Extended Lifting-Line Theory

Method of Weissinger The method of the extended lifting-line theory, as explained in its basic aspects in Sec. 3-2-3, has been developed into computational procedures for practical applications by Weissinger [95]. The basic equation for the

Figure 3-27 Circulation distribution 7 of a trapezoidal wing without twist of taper = 0, 1; aspect ratio л = 6; cb°° = 27r.

<*{y)

determination of the circulation distribution using this procedure is Eq. (3-52). With the dimensionless space coordinates %, r? from Eqs. (3-la) and (3-lb) and the dimensionless circulation distribution у from Eq. (3-59), Eq. (3-52) takes the form

with

3(£p> yi y) = 1 + 3(*p> V> y) = 2

As shown in Fig. 3-29, ^ = ^cC7? ) is the position of the lifting line at a distance c/4 from the leading edge and %p = |p(rj) the position of the control points. As explained in Sec. 3-2-3, the control points are arranged at three quarters of the local wing chord; thus xp — xc + c/2. This choice of the position of the control points (three-quarter points) results from two-dimensional skeleton theory for which c’Loo = 2it. To introduce another value for the position of the control point can be changed by setting (see [83]):

xp(y) = xc(y) + (3-89a)

WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 147

By introducing, according to [95], the function

K(,h „’) = K((„ rh (3-90)

Eq. (3-87) becomes

l

(x{rj) = 20Ci{i]) J K(rj,?]’)y(t]’) drj’ (3-91)

-l

where &i(v) is taken from Eq. (3-71 c)[16] The kernel function if(r?, n’) in Eq. (3-90) has been selected to be regular at r =7?, whereas the integrand in Eq. (3-87) is singular at this point. By a simple computation it can be shown that

(3-92)

The integral equation of the extended lifting-line theory now takes the following form, in analogy to Eq. (3-12b) of the simple lifting-line theory:

<*(*]) = 2[<*г(?1) + – civ)] (3-93)

і

where г(г() — J К {r, г}’) у (rj’) dr( (3-94)

-і

The kernel function K{p, r}’) depends exclusively on the geometry of the wing planform [83].

Wing with elliptic planform In Sec. 3-3-3 the wing with elliptic planform was treated by using the simple lifting-line theory. Now this wing shape will be computed using the extended lifting-line theory. A result of the simple lifting-line

———— Lifting line (quarter-point line) £ch?) Figure 3-29 Sketch for the extended lifting-

———— Line of the control points %p {77} line theory.

148 AERODYNAMICS OF THE WING theory, namely, that the elliptic wing without twist has an elliptic circulation distribution along the span, will be taken to apply here, too. Then the following study shows the difference in the total lift as determined from the simple and from the extended lifting-line theories, respectively. Since in the present case of an elliptic wing the circulation distribution along the span after the simple theory is assumed to apply, the kinematic flow condition can be satisfied only at one point of the three-quarter-point line. Following Helmbold [27], the three-quarter point of the wing half-span section will be chosen. The kinematic flow condition thus becomes, from Eq. (340),

* + <*»(£p> 0) = 0 (3-95)

Here = xp/s is the dimensionless distance of the control point from the c/4 line.

We shall not perform the calculation in detail, but the induced downwash angle at the wing, middle section becomes, according to Glauert [23], for elliptic circulation distribution,

Here E is the complete elliptic integral of the second kind with module l/Vlp + 1, and аг-= Cjr,/яЛ is the induced angle of attack introduced earlier. To simplify the computation, an approximate expression can be given for Eq. (3-96) that no longer contains the elliptic integral (see [27]). With Eq. (3-95), this expression becomes

The position of the three-quarter points is obtained from Eq. (3-895) with |c=0, and further with /1 =4blrtcr from Eq. (3-9) and к=їїЛ/с’ь„ from Eq. (3-755)* as

у Cl °° Cf _2_

*p 2 n b ттк

By introducing this expression into Eq. (3-97), the lift slope is found to be

dc l жЛ

da ■ і + і

In Fig. 3-25 the lift slope after this formula is presented for c’Lao = 2n, that is, к = /1/2, versus the aspect ratio. For comparison, the curve according to the simple lifting-line theory [Eq. (3-806)] is also shown. The difference between the two theories is similar to that for the rectangular wing of Fig. 3-32.

