Coefficient Of Lift
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In fluid dynamics, the lift coefficient () is a dimensionless quantity that relates the
lift Lift or LIFT may refer to: Physical devices * Elevator, or lift, a device used for raising and lowering people or goods ** Paternoster lift, a type of lift using a continuous chain of cars which do not stop ** Patient lift, or Hoyer lift, mobil ...
generated by a
lifting body A lifting body is a fixed-wing aircraft or spacecraft configuration in which the body itself produces lift. In contrast to a flying wing, which is a wing with minimal or no conventional fuselage, a lifting body can be thought of as a fuselage ...
to the fluid density around the body, the
fluid velocity In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
and an associated reference area. A lifting body is a
foil Foil may refer to: Materials * Foil (metal), a quite thin sheet of metal, usually manufactured with a rolling mill machine * Metal leaf, a very thin sheet of decorative metal * Aluminium foil, a type of wrapping for food * Tin foil, metal foil ...
or a complete foil-bearing body such as a fixed-wing aircraft. is a function of the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
of the body to the flow, its Reynolds number and its Mach number. The section lift coefficient refers to the dynamic lift characteristics of a
two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
foil section, with the reference area replaced by the foil chord. Abbott, Ira H., and Doenhoff, Albert E. von: ''Theory of Wing Sections''. Section 1.2


Definitions

The lift coefficient ''C''L is defined by :C_\mathrm L \equiv \frac = = , where L\, is the
lift force A fluid flowing around an object exerts a force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the force parallel to the flow directi ...
, S\, is the relevant surface area and q\, is the fluid
dynamic pressure In fluid dynamics, dynamic pressure (denoted by or and sometimes called velocity pressure) is the quantity defined by:Clancy, L.J., ''Aerodynamics'', Section 3.5 :q = \frac\rho\, u^2 where (in SI units): * is the dynamic pressure in pascals ( ...
, in turn linked to the fluid
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
\rho\,, and to the flow speed u\,. The choice of the reference surface should be specified since it is arbitrary. For example, for cylindric profiles (the 3D extrusion of an airfoil in the spanwise direction) it is always oriented in the spanwise direction, but while in aerodynamics and thin airfoil theory the second axis generating the surface is commonly the chordwise direction: :S_ \equiv c \, s resulting in a coefficient: :C_ \equiv \frac, while for thick airfoils and in marine dynamics, the second axis is sometimes taken in the thickness direction: :S_ = t \, s resulting in a different coefficient: :C_ \equiv \frac The ratio between these two coefficients is the thickness ratio: :C_ \equiv \frac c t C_ The lift coefficient can be approximated using the
lifting-line theory The Prandtl lifting-line theory is a mathematical model in aerodynamics that predicts lift distribution over a three-dimensional wing based on its geometry. It is also known as the Lanchester–Prandtl wing theory. The theory was expressed indepen ...
, numerically calculated or measured in a
wind tunnel Wind tunnels are large tubes with air blowing through them which are used to replicate the interaction between air and an object flying through the air or moving along the ground. Researchers use wind tunnels to learn more about how an aircraft ...
test of a complete aircraft configuration.


Section lift coefficient

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 c_\text. It is common to show, for a particular airfoil section, the relationship between section lift coefficient and angle of attack. It is also useful to show the relationship between section lift coefficient and
drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) 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 e ...
. 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 l, the lift force per unit span of the wing. The definition becomes :c_\text = \frac, where L is the reference length that should always be specified: in aerodynamics and airfoil theory usually the airfoil chord c\, is chosen, while in marine dynamics and for struts usually the thickness t\, is chosen. Note this is directly analogous to the drag coefficient since the chord can be interpreted as the "area per unit span". For a given angle of attack, ''c''l can be calculated approximately using the
thin airfoil theory 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. ...
, 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''l versus angle of attack show the same general shape for all airfoils, 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 π2/90 ≃ 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 angle of the airfoil, which is approximately 10 to 15 degrees on a typical airfoil. The stall angle for a given profile is also increasing with increasing values of the Reynolds number, at higher speeds indeed the flow tends to stay attached to the profile for longer delaying the stall condition. For this reason sometimes
wind tunnel Wind tunnels are large tubes with air blowing through them which are used to replicate the interaction between air and an object flying through the air or moving along the ground. Researchers use wind tunnels to learn more about how an aircraft ...
testing performed at lower Reynolds numbers than the simulated real life condition can sometimes give conservative feedback overestimating the profiles stall. Symmetric airfoils necessarily have plots of cl versus angle of attack symmetric about the ''c''l axis, but for any airfoil with positive
camber Camber may refer to a variety of curvatures and angles: * Camber angle, the angle made by the wheels of a vehicle * Camber beam, an upward curvature of a joist to compensate for load deflection due in buildings * Camber thrust in bike technology * ...
, 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''l = 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 In aerodynamics, the lift-to-drag ratio (or L/D ratio) is the lift generated by an aerodynamic body such as an aerofoil or aircraft, divided by the aerodynamic drag caused by moving through air. It describes the aerodynamic efficiency under gi ...
*
Drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) 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 e ...
*
Foil (fluid mechanics) A foil is a solid object with a shape such that when placed in a moving fluid at a suitable angle of attack the lift (force generated perpendicular to the fluid flow) is substantially larger than the drag (force generated parallel to the fluid flo ...
* Pitching moment *
Circulation control wing A circulation control wing (CCW) is a form of high-lift device for use on the main wing of an aircraft to increase the maximum lift coefficient. CCW technology has been in the research and development phase for over sixty years. Blown flaps were a ...
*
Zero lift axis A cambered aerofoil generates no lift when it is moving parallel to an axis called the zero-lift axis (or the zero-lift line.) When the angle of attack on an aerofoil is measured relative to the zero-lift axis it is true to say the lift coefficien ...


Notes


References

*
L. J. Clancy Laurence Joseph Clancy (15 March 1929 - 16 October 2014) was an Education Officer in aerodynamics at Royal Air Force College Cranwell whose textbook ''Aerodynamics'' became standard. He was born in Egypt to Alfred Joseph Clancy and Agnes Hunter. I ...
(1975): ''Aerodynamics''. Pitman Publishing Limited, London, {{ISBN, 0-273-01120-0 * Abbott, Ira H., and Doenhoff, Albert E. von (1959): ''Theory of Wing Sections'',
Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, book ...
New York, # 486-60586-8 Aerodynamics Aircraft wing design Dimensionless numbers of fluid mechanics