Category AERODYNAMICS OF THE AIRPLANE

7- 1 INTRODUCTION

6- 1-1 Function of the Stabilizers
and Control Surfaces

The main parts of an airplane are the wing, the fuselage, and the tail unit or empennage. The aerodynamics of the wing has been discussed in detail in Chaps. 2-4, and that of the fuselage alone and the interaction between the wing and fuselage in Chaps. 5 and 6, respectively. Now, in Chaps. 7 and 8, the aerodynamics of the stabilizing and control surfaces will be discussed. Generally, an airplane has (see Fig. 7-1) a horizontal tail consisting of a horizontal stabilizer (tail plane) with an elevator, a vertical tail consisting of a vertical stabilizer (fin) with a rudder, and two ailerons.

A primary purpose of the tail unit is the stabilization of the airplane. This means that the airplane should have the tendency to return to a stationary flight attitude after a small disturbance. This process should take place “by itself’; that is, the aerodynamic forces should move the airplane back to the original attitude without application of the control surfaces.

Another equally important function of the tail unit is control of the airplane. Whereas the horizontal stabilizer and the elevator control the motion about the lateral axis, the vertical stabilizer with the rudder and the ailerons control that about the vertical and longitudinal axes (Fig. 1-6). The control of the airplane requires establishment of an equilibrium of the moments about the three axes. Here,

Figure 7-1 The geometry of the tail sur­faces (empennage).

in addition to the moments of the aerodynamic forces, those of the inertia forces play a role.

As has already been pointed out in Sec. 1-3-3, the motion of the airplane about the lateral axis is termed longitudinal motion, that about the vertical and longitudinal axes lateral motion (side motion). Consequently, the horizontal stabilizer and the elevator stabilize and control the longitudinal motion. The vertical stabilizer and the rudder stabilize and, together with the ailerons, control the lateral motion.

Generally, each of the three control assemblies has the form of a wing with a control surface as shown in Fig. 2-24. It consists of a fixed and a movable part. The fixed part is termed a fin or vertical stabilizer at the vertical tail and a horizontal stabilizer or tail plane at the horizontal tail. The movable part is the control surface. It is termed a rudder at the vertical tail and an elevator at the horizontal tail. In Fig. 7-1, the horizontal stabilizer and the elevator and the vertical fin and the rudder are indicated by hatches. The changes of the moments required for control are effected by deflections of the control surfaces. At the horizontal and vertical tail assemblies, the moments may also be controlled by a stabilizer adjustment (stabilizer trim). The horizontal tail of many airplanes does not have a separate stabilizer and elevator. Here, the change of the moment about the lateral axis is achieved by displacement of the entire horizontal surface.

The aerodynamic effect of the horizontal tail is illustrated in Fig. 7-2 for an airplane with and without a horizontal tail. The lift coefficient is plotted against both the angle of attack and the moment coefficient. According to Fig. 7-2u, the contribution of the horizontal tail to the total lift is relatively small. Figure 7-26 gives the moment curves for several setting angles at the tail plane. Comparison
with the curves for the airplane without a horizontal tail shows that at all setting angles of the tail plane, the horizontal tail causes a considerable increase in the stability coefficient dcM/dcL, as defined-in Sec. 1-3-3. A change in the tail-plane setting angle zH causes a parallel shift of only the moment curve cM(cL)* When the moment reference axis passes through the center of gravity at a steady flight attitude, the equation cM = 0 describes the moment equilibrium about the lateral axis. Figure l-2b shows that this condition can always be satisfied by choosing the proper setting angle eH of the tail plane for a given lift coefficient. The results on wing-fuselage-tail systems at subsonic, transonic, and supersonic incident flows reported by Pitts et al. [26] should be pointed out. Schlichting [34] gives a summary of the importance of the interference among wing, fuselage, and tail unit for the stability coefficients of the airplane.

The aerodynamics of the tail units will be treated in two parts: The problems concerning the. tail surfaces without deflection of the control surfaces (stabilization) will be covered in Chap. 7, those concerning the effect of the control surfaces (control) and of the flaps (lift increase) will be discussed in Chap. 8.

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

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

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].

The following discussions on the interference in wing-fuselage systems in transonic flow will be restricted mainly to the drag problem. The drag of wing-fuselage systems near Ma„ = 1 is generally larger than the sum of the drags of the wing
alone and the fuselage alone. Here the wave drag at zero lift is the major factor.

Figure 6-37 shows drag measurements by Whitcomb [50] on wing-fuselage systems in the Mach number range from Max — 0.85 to Mz« = 1.1 with Ci = 0. The tested models are shown in Fig. 6-37a, their total drag in Fig. 6-376, and the drag remaining after subtraction of the friction drag in Fig. 6-37c. The curve for the fuselage alone (model 1) shows a strong drag rise near Mr» = 1. The simple combination of wing and fuselage (model 2) produces a particularly large drag in the transonic range. Whitcomb [50] showed that by contracting the fuselage within the wing range, the drag in the transonic range may be greatly reduced (model 3). ‘ This contraction of the fuselage has to be chosen such that the wing-fuselage system and the original fuselage (model 1) have approximately equal distributions of the cross-sectional areas normal to the fuselage axis. This rule for the distribution of cross-sectional areas of a wing-fuselage system is called the “area rule.” Figure 6-38 shows the application of this rule to an airplane, where Fig. 6-38<з gives the plan view of the airplane, Fig. 6-386 the contour of an axisymmetric body of equal cross-sectional area distribution Ap(x), and Fig. 6-38c the variation of this cross-sectional area along the fuselage axis dAp/dx. In Fig. 6-38c, the case without

Figure 6-37 Drag coefficients of wing- fuselage systems and axisymmetric fuse­lages in the transonic Mach number range, from Whitcomb. (a) Geometry. (b) Total drag coefficients Cjj at zero lift, (c) Coefficients of wave drag.

Figure 6-38 The area rule for transonic flow, (a) Airplane planform. (b) Distribu­tion of the cross sections Ap{x) of the equivalent body of revolution, (c) Varia­tion of the cross-sectional area distribution along the fuselage dAp/dx.

area contraction is drawn as a solid line, the case with area contraction as a dashed line. The fuselage area contraction has been chosen for as smooth a dApjdx variation as possible.

For the experimental proof, Whitcomb [50] also tested a fuselage whose cross-sectional area distribution is equal to that of the wing-fuselage system without contraction (model 4 of Fig. 6-37). This model indeed has the same drag rise as model 2 in the transonic range. The theoretical basis of this phenomenon has been studied by Jones [18] and Oswatitsch [35], as well as by Keune and Schmidt [19]. Finally, it may be seen in Fig. 6-39 that the advantage of the area rule is limited to

the transonic Mach number range. This figure gives the drag coefficients of 3 wing-fuselage systems in the Mach number range from = 0.8 to Max = 1.4. Model 1 is the fuselage without contraction, whereas models 2 and 3 are fuselages with two different contractions. The contraction of model 2 has been chosen for largest drag reduction at Mz*, = 1, whereas that of model 3 is for lowest drag at MZoo = 1.2. These tests show that contraction according to the area rule yields favorable results only in the transonic range. In the supersonic range, the results are even less favorable than for fuselages without contraction.

In this connection, the comprehensive experimental studies should be men­tioned that Schneider [42] conducted on wing-fuselage systems with three different wings (rectangular, swept-back, and delta wings). A computation of the pressure distribution on wing-fuselage systems at an incident flow of Max = 1 and a comparison with measurements have been conducted by Spreiter and Stahara [45]. Compare also the computational methods in [20].

