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Aerospace Engineering DISCUSSION
Thin airfoil theory gives a lift-curve slope of 2π per radian regardless of the airfoil shape. But the wing data I am comparing against, for a wing of aspect ratio 8, shows a clearly lower slope, and the measured drag has a part that grows with lift and is not in the airfoil data at all.
Is 2π a universal value, how do camber and thickness change it, and how do I correct two-dimensional airfoil data for a real wing?
2π per radian, about 0.11 per degree, is the result of an idealised model: a thin airfoil in inviscid, incompressible, two-dimensional flow at small angles. Under those assumptions the slope is the same for every shape. Camber does not change the slope; it shifts the whole line, so a cambered airfoil has lift at zero angle of attack and a negative zero-lift angle.
Real sections come close. Thickness raises the theoretical slope slightly and the boundary layer lowers it, so measured two-dimensional values are usually within about 10 percent of 2π in the linear range before stall.
The big reduction on your wing is a three-dimensional effect. A finite wing sheds trailing vortices from its tips, and they induce a downwash over the span that reduces the effective angle of attack of each section. For an unswept wing of moderate to high aspect ratio, lifting-line theory gives a = a0 / (1 + a0 / (π × e × AR)).
With a0 = 2π, AR = 8 and e = 0.9: a = 6.28 / (1 + 6.28 / 22.6) = 6.28 / 1.278 = 4.92 per radian, or 0.086 per degree. That is 22 percent below the two-dimensional value, which matches the kind of difference you describe.
The drag that grows with lift is induced drag, the other consequence of the same downwash: the lift vector is tilted slightly backward. Its coefficient is CDi = CL² / (π × e × AR). At CL = 0.5 with AR = 8 and e = 0.9 that is 0.25 / 22.6 = 0.011, which is of the same order as the profile drag of a clean wing.
Because it goes with CL squared, it dominates at low speed and in climb, where CL is high, and fades at high speed. That is why gliders and long-range aircraft have high aspect ratio wings.
Two more corrections to keep in mind. Compressibility raises the slope as speed increases: the Prandtl-Glauert factor is 1 / sqrt(1 - M²), about 1.25 at Mach 0.6, and it is usable only up to the point where the local flow goes sonic. Sweep lowers the slope, roughly with the cosine of the sweep angle, and for low aspect ratio or swept wings the simple lifting-line formula is no longer accurate, so a more general formula or a panel method is used instead.
And all of this applies to the straight part of the curve. Near stall the slope falls off and no linear theory describes it.
