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{{short description|Cross-sectional shape of a wing, blade of a propeller, rotor, or turbine, or sail}}
[[File:Examples of Airfoils.svg|thumb|350px|Examples of airfoils in nature and in or on various vehicles. The dolphin flipper at bottom left obeys the same principles in a different fluid medium; it is an example of a [[hydrofoil]].]]
[[File:Flow over aerofoils.webm|thumb|Streamlines on an airfoil visualised with a smoke [[wind tunnel]]]]

An '''airfoil''' ([[American English]]) or '''aerofoil''' ([[British English]]) is a streamlined body that is capable of generating significantly more [[Lift (force)|lift]] than [[Drag (physics)|drag]].{{sfn|Clancy|1975|loc=§5.2}} Wings, sails and propeller blades are examples of airfoils. [[Foil (fluid mechanics)|Foils]] of similar function designed with water as the working fluid are called [[hydrofoil]]s.

When oriented at a suitable angle, a solid body moving through a [[fluid]] deflects the oncoming fluid (for fixed-wing aircraft, a downward force), resulting in a force on the airfoil in the direction opposite to the deflection.<ref>{{harvnb|Halliday|Resnick|1988|p=378}}: "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."</ref><ref>{{cite web |last=Hall |first=Nancy&nbsp;R. |title=Lift from Flow Turning |url=http://www.grc.nasa.gov/WWW/K-12/airplane/right2.html |url-status=live |archive-url=https://web.archive.org/web/20110705131653/http://www.grc.nasa.gov/WWW/K-12/airplane/right2.html |archive-date=5 July 2011 |access-date=2011-06-29 |publisher=NASA Glenn Research Center |quote=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.}}</ref> This force is known as [[aerodynamic force]] and can be resolved into two components: lift ([[perpendicular]] to the remote [[freestream velocity]]) and drag ([[Parallel (geometry)|parallel]] to the freestream velocity).

The lift on an airfoil is primarily the result of its [[angle of attack]]. Most foil shapes require a positive angle of attack to generate lift, but [[Camber (aerodynamics)|cambered]] airfoils can generate lift at zero angle of attack. Airfoils can be designed for use at different speeds by modifying their geometry: those for [[subsonic flight]] generally have a rounded [[leading edge]], while those designed for [[Supersonic aircraft|supersonic flight]] tend to be slimmer with a sharp leading edge. All have a sharp [[trailing edge]].<ref>“It has been known from the very beginning of flight that wings with a sharp trailing edge must be used in order to obtain a well-defined lift.” [[Richard von Mises|von Mises, Richard]] (1945), ''Theory of Flight'', Section VIII.2, p.179, Dover Publications Inc. ISBN 0-486-60541-8</ref>

The deflection of air by an airfoil generates a lower-pressure region above and behind itself. 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.{{sfn|Weltner|Ingelman-Sundberg|1999}} 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 (fluid dynamics)|circulation]] and the [[Kutta–Joukowski theorem]].<ref>{{harvnb|Babinsky|2003|pp=497–503}}: "If a streamline is curved, there must be a pressure gradient across the streamline."</ref>

==Overview==
[[File:Streamlines around a NACA 0012.svg|frame|Streamlines around a NACA 0012 airfoil at moderate angle of attack]]
[[File:Lift drag graph.JPG|thumb|300px|Lift and drag curves for a typical airfoil]]

The wings and stabilizers of [[fixed-wing aircraft]], as well as [[helicopter]] rotor blades, are built with airfoil-shaped cross sections. Airfoils are also found in propellers, [[Fan (mechanical)|fans]], [[Axial compressor|compressors]] and [[turbines]]. Sails are also airfoils, and the underwater surfaces of sailboats, such as the [[centerboard]], [[rudder]], and [[keel]], are similar in cross-section and operate on the same principles as airfoils. Swimming and flying creatures and even many plants and [[Sessility (zoology)|sessile]] organisms employ airfoils/hydrofoils, common examples being bird wings, the bodies of fish, and the shape of [[sand dollar]]s. An airfoil-shaped wing can create [[downforce]] on an [[automobile]] or other motor vehicle, improving [[traction (engineering)|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 (physics)|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 are used 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 (aerodynamics)|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 [[Stall (fluid dynamics)|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 (aerodynamics)|camber]], which modifies the overall flow field so as to reduce the [[Circulation (fluid dynamics)|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 (flight)|spin]]–[[stall (flight)|stall]]. Thus a large range of angles can be used without [[Flow separation|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 (mathematics)|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 Mach number|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, [[flap (aircraft)|flaps]] and sometimes [[Leading edge slats|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.