General remarks Numerous contributions to the aerodynamics of the wing-fuselage interference at supersonic velocities have been published. However, the establish­ment of a simple, generally valid method for its computation, like the method already available for incompressible flow (Sec. 6-2), has not yet been devised. Summary presentations have been given by Lawrence and Flax [26], Ferrari [6], Pitts et al. [37], and Ashley and Rodden [2].

The earlier theories on the wing-fuselage interferences were based on the assumptions made for the theories of the wing and of the fuselage and were limited to specific wing-fuselage systems. The first work of this kind came from Kirkby and Robinson [5]. Here a wing of large aspect ratio, attached to a conical body, is treated by means of the stripe method. This method does not take into account the lift loss at the wing-fuselage interface and the effect of the wing on the fuselage due to the large ratio of wing span to fuselage diameter. According to Cramer [4], these two contributions cancel each other to a large extent and the total lift is obtained relatively well. Fundamental investigations in the field of wing-fuselage interference at supersonic velocities have been conducted by Ferrari [5]. He was concerned with the problem of a rectangular wing of large aspect ratio on a cylindrical fuselage with a pointed nose. The solution is accomplished through an iteration procedure. After having determined separately the potential functions for the wing and for the fuselage, these functions are combined and corrected step by step in such a way that the boundary conditions are satisfied exactly for one part of the system only, either for the wing or for the fuselage, whereas the boundary conditions of the other part are disregarded. This procedure converges after a few steps.

Browne et al. [3] investigated a delta wing with conical fuselage (wing and fuselage apex coincide) by the method of cone-symmetric flows. Results are given for wings with subsonic and supersonic leading edges. Although this method does not offer extensive practical applications, its exact solutions are valuable for com­parisons with approximation solutions. The results of Lock [28] illustrate this fact.

A rectangular wing with a cylindrical fuselage has been treated by Morikawa
[31] and Nielsen [37]. By applying the Laplace transform, an exact solution is obtained in the form of a series. Morikawa offers only an approximate solution, but Nielsen succeeded in re transforming the solution. The computational procedure of Woodward [52], applicable to subsonic and supersonic flows as well, should be mentioned.

Now it will be shown that the essential relationships may be gained through simple, physically plain considerations without lengthy mathematical derivations; see also Schrenk [43].

In analogy to incompressibly flow, the wing-fuselage interference for supersonic flow will be analyzed by first discussing the effect of the wing on the fuselage and then the effect of the fuselage on the wing.

Lift distribution of the fuselage As has been shown in Sec. 5-3-3, the lift distribution at supersonic incident flow of the fuselage alone may be determined from the relationship for incompressible flow, by setting a(x) = aoo = const in Eq. (5-28), as

In Fig. 6-32n, the supersonic flow for the simple wing-fuselage system of an axisymmetric fuselage and a rectangular wing in mid-wing position is demonstrated schematically. The absence of an influence of the wing on the fuselage portion before the wing in supersonic flow marks the important difference between supersonic incident flow and incompressible flow. Hence, the lift distribution for the front portion of the fuselage in a wing-fuselage system is identical to that of the fuselage alone as given by Eq. (6-35). Thus the lift of the fuselage front portion becomes

LFf=2’noLaaqmR (6-36)

For the remaining portion of the fuselage, a simple survey will be given of the lift distribution created in addition to Eq. (6-35) by the wing on the fuselage. To simplify the problem, a wing of infinite span has been assumed in Fig. 6-32. In this case the wing generates perturbation velocities only in the range between the Mach lines mі and m2 originating at its leading edge and its trailing edge. Under the simplifying assumption that the lift distribution of the wing is unchanged in the fuselage range, no additive lift force, caused by the wing, acts on the rear fuselage portion either. Hence the fuselage feels an additive lift force only within the range between the Mach lines тг and rn2.

This lift due to the wing influence is caused by the velocities induced by the wing in the x direction, that is, u(x), and in the z direction, that is, w(x). In Fig. 6-32Z? and c, the distribution of the induced velocities u(x) and w(x) is given. Their variation on the fuselage surface at z = 0 and # = 90°, respectively, is marked by the dashed curve, and that at у = 0 and $=0°, respectively, by the dash-dotted curve. These two curves are merely displaced from each other in the longitudinal

Figure 6-32 Computation of the inter­ference of wing-fuselage systems at supersonic incident flow, (a) Geometry of the wing-fuselage system. (b) Distri­bution of the longitudinal velocity u(x). (c) Distribution of the vertical velocity w(jc). (d) Lift distribution due to the longitudinal velocity, (e) Lift distribution due to the vertical veloc­ity. 00 Resultant lift distribution.

direction. The maximum values of the induced longitudinal and vertical velocities are given in Eqs. (4-41tz) and (441 b) as [31]

fuselage. The following expressions from Ferrari [5] are obtained for 0<x <x0, if in this range R(x) =R0 = const:

Here x0 =R0 ^/MaL — 1. Corresponding formulas are valid for с <x < c +• x0.

From these mean values of the induced velocities, two contributions are obtained to the lift distribution of the fuselage in the wing range, namely, (dLp/dx)і from the longitudinal velocity й(х), and {dbpldx)2 from the vertical velocity OieoUoo + w(x). For the case of a constant fuselage radius R(x) —R0 within the wing range 0 <x <c 4- xQ, these contributions are

Ш,-*-*•!? <«■>

(^ = 2m™R2°&c

The qualitative trend of these two contributions is shown in Fig. 6-32d and e. The resulting lift distribution as the sum of these two contributions is presented in Fig. 6-32/. Integration of the contribution of Eq. (6-396) results in LF2 = 0, because, according to Fig. 6-32c, w(x) is equal to zero before and behind the wing. From Fig. 6-326, the lift force is obtained through integration of Eq. (6-39a), because Lpi = Lp, as

This simple relationship for the lift of the fuselage due to the wing applies to the cases where the intersection of the front Mach line mi with the fuselage surface at у = 0 lies before the wing trailing edge (see Fig. 6-32a). There is another interpretation of this result of Eq. (640), namely, that the fuselage lift due to wing effects is equal to the lift in plane flow of the wing portion A A = 2R0c shrouded by the fuselage. Also, the case has been investigated by Ferrari [6] where the front Mach line intersects the fuselage upper edge behind the wing trailing edge. In this case the computation of the additive fuselage lift is considerably more difficult than explained above.

In Fig. 6-33 the lift distribution of the fuselage under the influence of the wing is shown for an example of Cramer [4]. The theoretical curve has been computed by Ferrari [5]. At the Mach number Ma^ = 2 used in this study and the chosen geometry of the wing-fuselage system, the Mach line from the leading edge of the wing – intersects the upper edge of the fuselage behind the wing trailing edge. Therefore, contrary to Fig. 6-32f, the lift distribution reaches far beyond the wing trailing edge. Agreement between theory and measurement is good.

As stated earlier, the above theoretical results apply to the case of wings of very large span. The effect of the wing aspect ratio can be seen in Fig. 6-34, where, from [6], the additive lift distributions of the wing are plotted for wing-fuselage systems with wings of several aspect ratios. The Mach number is also Max = 2. When the aspect ratio decreases, the additive lift force decreases considerably, as would be expected.