Equation (3-98) for the extended lifting-line theory evolves fromEq. (3-805) for the simple theory by formally replacing к by s/k2 + 1. In an analogous way, the Fourier coefficients for the circulation distribution of the twisted wing can be modified to comply with the extended lifting-line theory. Thus Eq.(3-77) takes the form

an — 1 ——- — f <x{$) sin # sin n& d & (3-99)

j/£2 + n2 n П J

The usefulness of this formula has been confirmed by numerous examples.

The rolling-moment coefficient cMx is obtained in closed form by introducing into Eq. (3-66b) the value for a2 from Eq. (3-99) and observing that 1? = cos $, Eq. (3-656), as

-f і

CMx = – ~==—T ~ f a(>?) }1 У1 – >f d}l (3-100)

+4 + 2 71 J

This is a quite simple equation for the determination of the rolling-moment coefficient.

Quadrature methods For the numerical evaluation of Eq. (3-93), Weissinger [95] presented a refined quadrature method analogous to that of the simple lifting-line theory (method of Multhopp). This method will not be presented here; instead, reference is made to [95]. Comprehensive sample computations using the Weissinger method have been conducted by de Young and Harper [103].

Further results of the extended lifting-line theory In Fig. 3-30 the circulation distribution versus the span at a = 1 is demonstrated for the rectangular wing without twist of aspect ratio Л — 6. For comparison, the curve using the simple lifting-line theory is also given. This figure shows that the extended theory produces a smaller lift than the simple theory for the same angle of attack. Furthermore, Fig.

3- 31 illustrates the lift distribution Cifci of the same wing. The extended lifting-line theory produces a somewhat less full distribution curve than the simple lifting-line theory. Both of these statements are typical for the extended lifting-line theory. The lift slope of rectangular wings after the extended and after the simple lifting-line theory are compared in Fig. 3-32. The difference between the curves is

Figure 3-30 Circulation distribution of the rectangular wing without twist of aspect ratio A = 6 for a — 1; С£оо = 2n. (1) Simple lifting-line theory. (2) Extended lifting-line theory.

Figure 3-31 Lift distribution c//cx of the rectangular wing without twist of aspect ratio A = 6; Cxoo = 2ir. (1) Simple lifting-line theory. (2) Extended lifting-line theory.

rather small for large aspect ratios A. It is considerable, however, for small values of.1. The limiting values of dc^/da for Л ->0 of the simple [Eq. (3-101<z)] and the extended [Eq. (3-lOlb)] lifting-line theory* are

(Л->0) (3-ЮІд)

d<x

^± = 2-Л (A 0) (3-101 b)

doi 2

The two limiting values are also indicated in Fig. 3-32; see also Fig. 3-25.

*For A -* 0, a(rj) = odj-(rj) in the simple lifting-line theory; for the extended theory, however, aCn) = 2az-(n) because K(r, r) = 0.

Figure 3-32 Lift slope dc^fda. of rectangular wings vs, aspect ratio A: c’iM = 27r. (1) Simple lifting-line theory. (2) Extended lifting-line theory.

In Fig. 3-33, results for a trapezoidal wing, a swept-back wing, and a delta wing with aspect ratios between A = 2 and 3 are presented. The geometric data for these three wings are compiled in Table 3-5. Figure 3-33 gives the circulation distribution for the wing without twist at a = 1. For the trapezoidal wing, the curve using the simple lifting-line theory has been added. In this case, too, it lies above the curve for the extended lifting-line theory. For all three wings, results are shown of the lifting-surface theory, which will be discussed in Sec. 3-3-5. Agreement between the extended lifting-line theory and the lifting-surface theory is good. The values for the lift slope and the neutral-point displacement, together with additional aerodynamic coefficients yet to be discussed, are compiled in Table 3-5.

