Lift coefficient

The lift coefficient ($c_\mathrm l\, ,$ $c_\mathrm a\,$ or $\!\ c_\mathrm z$) is a dimensionless coefficient that relates the lift generated by a lifting body, the dynamic pressure of the fluid flow around the body, and a reference area associated with the body. A lifting body is a foil or a complete foil-bearing body such as a fixed-wing aircraft.

Lift coefficient is also used to refer to the dynamic lift characteristics of a two-dimensional foil section, whereby the reference area is taken as the foil chord.^[1][2]

Lift coefficient may be described as the ratio of lift pressure to dynamic pressure where lift pressure is the ratio of lift to reference area.

Lift coefficient may be used to relate the total lift generated by a foil-equipped craft to the total area of the foil. In this application the lift coefficient is called the aircraft or planform lift coefficient $C_\mathrm L\, .$

Watercraft and automobiles equipped with fixed foils can also be assigned a lift coefficient.

The lift coefficient $C_\mathrm L\,$ is equal to:^[2][3]

$C_\mathrm L = {\frac{L}{\frac{1}{2}\rho v^2A}} = {\frac{2 L}{\rho v^2A}} = \frac{L}{q A}$
where

• $L\,$ is the lift force,
• $\rho\,$ is fluid density,
• $\vec v\,$ is true airspeed,(speed of the body relative to a static point on the earth's surface)
• $q\,$ is dynamic pressure, and
• $A\,$ is planform area.

The lift coefficient is a dimensionless number.

The aircraft lift coefficient can be approximated using the Lifting-line theory^[4] or measured in a wind tunnel test of a complete aircraft configuration.

Section lift coefficient

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 $c_\mathrm l\, .$ It is common to show, for a particular airfoil section, the relationship between section lift coefficient and angle of attack.^[5] 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 - the concept of a wing with infinite span and non-varying cross-section, the lift of which is bereft of any three-dimensional effects. It is not relevant to define the section lift coefficient in terms of total lift and total area because they are infinitely large. Rather, the lift is defined per unit span of the wing $l\, .$ In such a situation, the above formula becomes:

$c_l={l \over \frac{1}{2}\rho v^2c}$

where $c\,$ is the chord length of the airfoil.

The section lift coefficient for a given angle of attack can be approximated using the thin airfoil theory,^[6] or determined from wind tunnel tests on a finite-length test piece, with end-plates designed to ameliorate the three-dimensional effects associated with the trailing vortex wake structure.

Note that the lift equation does not include terms for angle of attack — that is because the mathematical relationship between lift and angle of attack varies greatly between airfoils and is, therefore, not constant. (In contrast, there is a straight-line relationship between lift and dynamic pressure; and between lift and area.) The relationship between the lift coefficient and angle of attack is complex and can only be determined by experimentation or complicated analysis. See the accompanying graph. The graph for section lift coefficient vs. angle of attack follows the same general shape for all airfoils, but the particular numbers will vary. The graph shows an almost linear increase in lift coefficient with increasing angle of attack, up to a maximum point, after which the lift coefficient reduces. The angle at which maximum lift coefficient occurs is the stall angle of the airfoil.

The lift coefficient is a dimensionless number.

Note that in the graph here, there is still a small but positive lift coefficient with angles of attack less than zero. This is true of any airfoil with camber (asymmetrical airfoils). On a cambered airfoil at zero angle of attack the pressures on the upper surface are lower than on the lower surface.

See also

• Fluid
• Density
• Foil (fluid mechanics)
• Drag coefficient
• Pitching moment
• Circulation control wing
• Zero lift axis

Notes

1. ^ Clancy, L. J.: Aerodynamics. Sections 4.15 and 5.4
2. ^ ^{a b} Abbott, Ira H., and Von Doenhoff, Albert E.: Theory of Wing Sections. Section 1.2
3. ^ Clancy, L. J.: Aerodynamics. Section 4.15
4. ^ Clancy, L. J.: Aerodynamics. Section 8.11
5. ^ Abbott, Ira H., and Von Doenhoff, Albert E.: Theory of Wing Sections. Appendix IV
6. ^ Clancy, L. J.: Aerodynamics. Section 8.2

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