The Wing-Fuselage System in Symmetric Incident Flow

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

yF=[28] /s y=R The total lift of the wing-fuselage system follows from Eqs. (6-2) and (6-3) as

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.