# SLENDER BODIES

In the previous sections of this chapter wing-fuselage systems with wings of large to moderately large aspect ratios have been discussed. Now systems with wings of small’ aspect ratios will be treated. Here the slender triangular wings (delta wings) with large sweepback play a special role. With flight velocities having increased from subsonic to supersonic speed ranges over the past decades, this kind of slender body (Fig. 6-40) has become most important. They are characterized by aerodynamic coefficients that are largely independent of the Mach number but depend, to a large extent nonlinearly, on the angle of attack (see Secs. 3-3-6 and 5-3-3).

The theory for lift computation developed by Munk [33] for slender fuselages and by Jones [17] for wings of low aspect ratio has been extended by Ward [49] and Spreiter [44] to wing-fuselage systems with wings of low aspect ratio; see also Jacobs [44]. The basic thought underlying this theory is the fact that changes in the perturbation velocities about slender bodies are small in the x direction (fuselage axis, wing longitudinal axis) compared with those of the perturbation velocities in

82Ф ду* |

the у and z directions normal to the x direction. This causes the potential equation, Eq. (4-8), to be reduced to that of two-dimensional flows in the yz plane:

where u = дФІду and w = ЭФ/Эг are the induced velocities in the lateral plane. Since Eq. (6-46) is valid for both incompressible and compressible flows, the results given below can be applied to both subsonic and supersonic incident flows.

The potential equation, Eq. (6-46), is to be solved for each cross section x = const (Fig. 6-4la), which can be accomplished by conformal mapping, for instance. The flow about a wing-fuselage system (Fig. 6-4lb) can therefore be determined from the flow about a flat plate at normal incidence (Fig. 641c).

Some results from Spreiter [44] and Ward [49] will now be discussed; see also Ferrari [6] and Haslet and Lomax [12].

Pressure distribution For wing-fuselage systems consisting of a delta wing and an infinitely long body of circular cross section, pressure distributions for two sections 1 and 2 normal to the axis are shown in Fig. 642. The load distribution on the wing is

In Eqs. (647д)-(648), у is the leading-edge semiangle of the wing, s(*) = л tan у is the local half-span, and R(x) is the body radius. The load-distribution curve in Fig. |

Figure 6-43 Load distribution in the longitudinal direction on the middle section (y = 0) and on the section at the wing root (y = R) for a wing – fuselage system with a delta wing (slender-body theory).

**6- **42 shows that the influence of the fuselage on the pressure distribution is greater at the front portion of the wing than at the rear portion. For comparison, the load distribution of the wing alone is also drawn for cross section 1 (curve l’).

in Fig. 6-43 the load distribution in the longitudinal direction along the wing-root section y=R is shown for the wing-fuselage system of Fig. 6-42. The influence of the body is seen in a somewhat smaller load decline in the axial direction than for the wing alone. The load distribution for the middle section (y — 0) is also given.

A procedure for computing the pressure distribution on slender bodies with arbitrary planform and cross-section shape is given by Hummel [14].

Lift distribution In Fig. 6-44, the lift distribution is shown versus the span of the wing-fuselage system of Figs. 6-42 and 6-43. The relative body width is rF = |. The effect of the body on the lift distribution is considerable. An example of the lift distribution over the body length is shown in Fig. 6-45. Note that the fuselage contributes to the lift only in the range of the wing. Close to the wing nose, the fuselage lift increases strongly; at the wing trailing edge, it drops abruptly to zero.

Total lift In Fig. 6-8, curve 2, the ratio of body lift Lp and total lift L(w+f) of a delta wing and an infinitely long body of circular cross section, according to this theory, was plotted as a function of the relative body width rip. Comparison of curves 2 and 1 in Fig. 6-8 shows that the slender-body theory yields almost the same values of Lp/L(W+p) as the theory of Lennertz [27], which is valid for arbitrary aspect ratios. We can conclude, therefore, that the values of Lp/L^+pj from the slender-body theory can also be used for wing-body systems with wings ol larger aspect ratios.

The total lift for wing-body systems from Fig. 6-44 is

L(w+P) = 2’nacaqoa(s2 — R2)2

Figure 6-44 Lift distribution over the span for a wing-fuselage system with a delta wing, = j (slender-body theory). Curve 1, У_ wing + fuselage. Curve 2, wing + flattened s fuselage. Curve 3, wing alone.

Figure 6-45 Lift distribution of the fuselage for a wing-fuselage system with a delta wing (slender-body theory).

