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The '''lift coefficient''' (''C''<sub>L</sub>, ''C''<sub>a</sub> or ''C''<sub>z</sub>) is a [[dimensionless]] coefficient that relates the [[Lift (force)|lift]] generated by a lifting body to the [[density]] of the fluid around the body, its velocity and an associated reference area. A lifting body is a [[foil (fluid mechanics)|foil]] or a complete foil-bearing body such as a [[fixed-wing aircraft]]. ''C''<sub>L</sub> is a function of the angle of the body to the flow, its [[Reynold number]] and its [[Mach number]]. The lift coefficient ''c''<sub>l</sub> is refers to the dynamic lift characteristics of a [[two-dimensional]] foil section, with the reference area replaced by the foil [[Chord (aircraft)|chord]].<ref name=Clancy>{{cite book|last=Clancy|first=L. J.|title=Aerodynamics|year=1975|publisher=John Wiley & Sons|location=New York|at=Sections 4.15 & 5.4}}</ref><ref name=TWS1.2>[[Ira H. Abbott|Abbott, Ira H.]], and Von Doenhoff, Albert E.: ''Theory of Wing Sections''. Section 1.2</ref> 
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==Definition of the lift coefficient ''C''<sub>L</sub>==
 
The lift coefficient ''C''<sub>L</sub> is defined by<ref name=TWS1.2/><ref>Clancy, L. J.: ''Aerodynamics''. Section 4.15</ref>
 
:<math>C_\mathrm L = {\frac{L}{\frac{1}{2}\rho v^2S}} = {\frac{2 L}{\rho v^2S}} = \frac{L}{q S}</math> ,
where
<math>L\,</math> is the [[Lift (force)|lift force]], <math>\rho\,</math> is [[fluid]] [[density]], <math>v\,</math> is [[true airspeed]], <math>S\,</math> is [[planform]] area and <math>q\,</math> is the fluid [[dynamic pressure]].
 
The lift coefficient can be approximated using the [[lifting-line theory]],<ref>Clancy, L. J.: ''Aerodynamics''. Section 8.11</ref> numerically calculated  or measured in a [[wind tunnel]] test of a complete aircraft configuration.
 
== Section lift coefficient ==
[[Image:Lift curve.svg|thumb|300px|right|A typical curve showing section lift coefficient versus angle of attack for a cambered airfoil]]
Lift coefficient may also be used as a characteristic of a particular shape (or cross-section) of an [[airfoil]]. In this application it is called the '''section lift coefficient''' <math>c_\text{l}</math>. It is common to show, for a particular airfoil section, the relationship between section lift coefficient and [[angle of attack]].<ref>Abbott, Ira H., and Von Doenhoff, Albert E.: ''Theory of Wing Sections''. Appendix IV</ref> It is also useful to show the relationship between section lift coefficient and [[drag coefficient]].
 
The section lift coefficient is based on two-dimensional flow over a wing of infinite span and non-varying cross-section so the lift is independent of spanwise effects and is defined in terms of <math>l</math>, the lift force per unit span of the wing.  The definition becomes
 
:<math>c_\text{l} = \frac{l}{\frac{1}{2}\rho v^2c},</math>
 
where <math>c\,</math> is the [[chord (aircraft)|chord]] of the airfoil.
 
For a given angle of attack, ''c''<sub>l</sub> can be calculated approximately using the [[thin airfoil theory]],<ref>Clancy, L. J.: ''Aerodynamics''. Section 8.2</ref> calculated numerically or determined from wind tunnel tests on a finite-length test piece, with end-plates designed to ameliorate the three-dimensional effects. Plots of ''c''<sub>l</sub> versus angle of attack  show the same general shape for all [[airfoil]]s, but the particular numbers will vary. They show an almost linear increase in lift coefficient with increasing [[angle of attack]] with a gradient known as the lift slope. For a thin airfoil of any shape the lift slope is π<sup>2</sup>/90 ≃ 0.11 per degree. At higher angles a maximum point is reached, after which the lift coefficient reduces. The angle at which maximum lift coefficient occurs is the [[Stall (flight)|stall]] angle of the airfoil.
 
Symmetric airfoils necessarily have plots of c<sub>l</sub> versus angle of attack symmetric about the ''c''<sub>l</sub> axis, but for any airfoil with positive [[camber (aerodynamics)|camber]], i.e. asymmetrical, convex from above, there is still a small but positive lift coefficient with angles of attack less than zero. That is, the angle at which ''c''<sub>l</sub> = 0 is negative. On such airfoils at zero angle of attack the pressures on the upper surface are lower than on the lower surface.
 
== See also ==
* [[Lift-to-drag ratio]]
* [[Drag coefficient]]
* [[Foil (fluid mechanics)]]
* [[Pitching moment]]
* [[Circulation control wing]]
* [[Zero lift axis]]
 
== Notes ==
{{reflist}}
 
== References ==
* Clancy, L. J. (1975): ''Aerodynamics''. Pitman Publishing Limited, London, ISBN 0-273-01120-0
* Abbott, Ira H., and Von Doenhoff, Albert E. (1959): ''Theory of Wing Sections''. Dover Publications Inc., New York, Standard Book Number 486-60586-8
 
[[Category:Aerodynamics]]
[[Category:Aircraft wing design]]
[[Category:Dimensionless numbers of fluid mechanics]]

Latest revision as of 22:22, 8 January 2015

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