For the test series of Fig. 6-34, the ratio of the fuselage lift Lp and the lift of

Figure 6-35 Ratio of fuselage lift Lpto wing lift L’yj vs. relative fuselage width rip for wing-fuselage systems at supersonic veloc­ities, from Ferrari (system as in Fig. 6-34). Mach number Ma«, = 2, angle of attack a» = 8°. Curve 1, theory from Lennertz. Curve 2, slender-body theory.

the wing portion not shrouded by the fuselage, JL’w, is plotted in Fig. 6-35. The data are compared with theoretical curves: Curve 1 reflects the theory of Lennertz [27] from Eq. (6-5), valid also for supersonic flow. Curve 2 gives the theory of wing-fuselage systems with wings of small aspect ratio of Sec. 6-4. The two theoretical curves are not very different. The measured data agree quite well with the theoretical curve (2).

The lift distribution leads to the pitching moment from Eq. (5-32). By introducing the two contributions of the lift distribution from Eqs. (6-39u) and (6-39b), the following two contributions to the pitching moment, referred to the middle of the wing, are obtained, where the contribution of the fuselage front portion has been disregarded:

Here a: is measured from the wing leading edge. From Fig. 6-32e, Mp2 is a free moment. The total moment (without ihe fuselage front portion), referred to the middle of the wing, thus becomes

Mp = —4 ті cj.-^ aoc RqC (6-43)

Note that this additive moment is independent of the Mach number.

The neutral-point displacement due to the wing influence on the fuselage with reference to the neutral point of the wing alone, xNW — cl2, becomes

O^JVOCW’ + F) – “ і

b(W+F)

For small relative fuselage widths r)F = 2R0/b, L(w+f) may be approximated by Lw. For the wing of large aspect ratio, from Eq. (4-46), this leads to

(4xn)(w+f) n 2 a rm.—————– 7

————————- ———————— = — rjpAyMaio – 1

This stabilizing neutral-point displacement due to the effect of the wing on the fuselage counteracts the destabilizing contribution of the fuselage front portion.

The above results on the effect of the wing are valid for the unswept wing of large aspect ratio, that is, for wings with supersonic leading edges. For wings with small aspect ratio, the discussions of Sec. 64 should be examined.

Lift distribution of the wing The effect of the fuselage on the lift distribution of the wing at supersonic velocities can be determined approximately by the method applied in Sec. 6-2-2 to incompressible flow. The additive angle-of-attack distribu­tion, caused by the cross flow over the fuselage as in Fig. 6-5b, creates additive lift locally on the wing. Under the assumption of an infinitely long fuselage, the additive angle-of-attack distribution for a given fuselage cross-section shape is the same as in incompressible flow, because the velocity of the cross flow of the fuselage is considerably lower than the speed of sound. Equations (6-16a) and (6-16b) give the distribution of the induced angle of attack for a fuselage of circular cross section (radius R) with a wing in mid-wing position. The computation of the approximate lift distribution along the span for the given angle-of-attack distribution may be conducted very easily with the so-called stripe method.* Hence, the local lift coefficient becomes

with Aa(y) from Eq. (6-16a).

An example for a wing-fuselage system of an axisymmetric fuselage and a rectangular wing is given in Fig. 6-36. It shows the lift distribution plotted against the span for the Mach number Маж = 2 at an angle of attack a„ = 8°. Curve 1 reflects the theory of the stripe method, Eq. (645), and curve 2 a theory of Ferrari [6]. Both theories agree quite well with the measurements, except for the stripe method in the immediate vicinity of the fuselage. For comparison, the theory for

Figure 6-36 Lift distribution on the wing due to the fuselage effect for a wing-fuselage system (mid-wing airplane) at supersonic velocities. Curve 1, theory, stripe method, from Eq. (6-45). Curve 2, theory from Ferrari. Curve 3, measurements from Ferrari. Curve 4, theory, wing alone.

the wing alone is added as curve 4. Obviously, the influence of the fuselage on the lift distribution of the wing is rather large.

The above results on the effect of the fuselage on the lift distribution of the wing apply to wings of large aspect ratios. For wings of small aspect ratios, reference should again be made to Sec. 6-4.

Wave drag The problem of the determination of the wave drag of wing-fuselage systems at supersonic velocities has been attacked by Vandrey [48], Lomax and Heaslet [30], Jones [18], and Keune and Schmidt [19]. Also, the experimental investigations of Schneider [42] should be mentioned.

6- 3-1 The Wing-Fuselage System in Subsonic Incident Flow

Fundamentals The following discussions on the flow about a wing-fuselage system at subsonic velocities will be limited to the case of straight flight. The effect of

Figure 6-28 Side force due to sideslip vs. lift coefficient. Measurements from Moller (system as in Fig. 6-27).

compressibility on the flow about a wing has been explained by means of the Prandtl-Glauert-Gothert rule in Sec. 4-4-2 for wings, and for fuselages in Sec. 5-3-2. This rule renders feasible the determination of a subsonic flow {Маж < 1) about wings and fuselages by means of a transformation to incompressible flow. By this means the incompressible flow will be computed for a transformed wing and a transformed fuselage. The transformation of the geometric quantities for wing and fuselage is given by Eqs. (4-66), (4-67a), (4-67Z?), (4-68д)-(4-68с), (5-51), (5-52<z), and (5-52b), where the quantities for incompressible flow are marked by the index “inc” and those for the compressible flow are given without the index. These quantities are as follows:

By computing the incompressible flow for the transformed wing-fuselage system at the angle of attack of the compressible flow, the transformation of the pressure coefficient from Eq. (4-69) becomes

cp = i-^—f-Cpinc C«inc=«) (6-32)[30]

Through an analogous transformation, the lift coefficients and the pitching-moment coefficients of wing-fuselage systems are obtained.

The discussions about the incompressible flow over wing-fuselage systems of Sec. 6-2-2 led to the conclusion that the lift slope of the wing-fuselage system is little different from that of the wing alone if the relative width of the fuselage is small to moderately large. Consequently, the relationship Eq. (4-74) for the wing alone applies directly to the wing-fuselage system, or

‘dcL_____________ 2rtA_______

У(1 – Mat)A~ + 44-2

The dependence of the neutral-point position on the Mach number of a wing-fuselage system follows immediately from the relationships just stated, because xN/c = —dcM/dcL, as

XN(W+F) XN(W+F) inc

~c " c (6’34)

Here xn(w+F) inc is the neutral-point position of the wing-fuselage system at incompressible flow as transformed according to Eqs. (6-29)-(6-31). As an example, the lift slopes and neutral-point positions are presented in Fig. 6-29 against the Mach number. These measurements of Schneider [42] are compared with theoretical results. The lift slope is little affected by the fuselage; the neutral-point displacement, however, shows a considerable fuselage influence. Results for wing-fuselage systems with a rectangular and a delta wing are also available in [42].

Investigation of the wing-fuselage system by means of the panel method As a result of utilizing efficient computers, methods have become more useful that are based on singularities distributed on the body surface, thus satisfying exactly the kinematic boundary conditions. Such generalizations have the advantage that geometric restrictions in the body shape are essentially eliminated. Based on the computational procedure for the displacement flow of Smith and Hess [13], the simultaneous treatment of the displacement flow and the lift flow of wing-fuselage systems has been presented independently by Kraus and Sacher [22] and Labrujere et al. [25]. In the method of Kraus and Sacher, the displacement flow is generated through a

Figure 6-29 Lift slope (a) and neutral-point shift (b) due to the fuselage effect vs. Mach number for axisymmetric fuselage with swept-back wing. Measurements from Schneider; theory for

incompressible flow from Sec. 6-2-2. (o———– o) Wing alone, л =2.15, A = 0.5, ^=52.4°.

