Integral Equation for the Circulation Distribution from the Extended Lifting-Line Theory

The lifting-surface theory of Sec. 3-2-2 can be transformed into a simpler theory of the kind given in Sec. 3-2-1 by replacing the continuously distributed circulation along the
chord by a vortex line, arranged at a suitably chosen station on the local chord (lifting-line theory). Let x’c = xc(y’) be the location of this lifting-vortex line which, from the results of Sec. 2-3-2 for the inclined flat plate, is expediently placed on the quarter-point line (Fig. 2-37). Then the function G(x, y; y’) of Eq. (342д) becomes

0(x, У У’) = Пу’) 11 + -7====^====-} (3-50e)

i{x – xcy2-f – (y – у)“ /

Here Г(у’) is the total circulation around the wing section у. Furthermore, for у — у and x > xc this function becomes

0(x, y y) = 2Г{у) (3-50b)

The kinematic flow condition [Eq. (3-40)] can be satisfied in this case at one point of the chord only. This control point has the coordinate xp(y): Expediently, it is placed on the three-quarter-chord station, measured from the leading edge (three-quarter point, theorem of Pistolesi), see Sec. 24-5. Hence, the expression in parentheses on the left-hand side of Eq. (343) becomes

where a(y) is the measured angle of attack relative to the zero-lift direction (Fig.

3- 18).

By introducing Eqs. (3-51) and (3-50) into Eq. (343), the integral equation for the circulation distribution from the extended lifting-line theory i§ obtained as

Compared with the simple lifting-line theory discussed in Sec. 3-2-1, Eq, (3-52) has the great advantage that it is also applicable to yawed and swept-back wings. This extended lifting-line theory is also called the three-quarter-point method. It was developed in detail and applied particularly by Weissinger [95]. Also Reissner [95] was engaged in the establishment of a solid foundation for this lifting-line theory.

For the swept-back wing a vortex arrangement as in Fig. 3-20 is obtained. In Fig. 3-20a the replacement of the wing by a system of elementary wings and in Fig.

3- 20b the equivalent vortex system according to PrandtTs concept (Fig. 3-9) are demonstrated.

In Prandtl’s lifting-line theory and in the three-quarter-point method described above, the wing is replaced by just one lifting line. Wieghardt [101] proposed the arrangement of several lifting lines in series. This method can be designated as a multiple-points method. Scholz [77] developed this method in more detail and applied it especially to the cambered rectangular wing.