Hence, when referring the lift coefficient cL to the wing area A — crs and the dynamic pressure of the incident flow qoo, setting Л =4sjcr and qF =R/s, the lift slope becomes

The second relationship applies to the wing alone (77^=0); see Eq. (3-101b). In Fig. 6-46, the ratio of the total lift to the lift of the wing alone, that is, L(w+F)ILw, is plotted as curve 1 against the relative fuselage width r)F. With increasing vF, the ratio L(w+f)ILw decreases strongly and becomes zero for Vf = I-

At a fuselage that is pointed in front, this finite front portion of the fuselage produces a lift additive to that of the infinitely long front portion from Eq. (6-6):

LFf = 2na.00q00Rl (6-51)

This means an increase of the lift slope over the value of Eq. (6-50c), and the lift slope becomes

{ da – )(w+F) = ‘2^(1 — Vf + Vf) (6-52)

The ratio L(w+f)/Rw f°r this case is plotted against the relative fuselage width T]F as curve 2 in Fig. 6-46.

Neutral-point position Finally, in Fig. 6-47, a few results are shown on the shift of the neutral point as caused by the fuselage. For the wing-fuselage system of Fig.

Figure 6-46 Ratio of total lift to wing lift for wing-fuselage systems with a delta wing (slender-body theory). Curve 1, infinitely long fuselage. Curve 2, fuselage of finite length (with fuselage nose).

Figure 6-47 Neutral-point shift of wing – fuselage systems with a delta wing (slender – body theory), from Spreiter, Curve 1, wing +• fuselage. Curve 2, wing + flattened fuselage. Curve 3, substitute – wing (with rectangular middle portion).

6-44, the shift of the neutral point caused by the fuselage, relative to the neutral point of the wing alone {AxN/c^w+F), is plotted as curve 1 of Fig. 6-47 against the relative fuselage width. From the theory for small aspect ratios, the neutral point of the wing alone lies at a distance cr from the wing nose. When rF increases, the neutral point moves rearward by an amount

(—^ = – -2^— (6-53)

/(W+-F) (1 + T? f)2

For r]F = 1, the shift of the neutral point becomes (AxN/Cfj)(w+F) — \ that is, in this case the neutral point of the wing-fuselage system is located at the wing trailing edge, as can easily be understood from inspection of Fig. 6-45. In Fig. 6-47, curve 2, the neutral-point shift is given for a “flat” fuselage (height zero). The difference to curve 1 is relatively small. For the case of a flat fuselage, the lift distribution over the span is also shown in Fig. 6-44 as curve 2. For comparison, the neutral-point shift for a wing with rectangular middle section (substitute wing) is given in Fig. 6-47, curve 3.

At last the case of a fuselage with a front portion of finite length will be discussed. The moment of the fuselage front portion, relative to the axis through the wing neutral-point, is given from Eqs. (6-6) and (5-32) as

Here, If is the length of the fuselage front portion from Fig. 6-48, and xNw is the distance of the wing neutral point from the fuselage nose. The distance (xNW — If) is easily determined as cr(2 — Зт}р). Evaluation of Eq. (6-54) for a fuselage with a parabolic nose yields

(6-55)*

where Сц =| cr.

For the wing-fuselage system with a fuselage front portion of finite length, the neutral-point shift relative to that of the wing alone is

_ V+F)°° + Mpf

Л JCjf j і r

l(W+F) « ^ LFf

The index 00 refers to wing-fuselage systems with infinitely long fuselages, where L(W+jp) is computed from Eq. (6-49) and M^w+p)«, = — AxnL(w + f’)<*> with ^xn from Eq. (6-53). The values of Lpf and Mpf are given by Eqs. (6-51) and (6-55), respectively.

*For a fuselage with elliptic nose section, the factor ^ of lf/cr must be replaced by 1.

Figure 6-48 Neutral-point shift of wing – fuselage systems with a delta wing and a fuselage of finite length (slender-body theory), from Eq. (6-56).

The shift of the neutral point according to Eq. (6-56) is plotted in Fig. 6-48 as a function of the relative body width rjp f°r various lengths of the fuselage front portion lflcr. These plots show that the shift of the neutral point AxN is positive (stabilizing) for small values of lf/cr, as in the case of an infinitely long fuselage (Fig. 6-47). At larger values of lfjcr, however, the unstable contribution of the fuselage front portion is predominant, making AxN negative.

Test results Finally, some test results will be presented that show the nonlinear lift characteristics cL(a) for slender bodies. In Fig. 6-49 the lift coefficients for three wings, three fuselages, and three wing-fuselage systems in incompressible flow are presented from Otto [36]. The lift coefficients of the fuselages (Fig. 6-49b) are referred to the wing area. For the wings alone, the results of linear theory for slender bodies according to Eq. (6-50&) are also shown. In all three cases (wing, fuselage, wing-fuselage system), the deviation from linear theory is considerable. Corresponding investigations on slender conical wing-fuselage systems in supersonic incident flow have been reported by Stahl [46]. Measurements on the vortex system of inclined wing-fuselage systems have been conducted by Grosche [10].

The design of slender, integrated airplanes for supersonic flight has been proposed by, among others, Kiichemann [23]. Design questions for airplanes in the transonic flight mode are discussed by Lock and Bridgewater [29].