The Fuselage in Asymmetric Incident Flow

General remarks Now the asymmetric inviscid flow about an inclined fuselage as in Fig. 5-12 will be considered.

First, it is important to state that, in inviscid flow, only a moment, not a resultant force, is acting on the – inclined fuselage. This is caused by the underpressures on the upper side of the body nose and the lower side of the tail and, conversely, the overpressures on the lower side of the nose and the upper side of the tail. This pressure distribution results in a moment Mp that attempts to turn the fuselage nose up (unstable moment). At small angles of attack a, this moment is proportional to the angle of attack.

The fuselage-wing interaction changes the magnitude of this moment greatly (see Chap. 6). However, the moment of the fuselage alone will be treated here, first in inviscid flow and later with consideration of friction. It should be mentioned that

Figure 5-12 Inviscid flow about an inclined fuselage.

Figure 5-13 Coefficient k for the computation of the moment of an inclined general ellipsoid of Eq. (5-186), from Vandrey.

the effect of friction on the aerodynamic properties of the fuselage is considerable. The moment in inviscid flow can be obtained from simple momentum considera­tions. Computation of the pressure distribution on the fuselage surface requires application of potential theory. As in the case of the fuselage in axial incident flow, exact solutions and approximate solutions to the singularities method are known. Finally, the effect of friction can be determined with the help of boundary-layer theory.

Fuselage moment by the momentum method of Munk An early account of the computation of the moment of an inclined fuselage was given by Munk [41]. It is based on an application of the momentum law. The momentum far behind a body moving at constant velocity in an inviscid fluid remains unchanged and no resultant force acts on the body, but this does not exclude the existence of a free force couple. According to the Munk theory, lift and pitching moment (free force couple) of a fuselage at an angle of incidence a and at free stream velocity Ux are

Lf = 0 (5-1 So)

MF = 2kqaoVFa (5-186)

Here, = (q{2)UL is the dynamic pressure of the incident flow, VF is the body volume, and A: is a factor describing the ratio of the volume of the fluid quantity moving with the body to the body volume. Values of к for general ellipsoids have been given by Zahm [36] and presented graphically by Vandrey [40]. The coefficients к for general ellipsoids of volume VF = mbc are given in Fig. 5-13 as a function of the axis ratios c/а and 6/c. Accordingly, the coefficient к for slender ellipsoids of revolution (b = c and c/a < 0.2) differs little from unity. Thus, from

Eq. (5-186), the moment of slender bodies of revolution is obtained from the

simple approximation formula

MF = 2qaaVFa (5-19)

Note that the unstable moment of the aerodynamic forces acting on an inclined slender fuselage of revolution is proportional to the angle of attack a and the body volume VF [Eq. (5-la)].

Pressure distribution by the method of dipole distribution The flow field of an inclined body of revolution can be computed by the singularities method. In the simplest approach, a distribution of spatial dipoles as in Fig. 5-14 is arranged on the body axis.* The axes of the dipoles are parallel to the z axis. The potential of the dipole distribution is

cos # m{x) 2л r

where m(x) stands for the dipole strength. The second relationship results from the expansion of the potential for small distances r from the axis, as required for slender bodies. The velocity components in axial, radial, and circumferential directions, respectively, are obtained from Eq. (5-206) as

8Ф и =—–	___	1	cos $ dm(x)	(5-21 а)
дх		2 л	г dx
дФ Щ = а			1 cos# . . „ , т{х)	(5-216)
8 г			2 л rs
1 гщ — г	дФ	=	1 sin# , . 2л тіХ)	(5-21 с)

The dipole strength is determined from the kinematic flow condition, which demands that, on the body, the velocity component normal to the surface is zero.

For a body with a cambered skeleton line as in Fig. 5-15, which is a generalization of Fig. 5-14, the kinematic flow condition becomes’

tx(x) U„з cos# – j – wr(it) = 0 for r = R (5-22)

Here a(x) is the local angle of attack of the skeleton line referred to the incident flow direction of U„ as given by

where a is the angle of attack of the fuselage axis and zF(x) is the skeleton line of the fuselage. Introduction of wr from Eq. (5-21 b) into Eq. (5-22) yields, for the dipole distribution,

m(x) = 2 я Ucq a (x) R2 {x) — 2 U^ a (x) AF(x) (5-24)

where Af(x) is the cross-sectional area of the fuselage.

