FLAPS ON THE WING OF FINITE SPAN AND ON THE TAIL UNIT
8-3-1 Flaps on the Wing in Incompressible Flow
Computational methods The aerodynamics of the flap wing of infinite span (plane problem) has been discussed in the previous section. Now the effect of a flap (control surface) on a wing of finite span will be treated. A further geometric parameter, the span of the flap, is added (see Figs. 7-1, 7-3, and 8-4a). Furthermore, in many cases the flap chord ratio varies over the flap span (see Fig. 8-1). To determine the lift distribution, a wing with a deflected flap is equivalent to a wing with an additional angle-of-attack distribution over the span (twist). For a flap covering only a portion of the span, this additional angle-of-attack distribution is discontinuous. The angle-of-attack distribution that is equivalent to a given flap deflection is obtained from the theory of the flap wing of infinite span as
where doijdrif is the local flap effectiveness from Eqs. (8-8a) and (8-9a) and from Figs. S-la and 8-9a. If the flap chord ratio f varies over the span, it is a function of the span coordinate 17 =y/s.
According to the procedure for the computation of the lift distribution on wings of Sec. 3-3, the additive circulation distribution caused by the flap deflection can be determined for such an angle-of-attack distribution. Special attention should be paid to the station of discontinuity in the angle of attack.
The case of a symmetric angle-of-attack distribution corresponds to a landing flap at the wing or an elevator at an all-wing airplane as shown in Fig. 7-3. The antimetric angle-of-attack distribution corresponds to the ailerons (Figs. 7-1 and
Following simple lifting-line theory (Sec. 3-3-3), Multhopp (Chap. 3, ) developed a method for handling the discontinuity in the angle-of-attack curve. In Fig. 8-29, a result of this method for a trapezoidal wing of aspect ratio A = 2.75 and taper A =0.5 is shown as curve 1. The station of discontinuity in the angle-of-attack distribution ay lies at щ = 0.5. In Fig. 8-29a it is symmetric, in Fig.
8- 29b it is antimetric. According to Fig. 8-29, the symmetric flap deflection at the wing outside generates a considerable lift, even in the wing middle section. The circulation distributions according to extended lifting-line theory (three-quarter – point method, Sec. 3-3-4) are also shown in Fig. 8-29 as curves 2. As should be expected, extended lifting-line theory gives a smaller lift than simple lifting-line
Figure 8-29 Circulation distribution over the span due to a discontinuous angle-of-attack distribution for a trapezoidal wing of aspect ratio л =2.75; taper Л. = 0.5. Curve 1, simple lifting-line theory. Curve 2, extended lifting-line theory. (a) Symmetric angle-of-attack distribution. (b) Antimetric angle-of-attack distribution.
theory. A computational method for the lift distribution on wings with flaps, based on lifting-surface theory (Sec. 3-3-5), is given in . This method requires the availability of the angle-of-attack distributions caused by the flap deflection on the cj4 line (I/) and on the trailing edge (%r). They are, considering Eq. (8-21),
where the coefficients da/drjf and dcmjdr]f from Eqs. (8-8д) and (8-8b) and from Eqs. (8-9a) and (8-9b), respectively, are known from the profile theory of the flap wing and depend only on the control-surface chord ratio.* An improved method for describing the effect of the angle-of-attack discontinuity has been given by Hummel . Lift distributions of wings with deflected flaps (angle-of-attack distribution with a break) have been computed by Bausch  from simple lifting-line theory for a wing of elliptic planform. For a wing with a trapezoidal planform, corresponding computations have been published by Richter . A large number of computations
have been conducted by de Young , who applied extended lifting-line theory; however, he did not exclude the station of discontinuity in his computations. Investigations, applying lifting-surface theory, have been conducted by Truckenbrodt and Gronau  on delta wings with deflected flaps.
A summary of American tests on wings of finite span with flaps that extend only over a portion of the span is given in . It includes the separation characteristics of such wings; compare the publications [31, 54].
Results of a few sample computations of the lift distribution of wings with flap and control-surface deflections will be given in the following section.
Landing flaps, elevators For the wing of elliptic planform, the change in the mean zero-lift angle caused by the flap deflection is obtained according to Sec. 3-3-3. For a sectionwise-constant, symmetric angle-of-attack distribution, Eq. (3-81) yields, after integration,
Here the flap (control surface), having a constant flap chord ratio, extends from —7?0 to + rjo. The relationship between aу and the flap angle i? y is given by the theory of the two-dimensional flap wings of Eq. (8-21). The coefficient Эа/Эоу is. shown in Fig. 8-30 as a function of the flap span. This result is obtained by both simple and extended lifting-line theories.
A further example, in which Truckenbrodt and Gronau  applied lifting – surface theory, is shown in Fig. 8-31. It deals with a delta wing of aspect ratio Л* = 2b*/cr = 2 equipped with a flap that is symmetrically deflected. The flap chord ratio = crfc, however, varies between X/= | at the wing root and Л/ = 1 at the wing tips. The local flap effectiveness was obtained by introducing Eqs. (8-9a) and (8-9b) into Eqs. (8-22a) and (8-22b). The changes of the mean zero-lift angle bajbrf and of the mean zero-moment coefficient bc^fbrf were computed first.
Figure 8-30 Change of the mean zero-lift angle due to flap deflection for an elliptic wing with various forms of the flap, from Bausch.
Figure 8-31 Measured aerodynamic coefficients of a delta wing with symmetrically deflected flap extending over the entire trailing edge. Aspect ratio л* = 2, profile NACA 0012; comparison of theory (y. = 0.75) and experiment, from Truckenbrodt and Gronau. (a) Geometry, (b) Lift coefficient vs. angle of attack, (c) Lift coefficient vs. pitching-moment coefficient.
Ailerons In Fig. 8-32, the rolling-moment coefficients are given for a wing of elliptic planform and antimetric control-surface deflection. Figure 8-32a gives the
Figure 8-32 Rolling-moment coefficient vs. flap deflection for an elliptic wing, from Bausch. (a) Flap extending over the entire half-span; curve 1, extended lifting-line theory; curve 2, simple lifting-line theory, (b) Effect of the flap span.
rolling moment of the ailerons plotted against the aspect ratio with each aileron extending over the entire half-span. The extended lifting-surface theory of Eq. (3-100) yields
^cMx _____ 1 —- (8-24z)
d(Hf У&2 – j~ 4 – f – 2
where k – TtAlc’Loo *=» A/2. For comparison, this coefficient according to simple lifting-line theory is added. The rolling moment of the ailerons for the case of an aileron extending over only a part of the wing half-span is shown in Fig. 8-32b. In this case, Eq. (3-100) yields
Цш = (&мЛ N/rr^3 (8-244)
where (dcMxjdaf)Vo=0 is given by Eq. (8-2Ad) and Fig. 8-32<z. For the delta wing of Fig. 8-3 a, the theoretical coefficients of the aileron rolling moment of antimetricaily deflected ailerons extending over the entire half-span are compared in Fig. 8-33 with measurements. Agreement between theory and experiment is good for small and moderate angles of attack.
Aileron investigations and comprehensive experimental results are summarized in [12, 45].