Transition from extended to simple lifting-line theory It should be shown that the extended lifting-line theory may be transformed into the simple lifting-line theory for large aspect ratio. In performing this limit operation, according to Truckenbrodt [83], the control-point line |p(p) for the kinematic flow condition of the extended lifting-line theory must be shifted toward the lifting line |z(p), %p |z, or S. -> 0 (Fig. 3-29). Thus the kinematic flow condition becomes

<xw(8 -> 0, rj) – f a(?j) =0 (A = large) (3-102)

where 8(rj) is defined by Eq. (3-895). The dimensionless induced downwash velocity according to Biot Savart of a lifting line normal to the incident flow becomes, for a control point |p = Xp/s = 5 that lies very close to the lifting line,

-<xw(8->0,t]) =<xi(rj) – F — (3-103)

71 О (rj)

The first term of the right-hand side signifies the contribution of the free vortex, the second term that of the bound vortex. Since, from Eqs. (3-895) and (3-lQa), 7г5(р)= l//(p), it follows from Eq. (3-102) that

a(p) = <*ї0?) + f(y) у (у) (3-104)

Table 3 -5 Geometric data and aerodynamic coefficients of a trapezoidal wing, a swept-back wing, and a delta wing, t Л ^ cLoо = 2 ir Geometric parameters (Fig. 3-2) I II III I II III Aspect ratio Taper Sweepback angle Reference chord Neutral-point position Л ct V cr XN-2a cr 2.75 0.5 0° 0.778 0.25 2.75 0.5 50° 0.778 0.796 2.31 0 52.4° 0.667 0.50 Trapezoidal wing Swept-back wing Delta wing 1 *514»., 1 . ^ J t у // V 60° , xNzs o’ Vc/4 XI —– Аэ —-

Aerodynamic coefficients	I	IT		III
		©	©	©	©	©	©	©	®	©
Lift slope	dcL doc	3.600	3.015	3.105 (3.078)		2.511	2.614 (2.440)		2.406	2.435 (2.378)
Roll damping	йсмх dQx	-0.649	– 0.480	-0.500		— 0.450	-0.465		-0.340	-0.342
Neutral-point displacement				-0.031			0.012			0.140
CM			(-0.028)			(0.057)			(0.156)
Induced drag	t‘Di			1.000			1.027			1.004
(CDihll			(1.000)			(1.056)			(1.012)

*(1) Simple lifting-line theory after Multhopp; (2) extended lifting-line theory after Weissinger; (3) lifting-surface theory after Truckenbrodt and Wagner (five-chord distributions, values in parentheses). Neutral-point displacement rixjy measured from the geometric neutral point.

where, from Eq. (2-62),

(3-111)

Here c(y) is the chord and Xf the position of the wing leading edge at section у.

For the values n = 0, 1, and 2, the distributions are given in Fig. 3-34; see Fig.

2- 27. The functions hQ and hi of Eq. (3-110) have been normalized to produce the local lift and moment coefficients relative to the c/4 point through integration over the chord after introduction into Eq. (3-107) [see Eqs. (2-54) and (2-55)]. The result is

ci(y) = c0(y) cm(y) = icl(y)

The explicit expression for the functions Hn(x, y;y’), which are dependent only on the wing planform, is obtained by introducing the distribution functions hn of Eq. (3-110) into Eq. (3-109).

Writing the function G of Eq. (3-108) in dimensionless form and considering Eq. (3-105a) leads to

g(£, v> vl = 2 Hn{£> r{) /„(??’)

Introducing this function g(£, p; r) into Eq. (3-106) finally yields

Figure 3-34 The functions h0, hl; and h2 for the lift distribution vs. wing chord, from [84]; see Eq. (2-88).