(?——– v) Wing and fuselage, Ipjd = 12.5, ejd = 7.25, b/d = 5.0. — ■ — • — ■ a) Wing and

fuselage, lp/d = 10, ejd = 6.5, bid = 3.33.

source-sink distribution on the surface of the body, the abrupt change of the potential of the circulation flow through a vortex distribution within the body. The total potential results from the superposition of the individual contributions. The defining equations for the as yet unknown singularities are established by means of the kinematic boundary condition, to be satisfied on the surface. The flow conditions for the displacement problem are expressed by the requirement that no flow is to penetrate the body surface, that is, that the velocity normal to the body surface is zero. For the lift problem (circulation flow), this condition requires smooth flow-off at the trailing edges of the lift-producing surfaces. The distribution of the singularities and the perturbation potentials are obtained from the solutions of the equations defining the singularities. Thus the total potential and the velocities and pressure coefficients are obtained in the entire flow field and on the body surface. The pressure coefficients are computed with all three of the components of the perturbation velocity.

A suitable approach to the solution of the defining equations is found in the panel method. The singularities, first assumed to be distributed continuously, are now assumed to be constant on small flat surfaces (panels) and thus are accessible to analytic integration of their defining equations. For the displacement flow, panels covered with a constant source-sink density as in Fig. 6-30a are distributed on the surface. For the circulation flow, panels on the inner surface are used, and on the edges of these panels a vortex filament is laid of constant vortex strength. This leads, as in Fig. 6-3Ob, to the well-known picture of lifting surfaces consisting of vortex ladders composed of individual horseshoe vortices forming elementary wings.

On each surface panel, carrying a singularity density assumed to be constant, a control point lies in its center of gravity at which the kinematic flow condition is to be satisfied. Hence there exist as many control points as surface panels with singularity densities individually assumed to be constant but as yet unknown in magnitude. To each vortex ladder (consisting of several inner panels with a vortex filament of constant circulation strength on their edges) a control point is assigned at the trailing edge of the wing in which the Kutta condition is satisfied. Thus there are as many Kutta control points as vortex ladders with unknown total circulation strength (the circulation distribution within each of the vortex ladders on the individual panels is assumed a priori to be a Bimbaum distribution).

The requirement of the defining equation that the kinematic boundary condition has to be satisfied at all control points for all perturbation potentials of all panels leads to a system of linear equations of a form similar to that derived for the lifting-surface method (see [22]). This, in most cases very extensive, system of linear equations is solved through iteration by means of a Gauss-Seidel procedure. By this means the singularity strength and thus the velocity and the pressure distribution at the control points are obtained. A typical result of a computation of a wing-fuselage system is shown in Fig. 6-31 by means of the pressure distributions on a few selected fuselages and wing sections in comparison with measurements by Schneider [42]. The effect of compressibility has been taken into account through the Gothert rule (see [22]). Further methods of general validity for the computation of the aerodynamics of wing-fuselage systems at subsonic flow have been developed by Woodward [52], Giesing et al. [9], and Korner [21].

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 [15] 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.

Following the above procedure, Jacobs [16] 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 [16] and by Moller [15]. 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 [32] 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 [32] 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 [16]. 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 [16] 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 [15]. 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 [15], 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 [16] and Geisten and Hummel [8] studied these phenomena theoretically.

Total lift of a wing-fuselage system The first attempt at a theoretical description of the interference of a wing-fuselage system was made by Lennertz [27]. First, only the lift distribution on the wing and fuselage of such a system will be investigated. For simplicity, let the fuselage be an infinitely long circular cylinder as shown in Fig. 6-7, whereas the unswept wing has an infinite span. For the portion of the wing not shrouded by the fuselage, let the lift distribution over the span be known and thus the circulation distribution Г(у). The vortex system of the wing can be composed, from Fig. 3-2Qa, of horseshoe vortices of width dy and vortex strength Г, as shown in Fig. в-lb. To determine the lift of this arrangement generated at the fuselage, the kinematic flow condition must be satisfied on the fuselage surface, thus making the fuselage surface a stream surface. In a cross section normal to the fuselage axis far behind the wing, the flow in the yz plane is two-dimensional. The kinematic flow condition can here be satisfied by means of the reflection principle; that is, for every free vortex outside of the fuselage, a vortex reflected with respect to a circle has to be placed into the fuselage that has the same vortex strength but

Figure 6-7 Determination of the total hit of a wing-fuselage system, (a) Rear view. (b) Plan view with vortex system. (c) Circulation distribution in span direction.

the opposite sense of direction of rotation. The reflected vortex belonging to the free vortex at station у is located at a distance yF=R2ly from the fuselage axis, where R is the radius of the fuselage cross section.* Thus, a circulation distribution is obtained on the fuselage as demonstrated in Fig. 6-lc.

The lift of the wing portion not shrouded by the fuselage L’w is obtained through integration of the circulation distribution over the span from Eq. (3-15) as

S

L’w=2qUooJ r(y)dy (6-2)

y=R

Analogously, the lift of the fuselage becomes, with dyF = —(R2/у2) dy and r(yF) = Г(у) for the bound vortex,

5

L(w+F) = L’w "h Lp = 2q Uoo j’ Г(у) ^1 + —j dy

y=R

For numerical evaluation of this equation, an assumption must be made about the circulation distribution Г {у). The simplest case is a constant circulation distribution Г(у)— Г0 = const. Here, Eqs. (6-2) and (6-3) yield for the ratio of fuselage lift to wing lift and for the ratio of fuselage lift to total lift:

Lf R Lf T)p

tt^-^Vf – j———– = ГГТГ (6‘5)

The latter ratio is presented in Fig. 6-8 versus the relative fuselage width rtF = R/s as curve 1. Lawrence and Flax [26] and Luckert [32] have shown that curve 1 of Fig. 6-8 may also be applied, in very good approximation, to different lift distributions. Curve 2 of Fig. 6-8, from Spreiter [44], applies to wings of small aspect ratio (cf. Sec. 6-4). The result of Eq. (6-3) for the lift of fuselages may also be obtained from the integral of the pressure over the body surface or by means of the momentum theorem.

The above considerations fail to give information about the distribution of the lift of the fuselage over its length. This problem will be treated in the following section.

Lift distribution of the fuselage To determine the lift distribution over the fuselage length under the influence of the wing, the corresponding considerations for the fuselage alone of Sec. 5-2-3 may be applied. It was shown there that the lift

Figure 6-8 Ratio of the fuselage lift Lp to the total lift of a wing-fuselage system L(W+F) ys• relative fuselage width rjp = R/s. Curve 1, theory from Lennertz (r = const). Curve 2, theory from Spreiter (slender-body theory).

distribution over the fuselage length for a fuselage as shown in Fig. 6-9 is given by Eq. (5-28) as

чІ- = ^17і[я{х)ь]г{хГі (6‘6)

Here cLLp is the lift force of a fuselage section of length dx, bj?(x) is the local fuselage width, a(x) is the local angle of attack of the fuselage axis, and = Oil’Ll2 is the dynamic pressure of the incident flow. To compute the lift – distribution of the fuselage alone, the angle of attack in this equation has to be

Figure 6-9 The lift distribution of an inclined fuselage.

taken as сфс) = ax = const. For the lift distribution over the fuselage under the effect of the wing, the angle of attack has to be expressed as

л(я) = «со – f ocw(x) (6-7)

where aw(x) represents the upwash and downwash angles induced by the wing at the location of the fuselage (see Fig. 6-5a). For bF(0) = 0 = bF(lF), the total lift of the fuselage under the influence of the wing is obtained from Eq. (6-6), in agreement with Eq. (5-29c), as LF = 0. As was shown in Sec. 5-2-3, this relationship is valid for inviscid flows.

