Nonlinear Wing Theory

The wing theory treated so far establishes a linear correlation between lift coefficient and angle of attack. It is designated, therefore, linear wing theory. It is known from experimental investigation that for wings of very small aspect ratio, A < 1, lift coefficients Ci are considerably larger than those obtained from linear theory when plotted against the angle of attack. Figure 349 illustrates this behavior for rectangular wings of aspect ratios Л =0.2, 0.5, 1.0, and 5.0 as compiled by Gersten [21]. The dashed theoretical curves represent linear theory as discussed earlier. Although linear theory produces the right lift slope {dcLlda)a~0 even for small aspect ratios, strong deviations of the measurements from linear behavior are already obvious for small angles of attack.

All wing theories discussed so far are based on the concept that bound and free vortices lie in the same plane. A linear relation between lift and angle of attack is the necessary consequence.

This much simplified vortex model must be abandoned for a theoretical explanation of the nonlinear relation between lift and angle of attack. A first trial in this direction was made by Bollay [8]. He used a vortex model similar to Fig.

3- 50tf in which the free vortices no longer lie in one plane but rather are shed in the downstream direction from the wing tips under the angle a/2 with the wing plane. Bollay assumes that the bound vortices are constant over the span. Gersten [21]

Figure 3-49 Measured lift coefficients vs, angle of attack a for rectangular wings of aspect ratios A = 0.2, 0.5, 1.0, and 5.0. Curve 1, linear theory of Scholz. Curve 2, nonlinear theory of Gersten.

refined this vortex model by prescribing a variable circulation distribution over the span (Fig. 3-50b). The cL(a) curves based on this theory are given in Fig. 349 as solid lines. They are in very good agreement with this theory (see Winter [102]). By the same theory, pitching moment, induced drag, and lift distribution along the span have also been determined. Agreement between tests and theory is good in these cases, too. Furthermore, the nonlinear theory has been extended by Gersten to arbitrary wing shapes. It represents an extension of the lifting-surface theory of Sec. 3-3-5 to the nonlinear angle-of-attack range. The cL(a) curves as determined

from this theory and the comparison with test data are shown in Fig. 3-51 for a swept-back and a delta wing.

It is known from test results that the aerodynamic coefficients of wings of small aspect ratio are strong functions of the wing leading-edge design. This is true particularly for swept-back and delta wings with sharp leading edges which, even at very small angles of attack (a — 3°), promote flow separation from the leading edge of the kind shown in Fig. 3-52. Starting at the wing tips, two vortex sheets form on the two leading edges that roE up into free vortices when floating downstream.

This process was first discussed by Legendre [55] and has been treated in

Figure 3-51 Lift coefficient of swept – back wings with sharp leading edge and small aspect ratio vs. angle of attack.

(——– ) Linear theory from Eq.

(3-lOli). (——— ) Nonlinear theory of

Gersten. (o) Measurements, (a) Swept – back wing.1 = 1, =1, (/3 = 45°. (b) Delta wing л = 0.78, К =

numerous other publications [4, 10, 33, 55, 98]. The roll-up of vortex sheets has been studied theoretically by Roy [71] and by Mangier and Smith [58, 80]. Roy established details through numerous flow-pattern photographs. Under certain circumstances, a striking change in the structure of the rolled-up vortex sheets can be observed that can be termed bursting of the vortices. Figure 3-52b-e shows smoke pictures of this phenomenon from Hummel [33]. The bursting of vortices occurs (1) at large angles of attack in symmetric incident flow (Fig. 3-52c), (2) at the vortex of the upstream-turned side of yawed wings (Fig. 3-52d), and (3) when an obstruction is placed into the vortex flow (Fig. 3-52e). Naturally, the bursting of vortices has a strong effect on the aerodynamic properties of the delta wing; compare [4, 33]. These processes affect lift and pitching moment as well as drag.

Further investigations of nonlinear effects on wings of small aspect ratios, especially on delta wings, are reported in [19, 32, 57, 67]. A very recent survey of the aerodynamic properties of slender wings with a «harp leading edge has been given by Parker [66].