Pressure distribution The inclination of the fuselage causes a pressure distribution on the body surface that, from Eq. (5-8), is given in first approximation as cp(x, $) = —2u(x, #)/f/oo. Introducing Eq. (5-24) into Eq. (5-2Їй) yields

cp (X, 0) = – 2 — £ [«(*) R – (*)] (5-25e)

If the angle of attack is constant along the fuselage axis, this equation takes the simpler form

dR(x)

cJx, #) = —4a cos $ —-— [a(x) = const] (5-25b)

и CiX

An example of these pressure distributions is given in Fig. 5-16 by means ol ellipsoids of revolution of thickness ratios SF = dFmaxllF = 0.1 and 0.2 and angle

"Here, the dipole distribution can be left on the body axis, as in the case of the plane skeleton theory (see Sec. 2-4-2).

Figure 5-16 Pressure distribution result­ing from asymmetric incident flow on ellipsoids of revolution of thickness ratios 5р = 0.1 and 0.2 from Eq. (5-26). Exact solution of Eq. (5-33).

of attack a= const. The following expression for the pressure distribution is easily found:

cp = — 2<x cos$ Sp (5-26)

fX(l – X)

Lift distribution The lift distribution is obtained from the pressure distribution by integration. A fuselage portion of length dx is supported by the lift force dbF of magnitude

2л

dLp = — qcoR(x)dx J cpcos&d& (5-27)

о

Observing Eq. (5-25a) and integrating over i3 yields the lift distribution,

j>И R2(ж)] (5-28)

CL tC G/2C

This relationship has been derived by Multhopp [40] from momentum considera­tions.

Equation (5-28) shows directly that the total lift force of a closed body vanishes, because

if i?(x) = 0 at the nose and tail of the body [see Eq. (5-18a)].

Pitching moment The pitching moment of the fuselage at constant angle of attack a(x) = const is obtained from Eq. (5-28) through integration by parts as

where VF is the fuselage volume from Eq. (5-1/2). In this way, the Munk approximation formula for the moment of slender bodies of revolution has also been obtained by means of the singularities method. Because LF = 0, the fuselage moment is independent of the location of the reference axis. It is a so-called free moment.

As an example, Fig. 5-17 illustrates, for theory and experiment, the lift distribution from [16] of an inclined ellipsoid of revolution of thickness ratio 8p=^. The theoretical lift distribution is obtained from Eq. (5-28) with a(x) — const as

(5-30)

Tills approximation is included in Fig. 5-17 as the solid line (line 1). The measurements agree well with theory in the front portion of the fuselage, but some deviations are found for the rear portion. For comparison, see Sec. 5-2-2.

The above. discussions apply to bodies of revolution. To determine the moments of bodies of noncircular cross sections at constant angle of attack, it should be realized that, essentially, only the fuselage width distribution bp(x) determines the

moment caused by the inclination. Equation (5-29b) can therefore be applied to fuselages of noncircular cross sections by substituting Dpi2 for R and introducing a correction factor k*. This leads to

(5-3lb)

о

Here Vp is the volume of a body of revolution that has the body width for its diameter. The correction factor can be determined by comparing Eq. (5-31 я) with the exact equation Eq. (5-18b) for general ellipsoids. Because Vp = (bjc)VF, we have k* = kc/b, where к is given in Fig. 5-13. The values of k* thus computed are presented in Fig. 5-18 as functions of fuselage width ratio 5p = bpm&x/lp and the cross-section ratio of the fuselage XF = hpmaxlbFmzx (see Fig. 5-1). It follows, therefore, that the factor k* is almost unity for slender fuselages of all practical cross-section ratios Лр. Thus, the above discussion has shown that for the computation of the moment of slender fuselages of noncircular ctoss sections, Eq. (5-29b) may be used in good approximation if the radius R is replaced by the semiwidth bpl2.