For many wings, the [[boundary layer]] of the air flow close to the upper surface rapidly becomes turbulent past the maximum thickness point, which increases the skin friction drag. A '''laminar flow wing''' moves the maximum thickness point well back along the chord from a typical 25% chord position from to 60% from the leading edge or more.<ref name=":0">{{Cite book |last1=Houghton |first1=Edward L. |title=Aerodynamics for engineering students |last2=Carruthers |first2=N. B. |date=1982 |publisher=Arnold |isbn=978-0-7131-3433-9 |edition=3. |location=London}}</ref> This maintains smooth laminar flow over a larger percentage of the wing and significantly reduces drag. However, surface contamination will disrupt the boundary layer, making it turbulent. Insects, for example, impacting and sticking onto the wing will cause the loss of wedge shaped regions of laminar flow across the wing's surface. This is a particular problem for aircraft with high take off speeds, since many insects are found near the ground making it unlikely that laminar flow can be sustained into the flight.<ref>{{Cite book|url = https://ntrs.nasa.gov/search.jsp?R=19850067951|title = Flight evaluation of an insect contamination protection system for laminar flow wings|last1 = Croom|first1 = C. C.|last2 = Holmes|first2 = B. J.|date = 1985-04-01}}</ref> [[Glider (sailplane)|Gliders]] have seen widespread uptake of laminar flow airfoils due to their low speeds and need for low drag aerodynamic structures.<ref name=":0" />

Before NASA's research in the 1970s and 1980s laminar flow wing designs were not practical using common manufacturing tolerances and surface imperfection, until new manufacturing methods were developed with machined metal and composite materials (e.g. laminar-flow airfoils developed by Professor [[Franz Xaver Wortmann|Franz Wortmann]] for use with wings made of [[fibre-reinforced plastic]]). 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.<ref>{{Cite journal|last1 = Holmes|first1 = B. J.|last2 = Obara|first2 = C. J.|last3 = Yip|first3 = L. P.|date = 1984-06-01|journal=NASA Technical Reports|title = Natural laminar flow experiments on modern airplane surfaces| page=18592 | bibcode=1984ntrs.rept18592H |url = https://ntrs.nasa.gov/search.jsp?R=19840018592}}</ref> 

Schemes have been devised to define airfoils – an example is the [[NACA airfoil|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.

== Airfoil terminology ==
[[File:Wing profile nomenclature.svg|500px|thumb|Airfoil nomenclature]]

The various terms related to airfoils are defined below:<ref>{{cite book
 | last = Hurt 
 | first = H. H. Jr.
 | orig-date = 1960
 | title = Aerodynamics for Naval Aviators
 |date=January 1965
 | publisher = U.S. Navy, Aviation Training Division
 | location = U.S. Government Printing Office, Washington, D.C.
 | id = NAVWEPS 00-80T-80
 | pages = 21–22
 }}</ref>

*The ''suction surface'' (typically upper surface) is generally associated with higher velocity and lower static pressure.
*The ''pressure surface'' (typically 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).{{sfn|Houghton|Carpenter|Collicott|Valentine|2012|p=18}}
*The ''[[trailing edge]]'' is the point on the airfoil most remote from the leading edge. The angle between the upper and lower surfaces at the trailing edge is the ''trailing edge angle''.
*The ''[[Chord (aircraft)|chord line]]'' is the straight line connecting leading and trailing edges. The ''chord length'', or simply '''chord''', <math>c</math>, is the length of the chord line. That is the '''reference dimension''' of the airfoil section.

[[File:Airfoil thickness definition.svg|thumb|Different definitions of airfoil thickness]]
[[Image:PSU-90-125.PNG|thumb|An airfoil designed for [[wingtip device|winglet]]s (PSU 90-125WL)]]
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.{{sfn|Houghton|Carpenter|Collicott|Valentine|2012|p=17}}{{sfn|Phillips|2004|p=27}} This is sometimes described as the "American convention";{{sfn|Houghton|Carpenter|Collicott|Valentine|2012|p=17}}
** Thickness measured perpendicular to the chord line.{{sfn|Bertin|Cummings|2009|p=199}} This is sometimes described as the "British convention".