This is a system of integral equations for the (N + 1) functions fn(y),(n = 0, 1, …, N). Choosing (N + 1) distribution functions by satisfying the kinematic flow conditions on (jV+1) control-point lines along the span, (N + 1) distribution functions can be determined. After having determined the functions/0(т?),/1(77), and so on, from this system of equations, the lift distribution is obtained from*

and the moment distribution (moment coefficient relative to the c/4 point) from

(3-115b)

The resultant of the pressure distribution of the lower and upper surfaces (load distribution) follows in analogy to the expression Eq. (3-107) and the relationship Eq. (3-44) as

(3-116)

In the following section this procedure will be explained through numerical execution.

Method of Multhopp and Truckenbrodt Multhopp [62] and Truckenbrodt [84] independently developed methods for the numerical evaluation of the method outlined above. In either publication the two distribution functions h0 and ht mark the basic approach. Multhopp puts the two control-point lines at 34.5 and 90.5% of the local chord. Truckenbrodt prefers positions of the control-point lines on the trailing edge and the c/4 line of the wing. A comparison of the best known lifting-surface theories is given in [18].

The explanation of the computational procedure now to be given follows closely [84]. As already stated, only two distribution functions over the chord are chosen, limiting the correlation functions of the method to H0 and Hi. With new designations,

Я0(|, r; 7]’) = rf)

4ЯХ(£ V> n) = H£> V’ y’)

Eq. (3-114) becomes

/0(17) is identical to the dimensionless circulation distribution у(ц) = Г/ЬІІ0, from Eq.

(3-59).

where /о and ft are replaced by 7 and д as in Eqs. (3-115c) and (3-115b). This equation must be satisfied for two values of |p, namely, for

= bs = £1 and ip ~ 1100 – £11 (3-119)

Here £2s(h) stands for the c/4 line and %wo(v) for the trailing edge.

The two functions 7(17) and ju(tj) of Eq. (3-118) are now to be determined. The angle-of-attack distribution aQp, 77) is given directly by the wing geometry, and the kernel functions і and / are given indirectly as functions of the wing planform. Only the angle-of-attack distribution values on the c/4 line and on the trailing edge are required in Eq. (3-118).

Wagner [91] expanded the described lifting-surface method to more than the two distribution functions over the span h0 and hx. Accordingly, the number of control-point lines must be increased. In selecting five distributions h0 through h4, in [91] the control-point lines are laid on the leading edge, the one-quarter, one-half, and three-quarter point lines, and on the trailing edge.

Quadrature methods The numerical solution of Eq. (3-118) is accomplished through an extended quadrature method, following Multhopp’s procedure for the lifting-line theory. Because of the considerable extent of the computations, use of an electronic computer is necessary. Further possible solution procedures for the equations of lifting-surface theory are reported by, among others, Kulakowski and Haskell [12], Cunningham [12], and Borja and Brakhage [9]. The panel method of Kraus and Sacher [46] should also be mentioned.

Lift distribution After having obtained the values 7(77) and p(r?) by solving the system of equations, the lift distribution along the span follows from Eq. (3-115д). The load distribution over the wing chord is consequently derived from Eq. (3-116) [compare also Eqs. (2-87) and (2-88)] as

Acp(X, V) = ^ [h0(X) y{n) + 4{Х)МчП (3-120)

where the functions h0 and hx are taken from Fig. 3-34.

Lift, rolling moment The total lift coefficient, the lateral distance of the lift center of a wing-half, the lift coefficient of a wing-half, and the rolling-moment coefficient may be determined from the formulas of Table 3-1.