The moment of the fuselage of variable angle of attack a(x) is obtained by using the semiwidth bp/2 in Eq. (5-28) instead of R. Hence, integration over the fuselage length yields for the pitching moment, from Eq. (5-29b),

lF

MF – Qoo^ j a(x)bp(x) dx (5-32)

о

This equation is applicable to the fuselage with cambered skeleton line from Eq. (5-23) and to fuselages in curved flow as encountered during rotation about the lateral axis [see Eq. (3-147)]. Furthermore, this relationship is important for the

Figure 5-18 Coefficient k* for the computation of the moment of an inclined fuselage of noncirculai cross sections, from Eq. (5-3la).

computation of the fuselage moment when a wing is attached to the fuselage (see Chap. 6).

The above considerations on the lift distribution and on the moment furnish, accordingly, the side force distribution and the yawing moment due to sideslip for a yawed fuselage.

Exact solutions A few data will now be given on the exact solutions for inclined ellipsoids of revolution. Maruhn [36] determined the pressure distribution of the inclined ellipsoid of revolution at small angles of attack as

(5-33)

Here, b/a = bp is the fuselage thickness ratio. The quantity В is defined as

where a0 is given by Eq. (5-16). The angle-of-attack-dependent pressure distribution for this exact solution is shown in Fig. 5-16 for 8F =0.2. In the vicinity of the nose and the tail, the exact solution gives somewhat smaller values of the pressure coefficient than the approximate solution by the method of singularities. This means that the correction factor k* for the moment in Eq. (5-3 la) is somewhat smaller than unity. In Fig. 5-17, too, the exact solution for the lift distribution is included as curve 2. Near the nose and the tail, the exact solution deviates somewhat from the approximation solution. In the vicinity of the nose, the measurements agree quite well with the exact solution. Larger differences remain, however, near the tail. They are caused by viscosity effects to be treated in the next section.

The values of the moment from the exact solution have already been given in Eq. (5-182?). From Eq. (5-3la), the theoretical moment coefficient cMF = Mpjq^Vp is obtained as

cMf — 2k*a

This theoretical value, with k* — 0.95, is compared in Fig. 5-3 with a measurement. The moment slope dcMFida from this theory is considerably steeper than that of the measurement. This difference is caused by viscosity effects. Viscosity effects are also responsible for the deviation of the measured lift from zero, as seen in Fig. 5-3.

Viscosity effects Qualitatively, viscosity affects the flow over the inclined fuselage (Fig. 5-12) in such a way that the pressure on the tail section is reduced, because the inviscid outer flow is forced outward by the boundary layer. Consequently, the negative lift of the tail section is somewhat smaller than the positive lift of the nose section. Overall, therefore, viscosity effects cause a positive lift, which is also termed friction lift. This fact may be seen in Fig. 5-17 for the lift distribution. The friction
lift changes the moment; specifically, it creates an additive nose-down moment with reference to the lateral axis through the fuselage center. Hafer [16] has described a method for the approximate computation of the viscosity effect by means of boundary-layer theory. Accordingly, the boundary-layer displacement thickness 6г is determined along the fuselage surface at axial incident flow. A rather strong growth of the boundary-layer thickness near the tail is found, as sketched in Fig. 5-19. Consequently, to compute the pressure distribution, the local fuselage radius R(x) in Eq. (5-25a) must be replaced by the radius [i?(x) 4- 5i(x)], that is, the radius R(x) enlarged by the displacement thickness. The lift distribution is obtained from the pressure distribution corrected for viscosity by integration. In Fig. 5-17, the lift distribution, computed in this way from [16], is also shown. Through the correction for viscosity, better agreement is reached with the measurements, particularly in the vicinity of the tad. Lift, pitching moment, and neutral-point position are determined through further integrations. In Fig. 5-20, the lift slope dcLp/da, the moment slope dcMF/da, and the neutral-point position xNFllF are plotted against the inverse fuselage thickness ratio lFldFmax for several axisymmetric fuselages. These measurements were taken by Truckenbrodt and Gersten [50]. Curves 1 are those from the inviscid theory, curves 2 from the viscous theory of Hafer [16]. The latter theory agrees quite well with measurements.