Some important parameters to describe an airfoil's shape are its ''[[Camber (aerodynamics)|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 location about which the pitching moment is independent of the lift coefficient and the angle of attack.
*The ''[[center of pressure (fluid mechanics)|center of pressure]]'', which is the chord-wise location about which the ''[[pitching moment]]'' is momentarily zero. On a cambered airfoil, the center of pressure is not a fixed location as it moves in response to changes in angle of attack and lift coefficient.

In two-dimensional flow around a uniform wing of infinite span, the slope of the [[Lift coefficient#Section lift coefficient|lift curve]] is determined primarily by the ''trailing edge angle''. The slope is greatest if the angle is zero; and decreases as the angle increases.<ref>Lyons, D.J., and Bisgood, P.L., (Jan 1945). ''An analysis of the lift slope of aerofoils of small aspect ratio''. Reports and Memoranda of the Aeronautical Research Council of Great Britain No 2308</ref><ref>Abbott, I.H., and Von Doenhoff, A.E. (1949) ''Theory of Wing Sections'', section 7.4(b)</ref> For a wing of finite span, the [[aspect ratio (aeronautics)|aspect ratio]] of the wing also significantly influences the slope of the curve. As aspect ratio decreases, the slope also decreases.<ref>Abbott, I.H., and Von Doenhoff, A.E. (1949) ''Theory of Wing Sections'', section 1.3</ref>

==Thin airfoil theory==
<!-- [[Thin airfoil theory]] and [[Thin-airfoil theory]] redirect here -->
[[Image:denney.kitfox.g-foxc.arp.jpg|thumb|An airfoil section is displayed at the tip of this [[Denney Kitfox]] aircraft, built in 1991.]]
[[Image:Helikopter forgószárnyának keresztmetszete 2.jpg|thumb|Airfoil of a [[Kamov Ka-26]] helicopter's lower rotor blade]]
'''Thin airfoil theory''' is a simple theory of airfoils that relates [[angle of attack]] to lift for incompressible, [[inviscid flow]]s. It was devised by German mathematician [[Max Munk]] and further refined by British aerodynamicist [[Hermann Glauert]] and others{{sfn|Abbott|Von Doenhoff|1959|loc=§4.2}} 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:{{sfn|Abbott|Von Doenhoff|1959|loc=§4.3}}{{sfn|Clancy|1975|loc=§8.1 to §8.8}}
# on a symmetric airfoil, the [[center of pressure (fluid mechanics)|center of pressure]] and [[aerodynamic center]] are coincident and lie exactly one quarter of the [[Chord (aircraft)|chord]] behind the leading edge.
# on a [[Camber (aerodynamics)|cambered]] airfoil, the aerodynamic center lies exactly one quarter of the chord behind the leading edge, but the position of the center of pressure moves when the angle of attack changes.
# the slope of the ''lift coefficient versus angle of attack'' line is <math>2 \pi\!</math> units per radian.

As a consequence of (3), the section [[lift coefficient]] of a thin symmetric airfoil of infinite wingspan is:
:<math> \ c_l = 2\pi \alpha</math>
:where <math>c_l\!</math> is the section lift coefficient,
:<math>\alpha\!</math> is the [[angle of attack]] in radians, measured relative to the [[Chord (aircraft)|chord]] line.
(The above expression is also applicable to a cambered airfoil where <math>\alpha\!</math>  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:
:<math> \ c_l = c_{l_0} + 2\pi\alpha</math>
:where <math> \ c_{l_0}</math> is the section lift coefficient when the angle of attack is zero.

Thin airfoil theory assumes the air is an [[inviscid fluid]] so does not account for the [[Stall (flight)|stall]] of the airfoil, which usually occurs at an angle of attack between 10° and 15° for typical airfoils.<ref>{{harvnb|Scott|2003}}: "The equation can only be used for aircraft with medium to large aspect ratio wings and only up to the stall angle, which is usually between 10° and 15° for typical aircraft configurations."</ref> 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.{{sfn|Morris|2009|p=}}  Morris's subsequent refinements contain the details on the current state of theoretical knowledge on the leading-edge stall phenomenon.{{sfn|Morris|Rusak|2013|pp=439-472}}{{sfn|Traub|2016|p=9}} 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.<ref>{{Cite journal|last1=Ramesh|first1=Kiran|last2=Gopalarathnam|first2=Ashok|last3=Granlund|first3=Kenneth|last4=Ol|first4=Michael V.|last5=Edwards|first5=Jack R.|date=July 2014|title=Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding|journal=Journal of Fluid Mechanics|volume=751|pages=500–538|doi=10.1017/jfm.2014.297|issn=0022-1120|bibcode = 2014JFM...751..500R |s2cid=121962230}}</ref> 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.

