# Flaps on the Wing in Compressible Flow

The flap wing of finite span in compressible flow may be treated according to the theory of the wing of finite span as discussed in Secs. 4-4 and 4-5.

Subsonic incident flow At subsonic velocities, the subsonic similarity rule (Prandtl – Glauert) of Sec. 44-1 applies. It requires the determination of a wing, to be computed for incompressible flow, that is transformed from the given geometry of the wing of finite span at compressible flow. These transformation formulas for the geometries of the wings are found as Eqs. (4-66)-(4-68). The influence of compressibility on the aerodynamic coefficients of the wing is obtained from the transformation formulas Eqs. (4-69)-{4-72). Here, the angle-of-attack distribution due to the flap deflection remains unchanged and is determined with lifting-surface theory from Eq. (8-22). Accordingly, Eqs. (8-15a) and (8-15h) give the changes of the angle of attack and of the momentum coefficient with the flap deflection. However, these equations for the incompressible reference flow now have to be evaluated for the transformed wing planform from Eq. (4-15). In Fig. 8-34, the results of sample computations for wings of finite span with deflected flaps are shown. They are the three wings discussed several times previously, namely, a trapezoidal, a swept-back, and a delta wing; see Table 34.

Supersonic incident flow The computation of the aerodynamic effect of a flap on a wing of finite span at supersonic velocities is in some respect simpler than at subsonic velocities. This becomes obvious from Fig. 8-35, which shows a rectangular

Figure 8-33 Measured rolling-moment coef­ficients of a delta wing as shown in Fig. 8-3la, with flaps extending over the entire half-span for several angles of attack a. Comparison of theory (« = 0.75) and mea­surements from Truckenbrodt and Gronau.

wing and a delta wing with flaps of constant chord extending over the entire trailing edge. When the flap is being deflected, an additive lift is generated only on this flap that is equal to the lift of a rectangular wing of span b and of the flap chord су. The lift of the wing lying before the flap is not changed by the flap deflection.

To compute the lift caused by flap deflection, the results for the rectangular wing of Sec. 4-5-4 may be recalled. From Eq. (4-112), the lift coefficient produced by the flap and referred to the total wing area A is given as

bcL _ Af 4 Эт?/ A iMa%. – 1

which is valid for b sjMaL — 1, but independent of the wing shape.

For the rectangular wing of Fig. 8-35, the change of the zero-lift angle caused by the flap deflection can easily be determined. Because Эй/Эту = — (Эс^/Згу)/ (.дсь/да), Eqs. (8-25) and (4-112) yield

ЛЁ = _ f 2Л (8-26)

dr? f 1 2Л ІМаІг – 1 – 1

where ‘hf = Cflc=AfjA is the control-surface chord ratio. In this equation, the fraction on the right-hand side, which is always greater than unity, gives the

Figure 8-35 Aerodynamics of the flap wing at supersonic incident flow, {a) Rectangular wing with flap extending over the entire trailing edge. (b) Delta wing with flap extending over the entire trailing edge.

correction of the value for the two-dimensional flap wing, as can be verified by comparison with Eq. (8-16a).

The pressure distribution on the flap of a wing in supersonic incident flow may also be established quite easily. Figure 8-36 shows a flap design in which the right-hand-side edge of the flap is an “outer edge,” the left-hand edge an “inner edge,” both of which are parallel to the incident flow direction. When the flap is deflected, Mach lines originate at either upstream edge. In the case of no intersection of these Mach lines on the flap, the pressure distribution in zone 1 is

і

Figure 8-36 Pressure distribu­tion due to flap deflection on a rectangular flap at supersonic incident flow.

that for plane flow. The resultant pressure coefficient on the upper and the lower side is, therefore, with Eq. (4-85) and Table 4-5,

(8-2 7a)

The flow in zones 2 and 3 is cone-symmetric. For zone 2, Eq. (4-111) yields

For zone 3, Tucker and Nelson  found the expression

cpз = — arccos (— Ocppi

In these expressions t – y/x tan ц = (y/x) sjMaL — 1, and у is measured from the upstream corners of the flap. In Fig. 8-36, the pressure distributions are shown for a section x = const. On the side of the inner edge, the flap deflection causes, within the range of the Mach cone, a lift on the undeflected wing that is equal to the lift loss at the adjacent portion of the flap.

Furthermore, Fig. 8-37 shows a flap arrangement with a swept-back outer edge of the flap such as, for example, is found in delta wings. In Fig. 8-37a, the outer, edge is a subsonic edge. If the two Mach lines originating at the two upstream flap corners do not intersect on the flap, zone 1 has again, as in Fig. 8-36, the pressure distribution of plane flow. In the case of the subsonic edge (m < 1) of Fig. 8-37a, the pressure distribution of zone 4 is of the kind given in Fig. 4-67 for a delta wing with a subsonic leading edge. In the case of the supersonic edge (Ma« > 1) of Fig. 8-37b, where the Mach cone from the right-hand upstream corner lies entirely on the flap, the pressure distributions of zones 5 and 6 are of the kind given in Fig.

Figure 8-38 Lift due to flap deflec­tion at supersonic incident flow. Curve 1, inner flap. Curve 2, tip flap. Curve 3, full-span flap.

4-69 for a delta wing with a supersonic leading edge. The pressure in zone 6 is constant, Eq. (4-89):

CP6 — —- cppi (8-28)

with m = tan 7/tan д. The pressure distributions in zones 4 and 5 have been determined by Tucker and Nelson .

Finally, a few data will be given, in the following two figures on the lift produced by the flap deflection and on the position of its center of application. Figure 8-38 gives the total lift of three rectangular flaps. Flap 1 has two inner edges (inner flap), flap 2 an inner and an outer edge (tip flap), and flap 3 two outer edges (full-span flap). Shown in this figure is the ratio of the total lift produced by the flap to the lift of the two-dimensional flap wing as a function of the quantity bfS/Malc — Ijcf. Flap 1 does not cause any lift loss compared with the two-dimensional flap wing; Eq. (8-25) applies to flap 3. The lift of flap 2 is the arithmetic mean of those of flaps 1 and 3. Figure 8-39 shows the position of the lift force of the flap (flap neutral point). Here, xy is the distance of the flap neutral point from the axis of rotation. For flap 1, the flap neutral point lies at the flap half-chord. It shifts forward for flaps 2 and 3.

The rolling moment due to aileron deflection can be computed very easily by realizing that the lift force at antimetrically deflected flaps acts, in very good approximation, on the half-span of the flap.

Further information on rectangular flaps is found in Schulz . Flaps on rectangular, delta, and swept-back wings have been investigated by Lagerstrom and Graham . Flaps with outer (horn) balances have been studied by Naylor .