes, able to generate more lift than similarly sized flat plates of the same area, and able to generate lift with significantly less drag. Airfoils have potential for use in the design of aircraft, propellers, rotor blades, wind turbines and other applications of aeronautical engineering.

A lift and drag curve obtained in wind tunnel testing is shown on the right. The curve represents an airfoil with a positive camber so some lift is produced at zero angle of attack. With increased angle of attack, lift increases in a roughly linear relation, called the *slope* of the lift curve. At about 18 degrees this airfoil stalls, and lift falls off quickly beyond that. The drop in lift can be explained by the action of the upper-surface boundary layer, which separates and greatly thickens over the upper surface at and past the stall angle. The thickened boundary layer's displacement thickness changes the airfoil's effective shape, in particular it reduces its effective camber, which modifies the overall flow field so as to reduce the circulation and the lift. The thicker boundary layer also causes a large increase in pressure drag, so that the overall drag increases sharply near and past the stall point.

Airfoil design is a major facet of aerodynamics. Various airfoils serve different flight regimes. Asymmetric airfoils can generate lift at zero angle of attack, while a symmetric airfoil may better suit frequent inverted flight as in an aerobatic airplane. In the region of the ailerons and near a wingtip a symmetric airfoil can be used to increase the range of angles of attack to avoid spin–stall. Thus a large range of angles can be used without boundary layer separation. Subsonic airfoils have a round leading edge, which is naturally insensitive to the angle of attack. The cross section is not strictly circular, however: the radius of curvature is increased before the wing achieves maximum thickness to minimize the chance of boundary layer separation. This elongates the wing and moves the point of maximum thickness back from the leading edge.

Supersonic airfoils are much more angular in shape and can have a very sharp leading edge, which is very sensitive to angle of attack. A supercritical airfoil has its maximum thickness close to the leading edge to have a lot of length to slowly shock the supersonic flow back to subsonic speeds. Generally such transonic airfoils and also the supersonic airfoils have a low camber to reduce drag divergence. Modern aircraft wings may have different airfoil sections along the wing span, each one optimized for the conditions in each section of the wing.

Movable high-lift devices, flaps and sometimes slats, are fitted to airfoils on almost every aircraft. A trailing edge flap acts similarly to an aileron; however, it, as opposed to an aileron, can be retracted partially into the wing if not used.

A **laminar flow wing** has a maximum thickness in the middle camber line. Analyzing the Navier–Stokes equations in the linear regime shows that a negative pressure gradient along the flow has the same effect as reducing the speed. So with the maximum camber in the middle, maintaining a laminar flow over a larger percentage of the wing at a higher cruising speed is possible. However, some surface contamination will disrupt the laminar flow, making it turbulent. For example, with rain on the wing, the flow will be turbulent. Under certain conditions, insect debris on the wing will cause the loss of small regions of laminar flow as well.^{ [4] } Before NASA's research in the 1970s and 1980s the aircraft design community understood from application attempts in the WW II era that laminar flow wing designs were not practical using common manufacturing tolerances and surface imperfections. That belief changed after new manufacturing methods were developed with composite materials (e.g. laminar-flow airfoils developed by F. X. Wortmann for use with wings made of fibre-reinforced plastic). Machined metal methods were also introduced. NASA's research in the 1980s revealed the practicality and usefulness of laminar flow wing designs and opened the way for laminar-flow applications on modern practical aircraft surfaces, from subsonic general aviation aircraft to transonic large transport aircraft, to supersonic designs.^{ [5] }

Schemes have been devised to define airfoils – an example is the NACA system. Various airfoil generation systems are also used. An example of a general purpose airfoil that finds wide application, and pre–dates the NACA system, is the Clark-Y. Today, airfoils can be designed for specific functions by the use of computer programs.

The various terms related to airfoils are defined below:^{ [6] }

- The
*suction surface*(a.k.a. upper surface) is generally associated with higher velocity and lower static pressure. - The
*pressure surface*(a.k.a. lower surface) has a comparatively higher static pressure than the suction surface. The pressure gradient between these two surfaces contributes to the lift force generated for a given airfoil.