=== Derivation ===
[[File:Aerofoils for different aeroplanes.svg|thumb|upright=1.2|From top to bottom:{{Bulleted list|Laminar flow airfoil for a RC [[park flyer]]|Laminar flow airfoil for a RC [[air racing|pylon racer]]|Laminar flow airfoil for a crewed propeller aircraft|Laminar flow at a jet airliner airfoil|Stable airfoil used for flying wings|Aft loaded airfoil allowing for a large main spar and late stall|Transonic supercritical airfoil|Supersonic leading edge airfoil}}
{{legend|black|laminar flow}}
{{Legend|red|turbulent flow}}
{{Legend|#939393|subsonic stream}}
{{Legend|#2ba3ff|supersonic flow volume}}]]

In thin airfoil theory, the width of the (2D) airfoil is assumed negligible, and the airfoil itself replaced with a 1D blade along its camber line, oriented at the [[angle of attack]] {{Math|&alpha;}}.  Let the position along the blade be {{Mvar|x}}, ranging from {{Math|0}} at the wing's front to {{Mvar|c}} at the trailing edge; the camber of the airfoil, {{Math|{{frac|''dy''|''dx''}}}}, is assumed sufficiently small that one need not distinguish between {{Mvar|x}} and position relative to the fuselage.<ref>{{harvnb|Auld|Srinivas|1995}}: "A simple solution for general two-dimensional aerofoil sections can be obtained by neglecting thickness effects and using a mean-line only section model....  This also means small changes in position are equivalent so that {{Math|''ds'' &approx; ''dx''}}."</ref>{{sfn|Batchelor|1967|p=467}}

The flow across the airfoil generates a circulation around the blade, which can be modeled as a [[vortex sheet]] of position-varying strength {{Math|&gamma;(''x'')}}.  The [[Kutta condition]] implies that {{Math|1=&gamma;(''c'')=0}}, but the strength is singular at the bladefront, with {{Math|&gamma;(''x'')&propto;{{frac|1|{{sqrt|''x''}}}}}} for {{Math|''x'' &approx; 0}}.{{sfn|Batchelor|1967|p=467-9}}  If the [[Freestream|main flow]] {{Mvar|V}} has [[density]] {{Math|&rho;}}, then the [[Kutta–Joukowski theorem]] gives that the total lift force {{Mvar|F}} is proportional to{{sfn|Auld|Srinivas|1995}}<ref>{{cite book|title=Elementary Fluid Dynamics|last=Acheson|first=D.&nbsp;J.|series=Oxford Applied Mathematics and Computing Science|publisher=Clarendon Press|location=Oxford|date=1990|publication-date=2009|pages=140–141, 143–145}}</ref> <math display="block">\rho V \int_{0}^{c} \gamma (x) \, dx</math>and its moment {{Mvar|M}} about the leading edge proportional to{{sfn|Batchelor|1967|p=467-9}} <math display="block">\rho V \int_{0}^{c} x \; \gamma (x) \, dx.</math>

From the [[Biot–Savart law]], the vorticity {{Math|&gamma;(''x'')}} produces a flow field <math display=block>w(x) = \frac{1}{2\pi} \int_{0}^{c} \frac{\gamma (x')}{x-x'}\, dx'\text{,}</math> oriented [[Normal (geometry)|normal]] to the airfoil at {{Mvar|x}}.  Since the airfoil is an [[Hydraulic conductivity|impermeable surface]], the flow <math>w(x)</math> must balance an inverse flow from {{Mvar|V}}.  By the [[small-angle approximation]], {{Mvar|V}} is inclined at angle {{Math|&alpha;-{{frac|''dy''|''dx''}}}} relative to the blade at position {{Mvar|x}}, and the normal component is correspondingly {{Math|(&alpha;-{{frac|''dy''|''dx''}})''V''}}.  Thus, {{Math|&gamma;(''x'')}} must satisfy the [[convolution]] [[Integral equation|equation]] <math display=block>\left(\alpha - \frac{dy}{dx}\right)V = -w(x) = -\frac{1}{2 \pi} \int_{0}^{c} \frac {\gamma(x')}{x-x'} \,dx'\text{,}</math> which uniquely determines it in terms of known quantities.{{sfn|Auld|Srinivas|1995}}{{sfn|Batchelor|1967|p=467-468}}

