An airfoil's aerodynamic force can be separated into lift and drag components that intersect with its chord line at its center of pressure.
Airfoil aerodynamic characteristics. Figure (a) shows the aerodynamic force acting on an airfoil. This force may be separated into lift and drag components, as shown in figure (b). Figure (c) illustrates lift, drag, and moment about the quarter-chord point-all a function of the angle of attack a while figure (d) shows the lift, drag, and moment about the aerodynamic center.
Coefficient of lift as a function of angle of attack. Lift increases until the angle of attack reaches about 16°, at which point a stall occurs.
This figure shows an airfoil whose angle of attack is being raised from 0° to past the stall angle of attack. Below the stall angle, the separation points on the airfoil move forward slowly but remain relatively close to the trailing edge. Near the stall angle, the separation points move rapidly forward and the pressure drag rises abruptly. Past the stall angle, the effect of the greatly increased separated flow is to decrease the lift.
Coefficient of drag as a function of angle of attack of airfoil section.
Subsonic Airflow Effects – The Two-Dimensional Coefficients
An airfoil's aerodynamic force may be separated into lift and drag components. This force intersects with its chord line at a point designated as its center of pressure. The lift, drag, and center of pressure for a cambered airfoil vary as its angle of attack is changed. No aerodynamic moments (the tendency of an airfoil to turn about its center of gravity) are present at the center of pressure because the line of action of the aerodynamic force passes through this point. If one has the airfoil mounted at some fixed point along the chord, for example, a quarter of a chord length behind the leading edge, the moment is not zero unless the resultant aerodynamic force is zero or the point corresponds to the center of pressure. The moment about the quarter-chord point is generally a function of the angle of attack.
At the aerodynamic center, the moment is independent of the angle of attack. This system of reporting is convenient for a number of aerodynamic calculations.
The data obtained by wind-tunnel testing of NACA families of airfoil sections is two-dimensional data. This means that since the airfoil was suspended in the wind tunnel from wall to wall, it essentially had no wingtips and simulated a section of a wing of infinite span. The data obtained is associated with just the airfoil and has no association with the span of the wing. Aerodynamic characteristics recorded include the lift coefficient cl , the drag coefficient cd, the moment coefficient about the quarter-chord point (cm)0.25c, and the moment coefficient about the aerodynamic center (cm)ac. These coefficients are obtained by measuring, in wind-tunnel tests, the forces and moments per unit length of the airfoil wing and expressing as follows:
cl = l/qc
where l is the measured lift per unit length of the airfoil wing, q is the testing dynamic pressure or 1/2 pV2 , and c is the chord length of the airfoil section. Similarly,
where d is the measured drag per unit length of the airfoil wing and,
cm = m/qc2
where m is the measured moment per unit length acting on the airfoil (whether at the quarter-chord point or at the aerodynamic center or any other point desired).
The aerodynamic coefficients are dependent on body shape (airfoil section chosen), attitude (angle of attack a), Reynolds number, Mach number, surface roughness, and air turbulence. For low subsonic flow, Mach number effects are negligible and air turbulence is dependent on the Reynolds number and surface roughness and need not be indicated as a separate dependency.
These aerodynamic coefficients vary with the angle of attack. At an angle of attack of 0°, there is a positive coefficient of lift, and, hence, positive lift. This is the case for most cambered airfoils. One must move to a negative angle of attack to obtain a zero lift coefficient (hence zero lift). This angle is called the angle of zero lift. A symmetric airfoil (one that is identical above and below the chord line) has an angle of zero lift equal to 0°.
From 0° up to about 10° or 12° there is a linear increase in the coefficient of lift with angle of attack. Above this angle, however, the lift coefficient reaches a peak and then declines. The angle at which the lift coefficient (or lift) reaches a maximum is called the stall angle.
The coefficient of lift at the stall angle is the maximum lift coefficient cl,max Beyond the stall angle, the airfoil is stalled and a change in the airflow pattern occurs. Below the stall angle, the separation points on the airfoil move forward slowly but remain relatively close to the trailing edge. Near the stall angle, the separation points move rapidly forward and the pressure drag rises abruptly. Past the stall angle, the effect of the greatly increased separated flow is to decrease the lift.
Although the "lift curve" continues through negative angles of attack and a negative stall angle occurs also, in general, an aircraft will operate at a positive angle of attack to obtain the lift necessary for flight.
Usually, the minimum drag coefficient occurs at a small positive angle of attack corresponding to a positive lift coefficient and builds only gradually at the lower angles. As the airfoil nears the stall angle, however, the increase in the drag coefficient cd is rapid because of the greater amount of turbulent and separated airflow occurring.
A wing of finite span (rather than infinite span as above) is said to be three-dimensional. The wing area is designated S and is the chord length c multiplied by the wing span b. Thus
S = bc
This is also known as the planform area. If one measures the lift, drag, and moment on this three-dimensional wing, one obtains the three-dimensional aerodynamic characteristics of the wing; CL, CD, and Cm where
Lift = qSCL
Drag = qSCD
Moment = qSc Cm
Notice that the coefficients for three-dimensional flow are capitalized whereas the coefficients for two-dimensional flow use lower case letters.
The important question now arises: How can one use experimental NACA two-dimensional airfoil characteristics data to obtain the lift, drag, and moments of a real, finite three-dimensional wing? Or, how are cl, cd, and cm related to CL, CD, and Cm. At first glance one might conclude that cl = CL, cd = CD, and cm = Cm. But this is wrong because the two-dimensional wing tested in the wind tunnel spanned the tunnel walls and did not allow for the possibility of airflow about the wing tips, that is, the spanwise flow of air. But the three-dimensional wing is freely exposed in the free stream and spanwise flow may occur. The two-dimensional results must be modified to account for the effects of three-dimensional flow.
Adapted from Talay, Theodore A. Introduction to the Aerodynamics of Flight. SP-367, Scientific and Technical Information Office, National Aeronautics and Space Administration, Washington, D.C. 1975. Available at http://history.nasa.gov/SP-367/cover367.htm
For Further Reading:
Abbott, I.H. and Von Doenhoff, A.E. Theory of Wing Sections. New York: McGraw-Hill, 1949. Also New York: Dover, 1959.
Anderson, Jr., John D. A History of Aerodynamics. Cambridge, England: Cambridge University Press, 1997.
Jacobs, Eastman N., Ward, Kenneth E., and Pinkerton, Robert. The Characteristics of 78 Related Airfoil Sections From Tests in the Variable-Density Wind Tunnel. National Advisory Committee on Aeronautics (NACA) Technical Report 460, 1933. Available at.
Smith, H.C. “Skip.” The Illustrated Guide to Aerodynamics, 2nd edition. Blue Ridge Summit, Pa.: TAB Books, 1992.
Wegener, Peter P. What Makes Airplanes Fly? New York: Springer-Verlag, 1991.