Pitching moment The local pitching moment about the local c/4 point is given by Eq. (3-115b). In Fig. 3-35, x25(y) designates the c/4 line and хг(у) the line of the local aerodynamic center. It follows, then, that the moment of a wing section у about the c/4 point is dM= —AxidL. By setting dM=cmq00c2dy and dL — cf[„cdy, the distance between the local aerodynamic force and the local c/4 point becomes, with the help of Eqs. (3-115й) and (3-115b),

AXi(y) = cm{rj) _ Xi(77) – X25(77) _ (jjrj)

c(v) ci(n) c( 7?) 7(h)

WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW 157

Figure 3-35 Computation of the pitching mo­ment. xi(y) = position of the local aerody­namic centers. x7S O’) = local c/4 line, TV = neu­tral point of total wing.

The pitching moment of the whole wing is obtained from the contributions of the individual sections to the moment about the у axis dM = xxdL, resulting in

S S

M = – / xfy)dL = – /[x35(y)+Ax, Cv)] dL

— S — -5

і cm — —klJ -i Finally, the neutral-point position of the whole wing is obtained from Eq. (1-29). Results of wing theory and comparison with tests The examples computed in this section include rectangular, trapezoidal, swept-back, and delta wings. Earlier, in Fig. 3-33, circulation distributions of a trapezoidal wing, a swept-back wing, and a delta wing, all without twist, were presented for several computational methods.* The geometric data of these three wings are compiled in

Hence the pitching-moment coefficient cM —Mfq^Ac^, withcM as the reference wing chord [Eq. (3-5b)], becomes

‘Computation of the lift distribution of delta wings has also been treated by, among others, Gamer [17].

Table 3-5. From Fig. 3-33 it was concluded that the difference between the extended lifting-line theory and the Ufting-surface theory is quite small. In Fig.

3- 36, the lift distribution of three wings without twist is illustrated in the form ciclcLcm versus the span coordinate. In this kind of presentation, the computational results are practically identical. The lift slopes dcL/da of these three wings, based on various theories, are compiled in Table 3-5.

Neither the simple nor the extended lifting-line theory allows determination of the local neutral-point position because these methods require that the local neutral point be fixed on the lifting line (c/4 line). Application of wing theory according to Eq. (3-121) is required for local neutral-point determination. In Fig. 3-37, the local neutral-point positions over the span are plotted for the three wings of Fig. 3-36; see also Table 3-5. The local neutral points of the unswept wing lie before the c/4 line over the whole span. On the other hand, the local neutral points of both of the swept-back wings lie behind the c/4 line near the wing root and before the c/4 line in the range of the wing tips. The resulting total wing neutral points and the geometric neutral points according to Eq. (3-7) are also shown in Fig. 3-37. The distance between aerodynamic and geometric neutral points is very large, particu­larly on the delta wing. The numerical data for this displacement are compiled in Table 3-5. Comparisons between theoretically and experimentally determined local neutral points of swept-back wings have been published by Hickey [29].

Additional test results on a series of delta wings from [85] are shown in Fig.

3- 38. They have aspect ratios from 1 to 4. Lift slope dcL/da and neutral-point displacement Лх^/Сц are plotted against the aspect ratio. Here, too, agreement between theory and experiment is good.

In Fig. 3-39, the theoretical lift distribution over the span of a delta wing is compared with measurements of Kraemer [85]. Agreement is very good for angles of attack up to about a = 5°. Flow separation from the outer parts of the wings

V—— jj——————————– v Figure 3-36 Lift distribution cic/cj^cm of three wings without twist of Table 3-5 and Fig. 3-33, C£oo = 2w; cm —A/b — mean wing chord. Curve 1, simple lifting-line theory of Multhopp. Curve 2, extended lifting-line theory of Weissinger. Curve 3, lifting-surface theory of Truckenbrodt. Curve 3a, lifting-surface theory of Wagner (five-chord distributions).

Figure 3-38 Lift and neutral-point positions of delta wings of various aspect ratios with taper A = |; c^ = 0.68cr. Comparison of theory and experiment from Truckenbrodt. Profile NACA 0012, {a) Lift slope, (b) Neu­tral-point displacement, J x_y = distance of aerodynamic neutral point Ar from geometric neutral point N2S.