The geometry of the airfoil is described with a variety of terms :

- The
*leading edge*is the point at the front of the airfoil that has maximum curvature (minimum radius).^{ [7] } - The
*trailing edge*is defined similarly as the point of maximum curvature at the rear of the airfoil. - The
*chord line*is the straight line connecting leading and trailing edges. The*chord length*, or simply**chord**, , is the length of the chord line. That is the**reference dimension**of the airfoil section.

The shape of the airfoil is defined using the following geometrical parameters:

- The
*mean camber line*or*mean line*is the locus of points midway between the upper and lower surfaces. Its shape depends on the thickness distribution along the chord; - The
*thickness*of an airfoil varies along the chord. It may be measured in either of two ways:- Thickness measured perpendicular to the camber line.
^{ [8] }^{ [9] }This is sometimes described as the "American convention";^{ [8] } - Thickness measured perpendicular to the chord line.
^{ [10] }This is sometimes described as the "British convention".

- Thickness measured perpendicular to the camber line.

Some important parameters to describe an airfoil's shape are its * camber * and its *thickness*. For example, an airfoil of the NACA 4-digit series such as the NACA 2415 (to be read as 2 – 4 – 15) describes an airfoil with a camber of 0.02 chord located at 0.40 chord, with 0.15 chord of maximum thickness.

Finally, important concepts used to describe the airfoil's behaviour when moving through a fluid are:

- The
*aerodynamic center*, which is the chord-wise length about which the pitching moment is independent of the lift coefficient and the angle of attack. - The
*center of pressure*, which is the chord-wise location about which the*pitching moment*is zero.

**Thin airfoil theory** is a simple theory of airfoils that relates angle of attack to lift for incompressible, inviscid flows. It was devised by German mathematician Max Munk and further refined by British aerodynamicist Hermann Glauert and others^{ [11] } in the 1920s. The theory idealizes the flow around an airfoil as two-dimensional flow around a thin airfoil. It can be imagined as addressing an airfoil of zero thickness and infinite wingspan.

Thin airfoil theory was particularly notable in its day because it provided a sound theoretical basis for the following important properties of airfoils in two-dimensional inviscid flow:^{ [12] }^{ [13] }

- on a symmetric airfoil, the center of pressure and aerodynamic center are coincident and lie exactly one quarter of the chord behind the leading edge.
- on a cambered airfoil, the aerodynamic center lies exactly one quarter of the chord behind the leading edge.
- the slope of the
*lift coefficient versus angle of attack*line is units per radian.

As a consequence of (3), the section lift coefficient of a symmetric airfoil of infinite wingspan is:

- where is the section lift coefficient,
- is the angle of attack in radians, measured relative to the chord line.

(The above expression is also applicable to a cambered airfoil where is the angle of attack measured relative to the zero-lift line instead of the chord line.)

Also as a consequence of (3), the section lift coefficient of a cambered airfoil of infinite wingspan is:

- where is the section lift coefficient when the angle of attack is zero.

Thin airfoil theory does not account for the stall of the airfoil, which usually occurs at an angle of attack between 10° and 15° for typical airfoils.^{ [14] } In the mid-late 2000s, however, a theory predicting the onset of leading-edge stall was proposed by Wallace J. Morris II in his doctoral thesis.^{ [15] } Morris's subsequent refinements contain the details on the current state of theoretical knowledge on the leading-edge stall phenomenon.^{ [16] }^{ [17] } Morris's theory predicts the critical angle of attack for leading-edge stall onset as the condition at which a global separation zone is predicted in the solution for the inner flow.^{ [18] } Morris's theory demonstrates that a subsonic flow about a thin airfoil can be described in terms of an outer region, around most of the airfoil chord, and an inner region, around the nose, that asymptotically match each other. As the flow in the outer region is dominated by classical thin airfoil theory, Morris's equations exhibit many components of thin airfoil theory.

