# THE FLAP WING OF INFINITE. SPAN (PROFILE THEORY)

8- 2-1 The Flap Wing in Incompressible Flow

For the coefficients of the flap effectiveness, the expressions of Eq. (2-82)[35] are |

The flap wing as a bent plate The fundamentals of the theory of the flap wing of infinite span in incompressible flow have been given in Sec. 2-4-2. In its simplest form, the wing with a deflected flap is replaced by a bent plate as shown in Fig. 2-24, on the chord of which, according to Glauert [16], a vortex distribution is arranged.

In Fig. 8-7 these theoretical coefficients have been given against the flap chord ratio f. The problem of the single-bent plate has been solved by Keune [24] with the method of conformal mapping. The most important result of this study is the confirmation of Glauert’s approximate solution for small flap angles. For larger flap

Figure 8-7 Flap effectiveness of several designs: theory and measurements, (a) Angle-of-attack change due to flap deflection Эа/Эт;/* vs. flap chord ratio Xf. (b) Pitching-moment change due to flap deflection ЪС]^1дт{^е vs. flap chord ratio Xf. |

angles, the deviations are more pronounced. In Fig. 8-7 these results are added to the results of comprehensive test series on wings of various flap shapes. The measured coefficients have been taken from test series for small flap angles. The coefficients thus obtained have been designated as Эа/Эт? уе and dcM/drife. Comparison of theory and experiment shows that the measured values are smaller than the theoretical ones for both the change in the angle of attack and the change in the moment. The curve for the wing with a split flap (spreader flap) shows the largest deviation from the theoretical curve. For larger flap deflections, the flap effectiveness declines. This trend is shown in Fig. 8-8 by assigning an effective flap angle rye to each geometric flap angle rf. This coordination applies approximately to the moment change as well.

The differences between the theoretical curves and the measurements in Fig.

8- 7д and b cannot be fully explained by the influence of the profile thickness. They should essentially be due to friction effects. For theoretical studies of the flap wing, it is advisable to apply empirical corrections to the coefficients of the flap wing as obtained from profile theory. This is accomplished simply by multiplying the effect of the camber on the coefficients Эа/Эру and Эсд^/Эру with an empirical factor «. Then the adjusted coefficients assume the form

(8-9 b)

Here the terms with the index к = 1 are the theoretical values from Eqs. (8-8a) and (8-8b). In Fig. 8-9, these coefficients for x = 0.75 are also shown; they agree satisfactorily with the measurements of Fig. 8-7.

In Fig. 8-10, the theoretical values for the position of the flap neutral point from Eq. (8-6) are plotted against the flap chord ratio with bc^jda = 2n. In this figure, the distance between the flap neutral point and the leading edge,

Figure 8-8 Correlation between the effective flap deflection туг and the geometric flap deflection ту for several flap designs (see Fig. 8-7).

Figure 8-9 Reduction of flap effectiveness from Eqs. (8-9a) and (8-96). (a) Change of angle of attack due to flap deflection. (6) Change of pitching moment due to flap deflection.

xNf= c/4 + (AxN)fb is given, where cf4 is the position of the wing neutral point. It is noteworthy that, for small flap chords, the flap neutral point lies at c/2. This is in consequence of the fact that the deflection of even a small flap strongly affects the pressure distribution on the front portion of the wing.

Computation of the flap loading (control-surface loading) and of the flap moment (control-surface moment) requires that the pressure distribution on the deflected flap be known. The theoretical pressure distribution on a bent plate is illustrated in Fig. 2-28, whereas Fig. 8-11 gives the experimentally determined pressure distribution on a wing with split flap from Schrenk ‘[40] (see also Seiferth [41]).

The aerodynamic force on the flap (flap loading), the knowledge of which is important for computation of the structural strength of the flap, is obtained from the pressure distribution on the flap as

L’f=bf I (pi~ pu) dx = Cifbfcfqoo

(fif)

Figure 8-10 Position of the flap neutral point vs. the flap chord ratio for incompressible flow.

In Fig. 8-12 the two coefficients have been plotted against the flap chord ratio f.

The flap moment (control-surface moment) of a wing portion of width bf, referred to the control-surface axis of rotation, is

Mf——bf f (Pi-Pu)(x-Xf) dx=cmfbfc}qOB (8-13)

(cf)

Figure 8-11 Pressure distribution on a wing with a slot flap, from Schrehk.

Figure 8-12 Flap loading; theory from Glauert. Curve 1, change of the coefficient of flap loading 0 with lift coefficient. Curve 2, change of the coefficient of flap loading with flap angle.

where Xf is the position of the axis of rotation as shown in Fig. 8-1 and су is the flap chord. The theory of the flap wing (bent flat plate, Sec. 2-4-2) yields the. following relationships for the control-surface moment coefficient cmf.

= " ЪЦ 1(3 _ 2Xf) ‘/V(1-V) – (3 – %) “resin V051 (8-14*)

In Fig. 8-13, these coefficients are plotted against the flap chord ratio Ay. Test results for simple cambered flaps are also shown. They lie considerably below the theoretical curves. These differences are caused by the influences of the profile thickness and, particularly, of the friction.

To reduce the control-surface moment Afy, several forms of control-surface balance arrangements have already been shown in Fig. 8-2. Of these, only the inner balance and the balance tab can be considered two-dimensional problems. At the inner balance, the control-surface moment is decreased by moving the axis of rotation rearward. Then, in deflecting the control surface, a control-surface “nose” protrudes from the profile, forming a contour that is hardly accessible to computation. To determine the aerodynamic coefficients of the flap wing with inner balance, mainly experimental studies have to be applied, such as, for example, those published by Gothert [18]. The aerodynamic coefficients of a flap wing with balance tab were first treated by Perring [16] using the theory of the multiple-bend plate. A comparison of his theoretical results with measurements is given by Gothert [18]. In this case the effect of friction is particularly strong.

The flap wing as a wing profile Several investigators have studied theoretically not only the flap wing of finite thickness but also the effect of flap arrangement and

Figure 8-13 Coefficient of control-surface moment vs. flap chord ratio Xf, theory from Gothert. (a) Change of the coefficient of control-surface moment with lift coefficient. (b) Change of the coefficient of the control-surface moment with flap deflection.

flap shape and, particularly, that of a slot between the fixed airfoil and the movable flap. In particular, the publications of Allen [2], Fliigge-Lotz and Ginzel [13], Keune [24], and Jacob and Riegels [22] should be pointed out. The results of these studies have been presented systematically by Gothert [18] within the framework of an experimental study. Furthermore, comprehensive test results on flap wings have been reported by Wenzinger [49] and by Keune [24]. Summary accounts of these studies are found in [7] and [45]; compare also [1, 35]. Theoretical investigations on the behavior of the boundary layer of flap wings and comparisons with measurements have been conducted by Goradia and Colwell [17].

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