The Wing-Fuselage System in Asymmetric Incident Flow
Rolling moment due to sideslip of a wing-fuselage system In asymmetric incident flow of a wing-fuselage system, the lateral component of the flow about the fuselage creates an additive antimetric distribution of the angle of attack of the wing as discussed in Sec. 6-2-1 and demonstrated in Fig. 6-6. It has reversed signs for high-wing and low-wing airplanes, and it is zero for mid-wing airplanes. This antimetric angle-of-attack distribution generates an antimetric lift distribution at the wing and thus a rolling moment due to sideslip. This additive rolling moment due to sideslip caused by the fuselage also has reversed signs for high-wing and low-wing airplanes.
For a theoretical assessment of the influence of the fuselage on the lift distribution of the wing, the antimetric angle-of-attack distribution as shown in Fig.
6- 6 must be determined as caused by the cross flow over the fuselage with velocity U„ sin j3« Uoofi. This angle-of-attack distribution Act — w/Uoo for an infinitely long fuselage with circular cross section (radius R) becomes
— = – 2№——– y—— (y >y0) (6-24)
p (yz – f za)2 V ^ j v J
where the fuselage cross section is given as in Fig. 6-22 as y + z = R2. Within the range of the fuselage, that is, for —yQ <y < +yQ, A a has to be taken as being zero, Aa = 0.
For the wing without dihedral, following Fig. 6-2a, z has to be replaced in Eq. (6-24) by z0 (z =zQ). Thus the’ angle-of-attack distribution may be expressed by the dimensionless coordinates y/s = r} and z0/s = with r}F—Rjs as the relative fuselage width. The angle-of-attack distributions computed by this method are shown in Fig. 6-22 for two values of ■ They have a very pronounced maximum near the fuselage axis (at rj = 0.578fo), which, however, in some cases lies within the fuselage, and thus does not contribute to the lift distribution.
To determine the angle-of-attack distribution of a fuselage of finite length, a consideration equivalent to that of Sec. 6-2-2 [see Eq. (6-17)] leads to
As will be shown later, it is sufficient in most cases, however, to assume an infinitely long fuselage.
In Fig. 6-23, the rolling moments due to sideslip дсмхШ of a low-wing, a mid-wing, and a high-wing fuselage system from measurements of Moller  are plotted against the lift coefficient cL. For comparison, the values for a wing without dihedral and for a wing with a dihedral of v = 3° are also shown. The fuselage causes a parallel shift of the curve for the wing alone. Thus the fuselage influence is reflected in a contribution to the rolling moment due to sideslip, independent of the lift coefficient, corresponding to the contribution of the
Figure 6-23 Coefficient of the rolling moment due to sideslip Эсд^/Э0 vs. Eft coefficient Ci of wing-fuselage systems, from Moller. Fuselage: ellipsoid of revolution of axis ratio 1:7. Wing: rectangle 4 = 5. L = low-wing airplane, M= midwing airplane, H — high-wing airplane, W = wing alone (v = angle of dihedral).
dihedral at the wing alone. Figure 6-23 shows that the effect of the fuselage on the rolling moment due to sideslip may be replaced by that of an “effective dihedral” of the wing. Here the high-wing airplane has a positive effective dihedral, the low-wing airplane a negative effective dihedral.
This fact is taken into account in airplane design: In order to obtain approximately the same rolling moment due to sideslip for different wing high positions, the low-wing airplane is given a considerably larger geometric dihedral than the high-wing airplane.
?f yT— rj
Following the above procedure, Jacobs  determined theoretically the fuselage influence on the rolling moment due to sideslip for an infinitely long fuselage. In Fig. 6-24, results are plotted of his computations for the additive rolling moment due to sideslip A(dcMx/d(3) as a function of the wing high position z0/R. Here the coefficient of the rolling moment due to sideslip is defined as Mx = cMxqxAs with s being the semispan of the wing. These theoretical results are compared with measurements by Bamber and House  and by Moller . Theory and measurements are carried to large wing high positions at which wing and fuselage no longer penetrate each other. Agreement between theory and measurement is very good. A closed formula may be obtained for the rolling moment due to sideslip caused by the fuselage by introducing into Eq. (3-100) the angle-of-attack distribution from Eq. (6-24) with z = z0 or y/s = r?, z0/s = and R/s = tf, respectively:
Figure 6-24 Additive rolling moment due to sideslip vs. wing high position, from Bamber and House and from Moller. Theory from Jacobs [the theoretical curves have been corrected considering the extended lifting-line theory ic’j^oo — 2rr) ]. Relative fuselage width rip = 1:7.5; aspect ratio A = 5.