To compute the pitching moment at a variable angle-of-attack distribution crfx), Eq. (5-32) is already available. This pitching moment is independent of the position of the reference axis because it is a free moment. The above method for the computation of the wing-fuselage interference was developed by Multhopp [32]. The computation of the lift distribution over the fuselage length from Eq. (6-6) and of the pitching moment from Eq. (5-32) requires the determination of the distribution along the fuselage axis of the angle of attack aw(x) induced by the wing. This is a problem of wing theory that has already been treated in Sec. 2-4-5 for the two-dimensional case and basically in Sec. 3-2 for the three-dimensional case. A comprehensive presentation of the computational procedures for the induced velocity fields of wings will be given in Chap. 7.

The fundamentals of the method for the computation of the lift distribution and of the pitching moment can be understood from the simple case of a wing-fuselage system with a wing of infinite span, as shown in Fig. 6-10. The induced angle of attack of the inclined flat plate is given by Eq. (2-116) with Aq = a« and An = 0 for n > 1 [see also Eq. (2-66)]. Hence, Eq. (6-7) yields, for the local angle of attack,

a(X) = a[29] for X > 1 and X < 0 (6-8<z)

where X — xjc is the dimensionless distance from the plate leading edge. This distribution is shown in Fig. 6-10&. Within the range of the wing, 0<X< 1, there is aw(x) = —o>oc and thus

a(X) — 0 for 0<X< 1 (6-87?)

The local angle of attack a(x) from Eqs. (6-8д) and (6-87?) is discontinuous at the wing leading edge: The quantity o(x) drops abruptly from an infinitely large positive value to zero. At this station, dajdx has an infinitely large negative value, requiring special attention when determining the lift distribution from Eq. (6-6). For clearness in the computation of the lift distribution, the discontinuity of the a(x) curve has been drawn in Fig. 6-107? as a steep but finite slope. With the local angle-of-attack change thus established, the lift distribution of Fig, 6-1 Oc is obtained.* It has a large negative contribution in the form of a pronounced peak

Figure 6-10 Computation of the lift distribution on the fuselage of a wing-fuselage system, (й) Geometry of the wing-fuselage system. (b) Angle – of-attack distribution a(x). (c) Lift distribution dLpjdx.

directly before the wing leading edge. This is caused by the large negative value of da/dx close to the wing nose. The magnitude of this negative contribution is easily found when one realizes that for the fuselage section from the fuselage nose to a station shortly behind the wing leading edge, the lift force must be zero according to Eq. (5-29<z), because bp — 0 at the fuselage nose and a=0 shortly behind the wing leading edge. Accordingly, the positive contribution LpX and the negative contribution Lp2 are equal.

On the other hand, the lift distribution of the wing alone (without fuselage interference) has a strongly pronounced positive peak in the vicinity of the wing leading edge. Actually, this positive lift peak of the wing is reduced by the negative lift peak of the fuselage Lp2 mentioned above. Hence, a lift distribution over the fuselage is obtained, including the shrouded wing area, given as the solid curve of Fig. 6-10c.

Finally, this analysis shows that the total lift of the fuselage in the wing-fuselage system is approximately equal to the lift of the shrouded wing portion.

An example of this computational procedure and a comparison with measure­ments is given in Fig. 6-11. The fuselage is an ellipsoid of revolution of axis ratio 1:7 that is combined with a rectangular wing of aspect ratio /1=5 in a mid-wing arrangement. Curve 1 shows the theoretical lift distribution from Eq, (6-6). It is in quite good agreement with the measurements in the ranges before and

Figure 6-11 Lift distribution on the fuselage of a wing-fuselage system (mid-wing airplane). Fuselage: ellip­soid of revolution of axis ratio 1:7. Wing: rectangle of aspect ratio л = 5. Measurements from [41]; theory: curve 1 from Multhopp, curve 2 from Lawrence and Flax, curve 3 from curve 2, from Adams and Sears.

behind the wing. No result is obtained by this computational procedure within the range of the wing. The measured lift distribution shows a pronounced maximum in the – vicinity of the wing leading edge. Curve 2 represents an approximation theory of Lawrence and Flax [26], which will be discussed later; it is in satisfactory agreement with the measurements in the range of the wing. Curve 3 will also be explained later.

The – influence of the wing – shape on the wing-fuselage – interference can – be assessed best by means of the angle-of-attack distribution induced on the fuselage axis. For unswept wings, Fig. 6-12 illustrates the effect of the aspect ratio on the distribution of the angle of attack. All the wings have an elliptic planform. The angle-of-attack distribution has been computed using the lifting-line theory. For an elliptic circulation distribution its value becomes, Eq. (3-97),

(6-9)

where £ = xfs and the coordinate origin x = 0 lies on the c/4 line. Because § = SXjirA with X = xjlr and with the relationship between and a«, of Eq. (3-98), Eqs. (6-9) and (6-7) yield

In Fig. 6-12, a/aeo is shown versus X * Hence, in the range before the wing, the upwash angles become markedly smaller when the aspect ratio A is reduced. In the range behind the wing, however, the downwash angles increase with decreasing aspect ratio. At the цс point, all curves have the value a = 0, as should be expected because of the computational method used (extended lifting-line theory = three- quarter-point method).

The effect of the sweepback angle on the distribution of the angle of attack is shown in Fig. 6-13 for a wing of infinite span, constant chord, and unswept middle section. This latter section represents the shrouding of the wing by the fuselage as shown in Fig. 6-1. The induced angle-of-attack distribution on the x axis is obtained from the lifting-line theory according to Biot-Savart as

where Г is the circulation of the lifting line, is the sweepback angle, and sF is the semiwidth of the unswept middle section.^ The relationship between the circulation

Г and the angle of attack сгот of a swept-back wing of infinite span is expressed by Eq. (6-1 lb), because Cp = 2Г(ижс and cL = cos from Eq. (3-123). Consequently, Eq. (6-11 a) may be written in the form

^ cos 9? Л – j – у A* – f apsmq> (6-12)

2X X cos <p – j – Op sin cp

with X — xjc and aF = sF/c. The angle-of-attack distributions computed by this equation are plotted in Fig. 6-13 for sweepback angles <p = 0, +45, and—45°, and for Op— 0 and 0.5.* From Fig. 6-13 it can be seen that the upwash before the wing is reduced in the case of a backward-swept wing and the downwash behind the wing is increased. In the case of a forward-swept wing, the reverse occurs. As would be expected, introduction of the rectangular middle section reduces the effect of sweepback. The distribution of the induced angle of attack on the fuselage axis for the swept-back wing without a rectangular middle section (Sp = 0) is given, from Eq. (6-1 Ід), as

Since aw = —rj2irU0,x for the unswept wing, Eq. (6-13) shows that the effect of the sweepback angle on the induced downwash angle may be expressed by a factor.