Figure 3-39 Lift distribution Cicjlba of a delta wing of aspect ratio л =2.3; profile NACA 65A005 according to measurements of Kraemer; comparison with lifting-surface theory of Truckenbrodt [84].

causes strong deviations of the measured lift distribution from theory for the large lift coefficients.

The local neutral-point positions are compared with theory in Fig. 3-40. Here again, satisfactory agreement is found. For the same wing, the measured pressure distributions for a few sections along the span are compared in Fig. 341 with theory according to Eq. (3-120). In general, the agreement is satisfactory. The

Jfco

Figure 340 Local neutral-point positions of a delta wing of aspect ratio A = 2; comparison of theory [84] and measurements {NACA TN 1650]. Profile NACA 0012.

Figure 3-41 Pressure distribution over wing chord for the delta wing of Fig. 3-40; theory [84] and measurements [NACA TN 1650]. cL = 0.585; profile NACA 0012.

deviations between theory and experiment can be partially explained by the fact that theory is valid for infinitely thin profiles and, therefore, does not account for the profile thickness.

Now, comparisons of experiment and theory will be made for unswept wings (rectangular wings). Figure 3-42 illustrates lift slope versus aspect ratio. The theoretical curve has been computed according to the multipoint method of Scholz [77]; it is in agreement with the curve for the extended lifting-line theory in Fig.

3- 32. The test points from several sources follow the theoretical curve well. In Fig.

3- 43, the neutral-point positions for the same series of rectangular wings are plotted against the aspect ratio. The neutral-point shifts considerably upstream of the c/4 line when the aspect ratio Л is reduced. Also included are measurements on rectangular plates that are in good agreement with theory.

Results for a series of swept-back wings of constant chord are presented in Fig.

3- 44. For both lift slope and neutral-point position, the measurements are in good agreement with theory. Note particularly that the lift slope of the swept-back wing, especially with a large aspect ratio, is considerably smaller than that of the unswept wing, <p=0. This reduction of the lift slope through sweepback can be assessed particularly well by considering the swept-back wing of infinite aspect ratio. Figure 345 depicts a span section b of an unswept and of a swept-back wing of infinite span. The section of the unswept wing produces the lift

L = Uibcc’io. a

Let the swept-back wing with sweepback angle $ be inclined to make, in the plane of the incident-flow direction U„, the angle of attack a equal to that of the

Figure 3-42 Lift slope of rectangular wings of various aspect ratios л; comparison of theory and experiment. Theory from Scholz (multiple-points theory) [77]. Measurements from Wieghardt, Scholz, and NACA Rept. 431.

unswept wing. Then, in the plane normal to the leading edge, the angle of attack is a* = a/cos (p. For the lift of the swept-back wing, only the velocity component normal to the leading edge, cos 9?, is effective. Thus, the cross-hatched surface portion of the swept-back wing has a lift

£*=-*- (U„ cos vfbcc-L. — = – f – U%bcc’L*a cos

2 cos 9 2

Hence, the lift coefficient of the swept-back wing is cL = L*lbc(pl2)U! = cLooacos ^, whereas that of the unswept wing is (с^)^=о ~ c’Laoa. The two lift slopes are thus related by

(3-123)

Figure 3-43 Neutral-point position of rec­tangular plates; comparison of measure­ments and theory, from Scholz [77].

Figure 3-44 Lift and neutral-point position of swept-back wings of con­stant chord and various aspect ra­tios; sweep-back angle <p = 45°; com­parison of theory and measure­ments from Truckenbrodt. Profile NACA 0012. (a) Lift slope. (b) Neutral-point displacement.

Figure 3-45 Geometry and velocity components ex­plaining the lift of swept-back wings of infinite span.

This relationship has been confirmed experimentally by Jacobs [37]. It is also valid, to good approximation, for the pressure distribution along the chord.

