Thin Airfoil Theory

Thin airfoil theory has been alluded to several times in earlier chapters. The gov­erning equation for thin airfoil theory is Laplace’s equation in two dimensions. The idea is that the airfoil is replaced by a camberline (which is a flat plate if the airfoil is uncambered) on which a vortex sheet singularity of unknown strength, y{x) is placed. The strength of this vortex sheet is determined by satisfying flow tangency on the camberline of the airfoil in conjunction with the Kutta condition at the trailing edge, that is, у {ТЕ) = 0. Solutions are usually obtained by expressing the vortex sheet strength as a Fourier series, and then calculating the coefficients in this series. See Glauert (1947) for details of the method. In most cases it is only the integrated lift and pitching moment on the airfoil that is required and not the chordwise loading. These can be determined using only the first three coefficients in the Fourier series. The general form of the chordwise loading on the airfoil is

where в — cos!(1 — 23c) and x = x/c. The coefficients Aq and An are given by the equations

where у = y(x) describes the camberline of the airfoil. The lift and pitching moment co­efficients are given by

Note that for a noncambered airfoil (dy/dx = 0), Aq — a and An = 0 if n > 0, giving

Thin airfoil theory is very powerful despite its simplicity and is particularly useful in helicopter analyses, especially during preliminary airfoil design, because it allows a rapid estimation of the pitching moment resulting from any given camberline – see Section 7.7.3. The theory also forms a basis for understanding the elements of unsteady airfoil aerody­namics, as has been examined extensively in Chapter 8. An interesting result is obtained by assuming that the vortex sheet strength is all lumped or “bound” to vortex of strength Г*, placed at the aerodynamic center (1/4-chord) of the airfoil section, that is, so that

Using the Kutta-Joukowski theorem, the lift per unit span

L = рУооГ» = X-pVlcCh

where Ci = 2not in 2-D flow. The only way Eq. 14.50 can be satisfied is if

Гь = TtVooCa.

This is equivalent to enforcing the flow tangency condition only at the 3/4-chord of the airfoil. (The 3/4-chord position is sometimes called the rear neutral point after Pistolesi and Kussner.) This result is equivalent to saying that the lifting properties of an airfoil, in steady flow at least, can be modeled by placing the bound circulation at the 1/4-chord and satisfying flow tangency at the 3/4-chord. This result is fundamental to the development of the 3-D lifting-line models described in the next section.