The airfoil is modeled as a thin lifting mean-line (camber line). The mean-line, *y*(*x*), is considered to produce a distribution of vorticity along the line, *s*. By the Kutta condition, the vorticity is zero at the trailing edge. Since the airfoil is thin, *x* (chord position) can be used instead of *s*, and all angles can be approximated as small.

From the Biot–Savart law, this vorticity produces a flow field where

is the location where induced velocity is produced, is the location of the vortex element producing the velocity and is the chord length of the airfoil.

Since there is no flow normal to the curved surface of the airfoil, balances that from the component of main flow , which is locally normal to the plate – the main flow is locally inclined to the plate by an angle . That is:

This integral equation can be solved for , after replacing *x* by

as a Fourier series in with a modified lead term .

That is

(These terms are known as the Glauert integral).

The coefficients are given by

and

By the Kutta–Joukowski theorem, the total lift force *F* is proportional to

and its moment *M* about the leading edge to

The calculated Lift coefficient depends only on the first two terms of the Fourier series, as

The moment *M* about the leading edge depends only on and , as

The moment about the 1/4 chord point will thus be

From this it follows that the center of pressure is aft of the 'quarter-chord' point 0.25 *c*, by

The aerodynamic center, AC, is at the quarter-chord point. The AC is where the pitching moment *M*′ does not *vary* with a change in lift coefficient, i.e.,

In physics, the **cross section** is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha-particle will be deflected by a given angle during a collision with an atomic nucleus. Cross section is typically denoted *σ* (sigma) and is expressed in units of transverse area. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

In mathematics, the **trigonometric functions** are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

In fluid dynamics, the **drag coefficient** is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equation in which a lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area.

**Flight dynamics** is the science of air vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of gravity (cg), known as *pitch*, *roll* and *yaw*.

The **lift coefficient** (**C _{L}**) is a dimensionless coefficient that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. A lifting body is a foil or a complete foil-bearing body such as a fixed-wing aircraft.

**Parasitic drag** is drag that acts on an object when the object is moving through a fluid. In the case of aerodynamic drag, the fluid is the atmosphere. Parasitic drag is a combination of **form drag** and skin friction drag. Parasitic drag does not result from the generation of lift on the object, and hence it is considered parasitic.

In physics and engineering, in particular fluid dynamics, the **volumetric flow rate** is the volume of fluid which passes per unit time; usually it is represented by the symbol Q. The SI unit is cubic metres per second (m^{3}/s). Another unit used is standard cubic centimetres per minute (SCCM). In hydrometry, it is known as *discharge*.

**Blade element theory** (**BET**) is a mathematical process originally designed by William Froude (1878), David W. Taylor (1893) and Stefan Drzewiecki to determine the behavior of propellers. It involves breaking a blade down into several small parts then determining the forces on each of these small blade elements. These forces are then integrated along the entire blade and over one rotor revolution in order to obtain the forces and moments produced by the entire propeller or rotor. One of the key difficulties lies in modelling the induced velocity on the rotor disk. Because of this the blade element theory is often combined with momentum theory to provide additional relationships necessary to describe the induced velocity on the rotor disk, producing blade element momentum theory. At the most basic level of approximation a uniform induced velocity on the disk is assumed:

In aeronautics and aeronautical engineering, **camber** is the asymmetry between the two acting surfaces of an airfoil, with the top surface of a wing commonly being more convex. An airfoil that is not cambered is called a *symmetric airfoil*. The benefits of cambering were discovered and first utilized by George Cayley in the early 19th century.

The **NACA airfoils** are airfoil shapes for aircraft wings developed by the National Advisory Committee for Aeronautics (NACA). The shape of the NACA airfoils is described using a series of digits following the word "NACA". The parameters in the numerical code can be entered into equations to precisely generate the cross-section of the airfoil and calculate its properties.

In aerodynamics, the **pitching moment** on an airfoil is the moment produced by the aerodynamic force on the airfoil if that aerodynamic force is considered to be applied, not at the center of pressure, but at the aerodynamic center of the airfoil. The pitching moment on the wing of an airplane is part of the total moment that must be balanced using the lift on the horizontal stabilizer. More generally, a pitching moment is any moment acting on the pitch axis of a moving body.