To show the effect of the sweepback angle on the lift slope, Fig. 3-46 illustrates, for swept-back wings of constant chord, the lift slope dcL jdct versus sweepback angle and aspect ratio according to de Young and Harper [103]. For large aspect ratios A, the decrease in lift slope with increasing sweepback angle is considerably stronger than for small aspect ratios. For /1= °°, the cosy? law of Eq. (3-123) is also shown for comparison.

The sweepback angle also strongly affects the circulation distribution over the span. Tills is apparent in Fig. 3-47, which demonstrates the circulation distribution along the span of a rectangular wing (<p> = 0) and a swept-back wing Qp =45°). The maximum value of the lift distribution of the swept-back wing is found at the outer wing portion. Sweepback causes a shift of the station of maximum local lift from the middle toward the outer end. Hence, the separation tendency of the swept-back wing is increased at large angles of attack compared with the, unswept wing. In this respect, sweepback produces unfavorable effects similar to a strong taper of an unswept wing (see Fig. 3-28).

Let us deviate from the wing theory discussed here. A swept-back wing theory has been developed by Kiichemann [48] that is not based on the Birnbaum normal distribution over the chord. This method, partially empirical, takes into account the wing thickness and the boundary layer, and also certain nonlinear effects. It therefore agrees very well with test results.

A cylindrical body in a flow that is inclined against its generatrix (yawed cylinder) may be subject to complex three-dimensional flow processes in the boundary layer. These are of considerable importance to the aerodynamic properties of swept-back wings. At larger lift coefficients, both yawed and swept-back wings undergo a strong pressure drop toward the rearward wing tip on the suction side

Figure 3-46 Lift slope of swept-back wings of constant chord vs. sweepback angle ^ and aspect ratio/1, from [103]; extended lifting-line theory. Curve for л = »; cbs <p law from Eq. (3-123).

near the wing nose, as shown in Fig. 3-48. In this figure, the isobars for the suction side of a yawed, inclined wing are seen. The fluid, decelerated in the boundary layer, follows this pressure gradient and consequently a strong cross flow in direction of the rearward wing sets in. Measurements of Jones [37] and Jacobs [37] have shown that, therefore, a marked thickening of the boundary layer is caused on the rearward wing tip and, as a consequence, a premature flow separation results. In airplanes with swept-back wings, this departure of the boundary layer toward the outside causes separation to occur first at the outer portion of the wing, in the

Figure 3-48 Evolution of cross flow in the boundary layer of a yawed wing (swept – back wing). Curves of constant pressure (isobars) on suction side of the wing.

range of the aileron. This in turn causes the feared “roll-off’ toward the stalled wing. This initiation of separation at the outer portion of the wing, and thus the undesirable “roll-off,” can be avoided by providing the wing with a boundary-layer fence (stall fence). This is a thin sheet-metal wall on the suction side of the front wing portion that prevents cross flows in the boundary layer. Liebe [13] describes the improvements in stall behavior by this provision. The work of Queijo et al. [13] includes results of comprehensive measurements on the improvement of the aerodynamic properties of a wing by means of boundary-layer fences. Compare also the basic studies of Das [13].

Poisson-Quinton [68] makes a contribution to the theoretical and experimental investigations on the problem of the aerodynamics of folding wings (wings with adjustable sweepback angle).

For wings of small aspect ratio, an essential simplification of wing theory is feasible, according to a proposal first made by Jones [36]. The basic concept of this theory is that the perturbation velocities in the x direction in the flow field about a slender wing are small compared to those in the transverse directions (y and z directions). The potential equation is then reduced to that of a two-dimensional flow in the yz plane (siender-body theory). In connection with this theory, the method of Lawrence [54] for the computation of the lift distribution of wings of small aspect ratio and the treatment of very strongly swept-back wings according to [59] should be mentioned. The application of slender-body theory to wings of extremely large thickness (covering of the wing contour with singularities) has been attacked by Hummel [34].