Figure 6-25 Effective dihedral angle v€ff for a wing-fuselage system of rp — R/s and wing high position zQ IR for fuselages of circular cross sections. Theory from Eq. (6-28).
Here, Vo =Wrjp — fo is the coordinate on the fuselage surface and lc = 7i/l/c’Loo Л/2. For a simplified integration in Eq. (6-26), Multhopp  gave the value of unity to the square root in the integrand and changed the upper integration limit from unity to 2/77. For ^0 ^ (2/tr)2, this leads to
which is valid for fuselages with circular cross sections and wing high positions —R <z0 <R.
Comparison of Eqs. (6-27) and (3-158) yields the following expression for the effective dihedral, corresponding to the additive rolling moment due to sideslip caused by the fuselage:
In Fig. 6-25, the computed effective dihedral angle is plotted against the relative fuselage width Vf f°r several wing high positions z0/R. The effective dihedral angle increases strongly with increasing relative fuselage width tjf an<3 increasing wing high position. For instance, for rF = 0.12 and z0jR = ± 1, its values are = +3 and —3°, respectively.
Multhopp  conducted that kind of computation for fuselages of elliptic cross sections. Computations of the rolling moment due to sideslip for other fuselages have been conducted by Maruhn . Some of his results are presented in Fig. 6-26. Fuselages with angular cross sections produce a particularly large rolling moment due to sideslip. All the theoretical results discussed so far are valid for
Figure 6-26 Additive rolling moment due to sideslip of wing-fuselage systems vs. wing high position for several shapes of the fuselage cross section, from Maruhn [the theoretical curves have been corrected considering the extended lifting-line theory (с^те = 2тг)]. Wing: ellipse л = 3.8. Relative fuselage width rp = ~. Fuselage cross- section ratios hp/bp = 1.0 and 1.5.
infinitely long fuselages. Braun and Scharn  computed the effect of fuselages of finite lengths.
Yawing moment due to sideslip and side force due to sideslip of a wing-fuselage system The wing-fuselage arrangement has a quite small effect on the yawing moment due to sideslip. Essentially, the right value for the yawing moment due to sideslip of a wing-fuselage system can be obtained by adding the stabilizing contribution of the wing (Sec. 3-5-3) to the destabilizing contribution of the fuselage (Sec. 5-2-3). Figure 6-27 shows the yawing moment due to sideslip of three different wing-fuselage systems (low-wing, mid-wing, and high-wing arrangements) from measurements of Moller . For comparison, the wing alone and the fuselage alone are also shown. Obviously, no substantial interference exists. Furthermore, it should be noted that, for the yawing moment of the entire airplane, the usually destabilizing contribution of wing and fuselage is much smaller than the stabilizing contribution of the vertical tail assembly (see Chap. 7). The interference of wing and fuselage is more pronounced, however, for the side force due to sideslip. Figure 6-28 shows the side force due to sideslip, again from measurements of Moller , for the three wing-fuselage systems of Fig. 6-27. Note that at Ci = 0.2, the side force due to sideslip for the high-wing and the low-wing airplanes is about twice as large as for the mid-wing airplane. Also, the coefficient of the side
Figure 6-27 Yawing moment due to sideslip of wing-fuselage systems vs. lift coefficient. Measurements from Moller. Fuselage: ellipsoid of revolution 1:7. Wing: rectangle л — 5. L = low-wing airplane, Ж = mid-wing airplane, H = high-wing airplane, W = wing alone, F = fuselage alone.
force due to sideslip of the high-wing and the low-wing airplanes depends strongly on the lift coefficients. The larger values of Зсу/Э/З and their dependence on the lift coefficient for low-wing and high-wing planes find their explanation in the induced side – wash. Puffert  and Geisten and Hummel  studied these phenomena theoretically.