The **Kutta–Joukowski theorem** is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil and any two-dimensional bodies including circular cylinders translating in a uniform fluid at a constant speed large enough so that the flow seen in the body-fixed frame is steady and unseparated. The theorem relates the lift generated by an airfoil to the speed of the airfoil through the fluid, the density of the fluid and the circulation around the airfoil. The circulation is defined as the line integral around a closed loop enclosing the airfoil of the component of the velocity of the fluid tangent to the loop. It is named after Martin Kutta and Nikolai Zhukovsky who first developed its key ideas in the early 20th century. Kutta–Joukowski theorem is an inviscid theory, but it is a good approximation for real viscous flow in typical aerodynamic applications.

The **Prandtl lifting-line theory** is a mathematical model that predicts lift distribution over a three-dimensional wing based on its geometry. It is also known as the **Lanchester–Prandtl wing theory.**

In fluid dynamics, the **Oseen equations** describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

In mathematics, **potential flow around a circular cylinder** is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and can be modeled as a potential flow. Unlike a real fluid, this solution indicates a net zero drag on the body, a result known as d'Alembert's paradox.

**Blade element momentum theory** is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade. Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor.

**Blade solidity** is an important design parameter for the axial flow impeller and is defined as the ratio of blade chord length to spacing.

In the theory of capillarity, **Bosanquet equation** is an improved modification of the simpler Lucas–Washburn theory for the motion of a liquid in a thin capillary tube or a porous material that can be approximated as a large collection of capillaries. In the Lucas–Washburn model, the inertia of the fluid is ignored, leading to the assumption that flow is continuous under constant viscous laminar Poiseuille flow conditions without considering effects of mass transport undergoing acceleration occurring at the start of flow and at points of changing internal capillary geometry. The Bosanquet equation is a differential equation that is second-order in the time derivative, similar to Newton's Second Law, and therefore takes into account the fluid inertia. Equations of motion, like the Washburn's equation, that attempt to explain a velocity as proportional to a driving force are often described with the term *Aristotelian mechanics*.

**Propeller theory** is the science governing the design of efficient propellers. A propeller is the most common propulsor on ships, and on small aircraft.

The **dynamic stall** is one of the hazardous phenomena on helicopter rotors, which can cause the onset of large torsional airloads and vibrations on the rotor blades. Unlike fixed-wing aircraft, of which the stall occurs at relatively low flight speed, the dynamic stall on a helicopter rotor emerges at high airspeeds or/and during manoeuvres with high load factors of helicopters, when the angle of attack(AOA) of blade elements varies intensively due to time-dependent blade flapping, cyclic pitch and wake inflow. For example, during forward flight at the velocity close to V_{NE}, *velocity, never exceed*, the advancing and retreating blades almost reach their operation limits whereas flows are still attached to the blade surfaces. That is, the advancing blades operate at high Mach numbers so low values of AOA is needed but shock-induced flow separation may happen, while the retreating blade operates at much lower Mach numbers but the high values of AoA result in the stall.

- ↑ "... the effect of the wing is to give the air stream a downward velocity component. The reaction force of the deflected air mass must then act on the wing to give it an equal and opposite upward component." (Halliday & Resnick 1988, p. 378)
- ↑ "... if a streamline is curved, there must be a pressure gradient across the streamline ..." (Babinsky 2003, pp. 497-503)