The procedure discussed so far for the determination of the wing influence on the angle-of-attack distribution of the fuselage does not give any information about

the distribution in the range of the wing, as may be seen from Fig. 6-11. Lawrence and Flax [26] developed a method allowing determination of the angle-of-attack distribution over the entire fuselage length, including the shrouded wing section. The basic concept of this method is indicated in Fig. 6-14. Contrary to the previous approaches, which were based on an undivided wing, now the fuselage is taken as being undivided and the wing as divided. Consequently, the effect of the two partial wings on the fuselage is determined, whereby both the x component and the z component of the induced velocity must be taken into account.

The first contribution to the lift distribution is generated by the longitudinal velocity components u(x) because they determine the pressure distribution on the fuselage surface by cp = —hi/Uoo. The induced velocities on the surface z = R cos $ can be expressed by

Here it has been taken into consideration that dujdz — dw/dx, because the flow is irrotational, and further that the simple relationship dawjdx = dajdx follows from Eq. (6-7). The second contribution to the lift distribution is generated by the

Figure 6-14 Computation of the lift distribution on the fuselage of a wing-fuselage system according to the theory of Lawrence and Flax.

upwash velocities on the fuselage axis resulting from the vortex system of the two wing parts. The corresponding pressure distribution is obtained from Eq. (5-25a). Thus, the resulting pressure distribution on the fuselage is

Cp(x, d) = ~4 cos # ^ [ФЖх)} (6-14)

Introduction of this expression into Eq. (5-27) and integration over 0<#<27r yield the total lift distribution

—1 = 4ttqxR(x) [ф)Д(х)j (6-15)

Note the difference from Eq. (5-28). For the case a = const (fuselage alone), the equations are identical. Lawrence and Flax [26] have evaluated Eq. (6-15) assuming that the circulation distribution is constant on either wing part. This result is given as curve 2 of Fig. 6-11. For the fuselage portions before the wing and within the wing range, agreement of this approximation theory with measurement is quite good. For the fuselage portion behind the wing, the deviations from measurement are considerable. Therefore, a correction for this range has been given by Adams and Sears [1], shown as curve 3. It should be mentioned in this connection that the computational procedure of Multhopp [32] leads to nearly the same results.

Lift distribution of the wing Since the effect of the wing on the fuselage has been discussed, the effect of the fuselage on the lift generation on the wing will now be investigated more closely. A typical test result on this problem is shown in Fig.

6- 15. For a mid-wing system consisting of a rectangular wing and an axisymmetric

Figure 6-15 Measured lift distributions on the span for a mid-wing system and for the wing alone at several angles of attack, from [41]. Fuselage: ellipsoid of revolution of axis ratio 1:7. Wing: rectangle of aspect ratio /1=5. The curves for the mid-wing airplane include only the fuselage lift within the wing range.

Figure 6-16 Induced angle-of – attack distribution of a wing-fuse­lage system. The fuselage is an infinitely long circular cylinder; R — c0 /2. Curve 1, angle-of-attack distribution a(x) = a„ = const over the entire fuselage length. Curves 2 and 3, angle-of-attack distributions a(x) before and be­hind the wing are constant, a(x) = 0 within the wing range. Curve 2 for the unswept wing, curve 3 for the swept-back wing = 45°. Curves 2 and 3 give the distribution of the induced angle of attack on the – point line of the wing.

fuselage, and for the wing alone, distributions of the local lift coefficients over the span are shown. These data have been extracted from comprehensive pressure distribution measurements of Moller [15] on wing-fuselage systems. In the case of the wing-fuselage system, the lift coefficients refer to the wing portion shrouded by the fuselage. The lift distributions on the wing portions outside of the fuselage at three different angles of attack are consistently little affected by the fuselage. However, within the fuselage range, a considerable drop in the lift coefficient occurs. This reduced wing lift within the fuselage range has previously been discussed in connection with Fig. 6-1 Oc.

For the theoretical determination of the influence of the fuselage on the lift distribution of the wing, the additive angle-of-attack distribution from Fig. 6-5b has to be determined that is the result of the cross flow over the fuselage. Figure 6-16 shows as cum 1 the additive angle-of-attack distribution induced by an infinitely long fuselage of circular cross section. Outside the fuselage, the induced angle of attack Aa=wfUca for mid-wing systems is given by

= — (y>R) (6-16a)

«00 yz

where R is the radius of the circular cylinder. For the range —R <y < 4-і?, A a is determined from the velocity component in the 2 direction on the fuselage surface, resulting in

The angle-of-attack distribution thus determined has a very sharp peak of Aajaoo = +1 on the fuselage side wall, whereas the value Aa/a^ = — 1 is reached on the fuselage axis, that is, the local angle of attack a = a» + A a= 0 on the axis. By using this angle-of-attack distribution of the wing according to curve 1 of Fig. 6-16, the fuselage influence on the wing is greatly overrated because it is based on the assumption that the angle of attack of the fuselage is a = ax within the wing range, too. Multhopp [32] computed lift distributions with angle-of-attack distributions of this kind. Compare also Liess and Riegels [32] and Vandrey [47].

A better approximation for the fuselage influence on the wing is obtained under the assumption that the wing turns the flow within the wing range parallel to the fuselage axis, that is, that a — 0 in this range. The corresponding distribution of the induced angles of attack over the span can be determined by arranging a dipole distribution on the fuselage axis that is dependent on x. This procedure has been given for the fuselage alone in Sec. 5-2-3. With rcos& — z and r2 — y2 + z2 and with m from Eq. (5-24), Eq. (5-20я) yields for Aa = w/Uoo = (Ъф1Ъг)/их in the wing plane z — 0,

(6-17)

valid for y>R. For an infinitely long fuselage of constant width whose angle of attack is constant before and behind the wing and zero (a = 0) within the wing range, the result is

Here, l0 is the wing chord at the fuselage side wall. The distribution of the induced angle of attack, computed with Eq. (6-18), is shown in Fig. 6-16 as curves 2 and 3

3 with curve 1 demonstrates that this refined computational method leads to a considerably smaller fuselage influence.

Neutral-point position of wing-fuselage systems Besides the changes of the lift distributions of fuselage and wing, the change of the neutral-point position is of particular importance for flight mechanical applications (see Sec. 1-3-3). The distance of the neutral point from the moment reference axis is generally given by xN — —dM/dL. Hence, for the wing-fuselage system it becomes

dM(W+F) dL(w+F)

dMw dMp dL ц/ dL ц>

where is the pitching moment and Z,(w + F) is the total lift of the

wing-fuselage system. The pitching moment of the wing-fuselage system may be composed of the contributions of the fuselage Mp and of the wing My. The fuselage contribution can be computed as described previously. The wing contribu­tion will be taken to be the moment of a wing with rectangular middle section (substitute wing). Since the fuselage influence on the wing is generally small, it can often be disregarded (see Hafer [11]). The lift of the wing-fuselage system L^w+f) is given approximately by the lift of the wing alone Lw, as was shown earlier. Because M(w+F) =MW 4- Mp and £(w+f) ~ Lp, Eq. (6-Ш) is obtained. The first term gives the neutral-point position of the wing with rectangular middle section, which can be determined through computation of the lift distribution of such a wing according to the lifting-surface method. The second contribution gives the neutral-point displacement caused by the fuselage including the influence of the wing on the fuselage.