An explicit solution can be obtained through first the [[change of variables]]<math display="block">x = c \cdot \frac{1 + \cos(\theta)}{2},</math>and then expanding both {{Math|{{frac|''dy''|''dx''}}}} and {{Math|&gamma;(''x'')}} as a nondimensionalized [[Fourier series]] in {{Math|&theta;}} with a modified lead term:<math display="block">\begin{align}
&\frac{dy}{dx}=A_0+A_1\cos(\theta)+A_2\cos(2\theta)+\dots \\
&\gamma(x)=2(\alpha+A_0)\left(\frac{\sin\theta}{1+\cos\theta}\right)+2A_1\sin(\theta)+2A_2\sin(2\theta)+\dots\text{.}
\end{align}</math>  The resulting lift and moment depend on only the first few terms of this series.{{sfn|Batchelor|1967|p=469-470}}

The [[lift coefficient]] satisfies <math display="block">C_L = 2\pi\left(\alpha+A_0+\frac{A_1}{2}\right)=2\pi\alpha+2\int_0^\pi{\frac{dy}{dx}\cdot(1+\cos\theta)\,d\theta}</math> and the moment coefficient{{sfn|Batchelor|1967|p=470}}  <math display=block>C_M = -\frac{\pi}{2}\left(\alpha+A_0+A_1-\frac{A_2}{2}\right)=-\frac{\pi}{2}\alpha-\int_0^\pi{\frac{dy}{dx}\cdot\cos(\theta)(1+\cos\theta)\,d\theta}\text{.}</math>  The moment about the 1/4 chord point will thus be <math display=block>C_M(1/4c) = - \pi /4 (A_1 - A_2)\text{.}</math>  From this it follows that the [[center of pressure (fluid mechanics)|center of pressure]] is aft of the 'quarter-chord' point {{Math|0.25''c''}}, by <math display=block>\Delta x /c = \pi /4 ((A_1-A_2)/C_L)\text{.}</math>  The [[aerodynamic center]] is the position at which the pitching moment {{Math|''M''{{prime}}}} does not ''vary'' with a change in lift coefficient:{{sfn|Auld|Srinivas|1995}} <math display=block>\frac{ \partial (C_{M'}) }{ \partial (C_L)} = 0\text{.}</math>
Thin-airfoil theory shows that, in two-dimensional inviscid flow, the aerodynamic center is at the quarter-chord position.

==See also==
* [[Circulation control wing]]
* [[Hydrofoil]]
* [[Kline–Fogleman airfoil]]
* [[Küssner effect]]
* [[Parafoil]]
* [[Wing configuration]]