- ↑ Clancy 1975, §5.1.
- ↑ "If the body is shaped, moved, or inclined in such a way as to produce a net deflection or turning of the flow, the local velocity is changed in magnitude, direction, or both. Changing the velocity creates a net force on the body" "Lift from Flow Turning". NASA Glenn Research Center. Archived from the original on 5 July 2011. Retrieved 2011-06-29.
- ↑ Weltner & Ingelman-Sundberg n.d.
- ↑ Croom, C. C.; Holmes, B. J. (1985-04-01).
*Flight evaluation of an insect contamination protection system for laminar flow wings*. - ↑ Holmes, B. J.; Obara, C. J.; Yip, L. P. (1984-06-01). "Natural laminar flow experiments on modern airplane surfaces".
*NASA Technical Reports*. - ↑ Hurt, H. H., Jr. (January 1965) [1960].
*Aerodynamics for Naval Aviators*. U.S. Government Printing Office, Washington, D.C.: U.S. Navy, Aviation Training Division. pp. 21–22. NAVWEPS 00-80T-80. - ↑ Houghton et al. 2012, p. 18.
- 1 2 Houghton et al. 2012, p. 17.
- ↑ Phillips 2004, p. 27.
- ↑ Bertin & Cummings 2009, p. 199.
- ↑ Abbott & Von Doenhoff 1959, §4.2.
- ↑ Abbott & Von Doenhoff 1959, §4.3.
- ↑ Clancy 1975, §8.1 to §8.8.
- ↑ Aerospaceweb's information on Thin Airfoil Theory
- ↑ Morris 2009.
- ↑ Morris & Rusak 2013, pp. 439-472.
- ↑ Traub 2016, p. 9.
- ↑ Ramesh et al. 2014, pp. 500-538.

- Abbott, Ira Herbert; Von Doenhoff, Albert Edward (1959).
*Theory of Wing Sections, Including a Summary of Airfoil Data*. Dover. ISBN 978-0-486-60586-9. - Babinsky, Holger (November 2003). "How do wings work?" (PDF).
*Physics Education*.**38**(6): 497–503. Bibcode:2003PhyEd..38..497B. doi:10.1088/0031-9120/38/6/001. - Bertin, John J.; Cummings, Russel M. (2009).
*Aerodynamics for Engineers*(5th ed.). Pearson Prentice Hall. ISBN 978-0-13-227268-1. - Clancy, L.J. (1975).
*Aerodynamics*. London: Pitman. ISBN 0-273-01120-0. - Halliday, David; Resnick, Robert (1988).
*Fundamentals of Physics*(3rd ed.). John Wiley & Sons. - Houghton, E. L.; Carpenter, P. W.; Collicott, Steven H.; Valentine, Daniel (2012).
*Aerodynamics for Engineering Students*(6th ed.). Elsevier. ISBN 978-0-08-096633-5. - Morris, Wallace J., II (2009).
*A universal prediction of stall onset for airfoils at a wide range of Reynolds number flows*(PhD). Harvard University. Bibcode:2009PhDT.......146M. - Morris, Wallace J.; Rusak, Zvi (October 2013). "Stall onset on aerofoils at low to moderately high Reynolds number flows".
*Journal of Fluid Mechanics*.**733**: 439–472. Bibcode:2013JFM...733..439M. doi:10.1017/jfm.2013.440. ISSN 0022-1120. - Phillips, Warren F. (2004).
*Mechanics of Flight*. John Wiley & Sons. ISBN 978-0-471-33458-3. - Ramesh, Kiran; Gopalarathnam, Ashok; Granlund, Kenneth; Ol, Michael V.; Edwards, Jack R. (July 2014). "Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding".
*Journal of Fluid Mechanics*.**751**: 500–538. Bibcode:2014JFM...751..500R. doi:10.1017/jfm.2014.297. ISSN 0022-1120. - Traub, Lance W. (24 March 2016). "Semi-Empirical Prediction of Airfoil Hysteresis".
*Aerospace*.**3**(2): 9. doi: 10.3390/aerospace3020009 . - Weltner, Klaus; Ingelman-Sundberg, Martin (n.d.). "Physics of flight - revisited".
*angelfire.com*. Retrieved 25 April 2021.

- Anderson, John, D (2007).
*Fundamentals of Aerodynamics*. McGraw-Hill. - Batchelor, George. K (1967).
*An Introduction to Fluid Dynamics*. Cambridge UP. pp. 467–471.

Wikimedia Commons has media related to Airfoils and wing cross-sections . |

- UIUC Airfoil Coordinates Database
- Airfoil & Hydrofoil Reference Application
- FoilSim An airfoil simulator from NASA
- Joukowski Transform Interactive WebApp
- Desktopaero
- University of Sydney, Aerodynamics for Students
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