It is advantageous to refer the neutral-point position of the wing-fuselage system to the position of the neutral point of the wing alone, that is, of the original wing (Fig. 6-1). As reference chord, that of the original wing is chosen likewise. The neutral-point displacement of the wing-fuselage system from the aerodynamic neutral point of the wing alone becomes, from Eq. (6-19&),

the wing (introduction of the rectangular middle section into the range shrouded by the fuselage) and (AxN)p is the neutral-point displacement because of the fuselage. Obviously, the first contribution can be of real importance for only swept-back and delta wings. By considering, as a first approximation, the displacement of the geometric neutral point only, the neutral-point displacement of the swept-back wing of constant chord becomes

with 7]p as the relative fuselage width from Eq. (6-1).

The second contribution in Eq. (6-20), that is, the neutral-point displacement due to the fuselage, is obtained from the fuselage moment Mp by the relationship

where dcLldais the lift slope of the wing (see Sec. 3-5-2).

The neutral-point displacement caused by the fuselage of Eq. (6-22) depends mainly on the following geometric parameters, as intuitively plausible: wing rearward position, fuselage width ratio, and sweepback angle. In Figs. 6-17-6-19, a few computational results from Hafer [11] on the influence of these parameters are presented and compared with measurements.

The neutral-point displacement due to the wing rearward position for an

unswept wing is given in Fig. 6-17 as a function of the widely varied wing rearward position. The fuselage causes an upstream displacement of the neutral point (destabilizing fuselage effect) that increases with the rearward wing position. The wing high position, also varied in these measurements, has no marked effect.’ Agreement between theory and experiments is good.

Figure 6-18 illustrates the effect of the sweepback angle on the neutral-point

Figure 6-19 Neutral-point shift of wing-fuselage systems due to the fuselage effect vs. wing rearward position, from Hafer. (I) Swept – back wing; л = 2.75; = 0.5; <p — 50°. (II) Delta wing: л = 2.33; K = 0.125. Curve 1, fuselage with pointed nose. Curve 2, fuselage with rounded nose.

position caused by the fuselage. The measurements are for wing-fuselage systems with wings of constant chord (A = 1) and with trapezoidal wings (A = 0,2). The neutral-point displacement becomes markedly smaller when the sweepback angle increases. It is noteworthy that the neutral-point displacement is almost zero for strong sweepback (<p « 45°). Here, too, agreement between theory and measurement is quite good. The first theoretical studies about the effect of the sweepback angle on the neutral-point displacement caused by the fuselage was conducted by Schlichting [40].

Finally, in Fig. 6-19, results are given on the influence of the wing rearward position of a swept-back wing and a delta wing. The swept-back wing has the aspect ratio A = 2.75, the taper A= 0.5, and the sweepback angle of the quarter-point line іp— 50°. The neutral-point position of this wing has been shown in Fig. 3-37b. The delta wing has the aspect ratio A= 2.33 and the taper X = 0.125. The results of Fig. 6-19 are given for two different fuselage shapes, namely, a pointed and a rounded fuselage front portion. For either wing, in agreement with Fig. 6-17, a considerable increase in the neutral-point displacement is caused by the fuselage when the wing is moved rearward. Here, too, agreement between theory and measurement is good. Important contributions to the interference between a swept-back wing and a fuselage are also due to Kuchemann [24].

Drag and maximum lift of wing-fuselage systems The interference effect of wing-fuselage systems on drag and maximum lift lies mainly in the altered separation behavior when wing and fuselage are put together. These effects are hardly accessible to theoretical treatment, however, and their study must be limited to experimental approaches. The first summary report hereof comes from Muttray [34]; compare also Schlichting [38]. Very comprehensive experimental investiga­tions on the interaction of wing and fuselage, particularly concerning the drag
problem, have been conducted by Jacobs and Ward [15] and by Sherman [15].

For drag and maximum lift of a wing-fuselage system, the low-wing arrange­ment is particularly sensitive, because the fuselage lies on the suction side of the wing, strongly influencing the onset of separation at larger lift coefficients. Through careful shaping of the wing-fuselage interface by means of so-called wing-root fairings, the flow can be favorably affected in this case, that is, the onset of separation can be shifted to larger angles of attack.

The investigations of Jacobs and Ward [15] and of Sherman [15] cover a comprehensive program on two different fuselages (circular and rectangular cross sections) and two wings of different profiles (symmetric and cambered). Varied were the wing rearward position, the wing high position, and the wing setting angle. Included in the study was the effect of wing-root fairings.

The drag of a wing-fuselage system depends predominantly on the wing high position, and very little on its rearward position and its setting angle. In Fig. 6-20, the lift coefficient cL is plotted against the coefficient of the form drag

2

cDe=cD-^ (6-23)

of several wing-fuselage systems. The coefficient of the form drag is obtained as the difference of the coefficients of total drag and induced drag. These wing-fuselage systems are a mid-wing airplane with round fuselage and low-wing airplanes with round and square fuselages. For comparison, the wing alone is added as curve 1. A strong drag increase above a certain lift coefficient is characteristic for wing-fuselage systems. It is the result of the onset of separation caused by the fuselage. This

Figure 6-20 Lift coefficients of wing-fuselage systems vs. drag coefficients, from Jacobs and Ward. Cjrjg = coefficient of the form drag from Eq. (6-23). Fuselages with circular and square cross sections, wing profile NACA 0012.

Figure 6-21 Maximum lift coefficients of wing-fuselage systems, from [38]. Fuselages with circular ctoss sections, wing profile NACA0012. {a) Maximum lift coefficient vs. wing rearward posi­tion, z0//o = 0. (b) Maximum lift coeffi­cient vs. wing high position, e0//0 =0.

phenomenon is most pronounced in the low-wing system with round fuselage, curve 3, where separation begins very early at cL — 0.6. Here fuselage side wall and wing upper surface form an acute angle that particularly promotes boundary-layer separation. Considerably more favorable than the low-wing airplane is the mid-wing airplane, curve 2, because here the wing is attached to the fuselage at a right angle. By going from a round to a square fuselage, the conditions may be further improved, as shown by curve 4 for the low-wing airplane.

Theoretical results on the pressure distribution at the wing-fuselage interface are given by Liese and Vandrey [47] for the case of a symmetric wing-fuselage system (mid-wing) in symmetric incident flow (c^ = 0).

The maximum lift of wing-fuselage systems depends on both the wing high position and the wing rearward position. A survey of the cLmax values for several high and rearward positions is given in Fig. 6-21. From Fig. 6-21 a, the maximum lift coefficient cLmax decreases with increasing rearward position. In the most favorable case, cLm^x of a wing-fuselage system is equal to that of the wing alone. With regard to the wing high position, the mid-wing arrangement is least favorable, as shown by Fig. 6-2lb (compare also Fig. 6-3). From this value for the mid-wing arrangement, cLmax increases when the wing is shifted to both high – and low-wing positions.

6- 2-1 Fluid Mechanical Fundamentals of the Wing-Fuselage Interference

In the next section the quantitative computation of the interactions between wing and fuselage will be discussed, but a few physical explanations will be given here first. When putting together a wing and a fuselage, a flow about the wing-fuselage system results, with the fuselage lying in the flow field of the wing and the wing in the flow field of the fuselage. Thus, an aerodynamic interference exists between the fuselage and the wing, in that the presence of the fuselage changes the flow about the wing and the presence of the wing changes the flow about the fuselage. Consequently, the computation of the flow about a wing-fuselage system can be accomplished by first computing the flows about the wing and fuselage separately, and then adding the interference effects of the wing on the fuselage and the

fuselage on the wing. These interference effects are obtained by satisfying the

kinematic flow condition (zero normal component of the velocity on the surface of the wing-fuselage system).

