Effective Aspect Ratio

It was stated earlier that the profile drag of an airfoil section increases approximately with the square of the section Q. Combined with the induced drag, given by Equation 4.20, the total CD for a wing can be written approximately as

c*2 H—(кжА + 1 + 5)

Си = + kCi2 + (1 + S)

where к us the constant of proportionality giving the rate of increase of Cd with C2.

Equation 4.31 can be rewritten as

where

1

1 + 8 + кттА

The factor e is known as Oswald’s efficiency factor (see Ref. 4.2). The Ifoduct Ae is referred to as the “effective aspect ratio” and is sometimes written as Ae.

Consider data from References 3.1 and 3.27 in light of Equation 4.32. Figure 4.25 presents CD as a function of CL for the finite wing tested in Reference 3.27 and Cd versus Q from Reference 3.1 for the 65-210 airfoil.

This particular airfoil section is conducive to laminar flow for Q values between approximately 0.2 and 0.6, as reflected in the “drag bucket” in the lower curve of this figure. The “bucket” is not evident in the wing test results of Reference 3.27, either as the result of wing. surface roughness or wind tunnel flow disturbances. Neglecting the bucket in the airfoil section Cd curve, the constant, k, is found to be 0.0038. From Figure 4.22, 8 = 0.01. Thus, from the airfoil Cd curve and lifting line theory, the wing CD curve is predicted to be –

CD = 0.0055 + 0.0394CL2 (4^3)

This equation is included on Figure 4.25, where it can be seen to agree closely with the test results. Thus one can conclude that the difference in the drag between an airfoil and a wing is satisfactorily explained by the induced drag. In this particular Case, Oswald’s efficiency factor is 0.89.

Generally, for a complete airplane configuration, e is not this high because of wing-fuselage interference and contributions from the tail and other components.

Figure 4.25 Comparison between predicted and measured drag polar for a wing having a finite aspect ratio. .

High-wing and low-wing airplanes show a measurable difference in Oswald’s efficiency factor. Most likely as the result of interference between the boundary layer on the wing’s upper surface with that on the fuselage, e values for low-wing airplanes are lower than those for high-wing airplanes. The boundary layer on the upper surface of a wing is considerably thicker than the one on the lower surface. Combining with the boundary layer over the sides of the fuselage, the wing’s upper surface boundary layer, for the low-wing airplane, can cause a rapid increase in the wing and fuselage parasite drag as the angle of attack increases. For a high-wing airplane, the relatively thin boundary layer on the lower surface of the wing interferes only slightly with the fuselage boundary layer. Typically, e is equal approximately to 0.6 for low-wing airplanes and 0.8 for high-wing airplanes. These values are

confirmed by the flight tests reported in Reference 4.3 and in other data from isolated sources.