==References==

===Citations===
{{Reflist}}

===General Sources===
{{refbegin|30em|indent=yes}}
*{{cite book|last1=Abbott|first1=Ira Herbert |author-link1=Ira H. Abbott|last2=Von Doenhoff|first2=Albert Edward |title=Theory of Wing Sections, Including a Summary of Airfoil Data|url=https://books.google.com/books?id=T-22wxF60tMC|year=1959|publisher=Dover |isbn=978-0-486-60586-9}}
*{{cite book |last1=Auld |first1=Douglass |url=http://www.aerodynamics4students.com/ |title=Aerodynamics for Students |last2=Srinivas |publisher=University of Sydney |year=1995 |chapter=2-D Thin Aerofoil Theory |chapter-url=http://www.aerodynamics4students.com/subsonic-aerofoil-and-wing-theory/2-d-thin-aerofoil-theory.php}}
*{{cite journal
  | journal=Physics Education
  | volume=38
  | issue=6
  | pages=497–503
  | first=Holger
  | last=Babinsky
  | date=November 2003
  | url=http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf 
  | title=How do wings work?
  | doi=10.1088/0031-9120/38/6/001
  | bibcode=2003PhyEd..38..497B
  | s2cid=1657792
 }}
*{{cite book |last=Batchelor |first=George. K |author-link=George Batchelor |year=1967 |title=An Introduction to Fluid Dynamics |publisher=Cambridge UP |pages=467–471}}
*{{cite book | title=Aerodynamics for Engineers |last1=Bertin|first1= John J. |last2=Cummings|first2= Russel M. | edition=5th | year=2009 | publisher=Pearson Prentice Hall | isbn=978-0-13-227268-1}}
*{{cite book|last=Clancy|first= L.J. |date=1975|title=Aerodynamics|publisher=Pitman |location= London|isbn=0-273-01120-0}}
*{{cite book
 | first1=David
 | last1=Halliday
 | first2=Robert
 | last2=Resnick
 | title=Fundamentals of Physics |edition=3rd
 | publisher=John Wiley & Sons
 | date=1988|url=https://books.google.com/books?id=f04TzQEACAAJ
}}
*{{cite book|last1=Houghton|first1=E. L. |last2=Carpenter|first2=P. W. |last3=Collicott|first3=Steven H. |first4=Daniel |last4=Valentine|title=Aerodynamics for Engineering Students|url=https://books.google.com/books?id=rA6mIXaF69kC&pg=PA18|edition=6th|date= 2012|publisher=Elsevier|isbn=978-0-08-096633-5}}
*{{Cite thesis|type=PhD|last=Morris|first=Wallace J. II|date=2009|title=A universal prediction of stall onset for airfoils at a wide range of Reynolds number flows|bibcode=2009PhDT.......146M|publisher=Harvard University}}
*{{Cite journal|last1=Morris|first1=Wallace J.|last2=Rusak|first2=Zvi|date=October 2013|title=Stall onset on aerofoils at low to moderately high Reynolds number flows|journal=Journal of Fluid Mechanics|volume=733|pages=439–472|doi=10.1017/jfm.2013.440|issn=0022-1120|bibcode = 2013JFM...733..439M |s2cid=122817884}}
*{{cite book|last=Phillips|first=Warren F. |title=Mechanics of Flight|url=https://books.google.com/books?id=6-_iGbJHM-8C&pg=PA27|year=2004|publisher=John Wiley & Sons|isbn=978-0-471-33458-3}}
*{{cite web|url=http://www.aerospaceweb.org/question/aerodynamics/q0136.shtml|last=Scott|first=Jeff|date=10 August 2003|department=Ask a Rocket Scientist: Aerodynamics|title=Question #136: Lift Coefficient & Thin Airfoil Theory|publisher=Aerospaceweb.org}}
*{{Cite journal|last=Traub|first=Lance W.|date=24 March 2016|title=Semi-Empirical Prediction of Airfoil Hysteresis|journal =Aerospace|volume=3|issue=2|page=9|doi=10.3390/aerospace3020009|bibcode=2016Aeros...3....9T |doi-access=free}}
*{{Cite web<!--Not actually published in a journal (Google Scholar only shows it as "submitted" to a garbled journal name)-->|title=Physics of flight - revisited |last1=Weltner |first1=Klaus |last2=Ingelman-Sundberg |first2=Martin |date=1999 |access-date=25 April 2021 |archive-url=https://web.archive.org/web/20110929025540/http://user.uni-frankfurt.de/~weltner/Flight/PHYSIC4.htm |archive-date=29 September 2011 |url=http://user.uni-frankfurt.de/~weltner/Flight/PHYSIC4.htm }}
{{refend}}

===Further reading===
* {{cite book |last=Anderson |first=John, D |year=2007 |title=Fundamentals of Aerodynamics |publisher=McGraw-Hill}}
* Ali Kamranpay, Alireza Mehrabadi. [https://engineeringjournals.stmjournals.in/index.php/JoMA/article/view/3639 Numerical Analysis of NACA Airfoil 0012 at Different Attack Angles and Obtaining its Aerodynamic Coefficients.] Journal of Mechatronics and Automation. 2019; 6(3): 8–16p.
* {{cite journal |last1=Bearman |first1=Matt |title=Going with the Flow? Britain's Contribution to Laminar-Flow Research, 1930–1947 |journal=The Aviation Historian |date=2019 |issue=29 |pages=74–87 |issn=2051-1930}}

==External links==
{{Commons category}}
* [https://m-selig.ae.illinois.edu/ads/coord_database.html UIUC Airfoil Coordinates Database]
* [https://web.archive.org/web/20110109230632/http://www.skias-engineering.gr/index.php?option=com_content&task=view&id=19&Itemid=47 Airfoil & Hydrofoil Reference Application]
* [https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/foilsimstudent/ FoilSim] An airfoil simulator from NASA
* [http://airfoil.dimanov.com/ Airfoil Playground - Interactive WebApp]
* [https://web.archive.org/web/20070222145228/http://www.desktopaero.com/appliedaero/airfoils1/tatderivation.html Desktopaero]
* [https://m.youtube.com/watch?v=UqBmdZ-BNig Airflow across a wing (University of Cambridge)]
* [https://www.dreesecode.com DesignFOIL] An airfoil generation & analysis tool that no longer requires registration.

{{Machines}}
{{Authority control}}

[[Category:Aerodynamics]]
[[Category:Aircraft wing design]]