The, flow field of a wing-fuselage system at subsonic velocity in symmetric incident flow (angle of sideslip /3=0) is illustrated in Fig. 6-5. Figure 6-5a shows the flow about the fuselage as affected by the wing. Along the fuselage axis, additive velocities normal to the fuselage axis are induced by the wing, which are

directed upward before the wing and downward behind it. In the range of the

wing-fuselage penetration, the flow is parallel to the wing chord, corresponding to a constant downwash velocity along the wing chord. The fuselage is therefore in a curved flow with an angle-of-attack distribution сфс) varying along the fuselage axis as shown in Fig. 6-5a. This angle-of-attack distribution, induced by the wing, shows that the fuselage is subjected to an additive nose-up pitching moment.

The effect of the fuselage on the flow about the wing is sketched in Fig. 6-5b. The component of the incident flow velocity normal to the fuselage axis Uoc sin Qoc ~ U0„а» generates additive upwash velocities in the vicinity of the fuselage. The effect on the wing of these induced velocities normal to the plane of the wing is equivalent to an additive symmetric angle-of-attack distribution over the wing span (twist angle).

The flow field of a wing-fuselage system at asymmetric incident flow is shown schematically in Fig. 6-6. The flow about the wing-fuselage system with the angle of sideslip /3 can be thought to be divided into an incident flow parallel to the plane of symmetry of the velocity Um cos @ ~ and an incident flow normal to the plane of symmetry of the velocity £/«, sin j3 ~ С/«Д The latter component of the incident flow generates a cross flow over the fuselage as illustrated in Fig. 6-6b, c, and d for a high-wing, a mid-wing, and a low-wing system, respectively. This cross flow over the fuselage results in an additive antimetric* distribution of the normal velocities along the span that is equivalent to an antimetric angle-of-attack distribution a(y).

The lift distributions over the wing span generated by this angle-of-attack distribution have reversed signs for high-wing and low-wing airplanes. The rolling moment (rolling moment due to sideslip), as affected by this antimetric lift distribution, is zero for the mid-wing airplane, positive for the high-wing airplane, and negative for the low-wing airplane. These findings are confirmed by the test results of Fig. 6-4, which show that the rolling moment due to sideslip ЬсМх/д& of a high-wing airplane is larger than for the wing alone.

The effect of the fuselage on the wing in yawing motion may be interpreted, therefore, as the effect of a positive dihedral of the wing on the high-wing airplane, and as that of a negative dihedral on the low-wing airplane.

Figure 6-6 Asymmetric flow over a wing-fuselage system (schematic), (a) Wing planform.

(b) High-wing airplane with angle-of-attack distribution a(y).

(c) Mid-wing airplane. (d) Low – wing airplane with angle-of – attack distribution a(y).

It is advantageous and generally customary to refer the aerodynamic coefficients of a wing-fuselage system to the geometric quantities of the original wing. A summary of the aerodynamic coefficients of the wing has been given by Eq. (1-21). These definitions of the aerodynamic coefficients are applicable directly to the wing – fuselage system when the forces and moments of the wing-fuselage system are
substituted. The reference axes and the signs of forces and moments are shown in Fig. 1-6.

To convey a feeling for the magnitude of the interference effects on wing-fuselage systems, a few test results are given in Figs. 6-3 and 6-4. In Fig. 6-Зд the lift coefficient cL is shown plotted against the angle of attack a for a simple mid-wing system of a rectangular wing and an axisymmetric fuselage, and for the wing alone. In the range of moderate angles of attack, the fuselage does not noticeably affect the trend of the cL(a) curve. The coefficient of maximum lift cLmax, however, is markedly reduced by the fuselage. This can be understood by realizing that the flow about the wing of a mid-wing airplane is strongly disturbed by the fuselage, leading to premature flow separation. The lift coefficient c£, versus the pitching moment coefficient cM for the wing alone and the wing-fuselage system is plotted in Fig. 6-36. Here the fuselage causes a strong increase in the pitching-moment slope йсмІйс^. The inclined fuselage alone has a pitching moment that tends to turn it into a crosswind position (see Sec. 5-2-3), and this pitching moment obviously is greatly increased by the effect of the wing.

In Fig. 6-4, the rolling-moment coefficient cMx is plotted against the angle of

Figure 6-4 Rolling moment due to side­slip of a high-wing system and of the wing alone, from Moller; fuselage and wing of Fig. 6-3.

sideslip for a high-wing system also consisting of a rectangular wing and an axisymmetric fuselage. The difference between the trends of the curves cMx(j3) for the wing alone and for the wing-fuselage system is quite large. The effect of the fuselage of a high-wing airplane consists of a strong increase in the rolling moment due to sideslip дсМх/Ъ&. This effect is caused by the cross flow over the fuselage. The interference effects shown in Figs. 6-3 and 6-4 can be treated theoretically. Other interference problems, particularly those of the drag of wing-fuselage systems, are hardly accessible to theoretical determinations. Therefore, in these cases experimental studies are indispensible [11, 15].

Summary reports about the interactions between the wing and fuselage in incompressible, and to some extent in compressible flow, have been published by Wieselsberger [51], Muttray [34], Schlichting [39, 41], Ferrari [6], and Lawrence and Flax [26], as well as Ashley and Rodden [2]. Surveys of the aerodynamics of slender bodies have been given bv Adams and Sears [1] and Gersten

For a better understanding of the aerodynamics of the wing-fuselage system to be discussed below, the geometry of such a system will be discussed first. The geometry of the wing has been described in Sec. 3-1 (Figs. 3-1 and 3-2), that of the fuselage in Sec. 5-1 (Fig. 5-1). The geometry of the wing-fuselage system is illustrated in Figs. 6-1 and 6-2. Figure 6-1 gives the plan view and the side view of a wing-fuselage system, Fig. 6-2 the rear view of two wing-fuselage systems. The position of the wing relative to the fuselage is defined by the wing rearward position e, the wing high position z0, and the angle of wing setting є0. As shown in Fig. 6-1, the wing rearward position e is the distance between the geometric neutral

point of the wing (Sec. 3-1) and the fuselage nose. According to Fig. 6-2a, the wing high position Zq is the distance between the wing and the fuselage axis. Its values are

High-wing airplanes: Mid-wing airplanes: Low-wing airplanes:

A typical mid-wing airplane with dihedral is sketched in Fig. 6-2b. The angle of wing setting є0 is, from Fig. 6-1, the angle between the chord of the wing root section and the fuselage axis. When the wing penetrates the fuselage, the portion of the wing shrouded by the fuselage requires special explanation. In the case of a swept-back trapezoidal wing, it is advantageous to replace the portion of the wing shrouded by the fuselage by a rectangular wing section. This rectangle is formed by the length of the root section /0 and by the mean fuselage width in the range of the wing Ьр0. For conventional wing-fuselage systems, bp0 is almost equal to the maximum fuselage width bpmax, according to Fig. 5-1. The wing thus defined will be termed the “substitute wing,” whereas the wing from which it has been derived will be termed the “original wing.”

Another important geometric parameter of a wing-fuselage system is the ratio of fuselage width bp0 and wing span b:

(6-1)