Category AERODYNAMICS OF THE AIRPLANE

Application of the Kutta-Joukowsky theorem The induced drag of an unswept wing of finite span from the Prandtl lifting-line theory, Eq. (3-18), is

S

Di = g J Г(у) Wiiy) dy (3-126)

— S

Here Г(у) is the circulation distribution and Wi(y) is the distribution of the induced downwash velocity over the span — <5, Eq. (3-19).

Now it shall be shown that Eq. (3-126) is also valid for arbitrary wing
planforms. Following Fig. 3-16, let the wing be replaced by so-called elementary wings of the infinitesimal span dy and wing chord c(y). The vortex system of an elementary wing (see Fig. 3-17) consists of a number of horseshoe vortices of width dy arranged in series, one behind the other. In Fig. 3-56, two horseshoe vortices are drawn that originate at the stations xx, yx and x2, у2 of the wing. Their respective widths and circulation strengths are dyx, dy2 and dT 1, dr2. Horseshoe vortex dTг induces at station x2,y2 the upward velocity d2w2X, whereas horseshoe vortex dT2 induced at station xx, yx the upward velocity d2wx2. In analogy to the Kutta-Joukowsky theorem [see Eq. (3-14)], the lifting-circulation elements dFx and dT2 produce forces normal to the upward flow that are the result of the upward-flow velocities d2wn and d2w2X, respectively. These forces represent contributions to the induced drag. The vortex system of Fig. 3-56 produces the partial induced drag

d*Di = —gdrx d2wn dyx — qdT2 d2w2X dy2

where the sense of rotation of the circulation elements has been taken into account. Because dr=kdx, the induced upward velocities of the exterior induction (уx —Уг) are found from Eq. (341) with Eq. (3-50tz), as long as yx Фу2, as

Figure 3-56 Explanatory sketch for computa­tion of induced drag.

It can be seen that the second terms in the square brackets differ by their signs only. When introducing Eq. (3-128) into Eq. (3-127), these terms do not contribute to the induced drag, leading to:

(3-129)

From these considerations there follows immediately that the location of the lifting-circulation elements in the x direction does not affect the drag. Hence, the relationship found earlier [Eq. (3-126)] for the unswept wing is valid for the total induced drag of a wing of arbitrary planform. Since the induced downwash velocity W/(y) from Eq. (3-19) depends only on the circulation distribution over the span, the total value of the induced drag also depends only on the circulation distribution over the span. It is independent of the arrangement of the elementary horseshoe vortices in the. chord direction (flight direction). This result was realized very early by Munk [63, 64] and is known as the Munk displacement theorem. Thus, it is immaterial for the magnitude of the induced drag whether the circulation distribution is caused by the wing planform (aspect ratio, sweepback, taper), by a wing twist, or by camber of the wing surface.

Application of the energy law Although the total lift of a wing can easily be computed by using the momentum law (Sec. 3-3-2), computation of the induced drag by means of the momentum law is considerably more difficult because the inclination of the vortex sheet has to be considered.[17] However, when using the energy law, the inclination of the free vortex sheet (Fig. 3-21), relative to the incident flow direction, can be disregarded. Since the induced velocities on surface I (Fig. 3-21) are zero, the mass gUoodydz permeating the area element dydz of surface II per unit time undergoes an energy increase dEu = (g/2)Uoo(vL + wlo) dy dz. Here u« and vv„ are the induced velocities in the у and z directions, respectively, and the area integral over dEu is the work done by the induced drag UooDi per unit time. Hence, after division by [/«,,

(3-130)

This relationship is valid for both not-rolled-up and rolled-up vortex sheets behind the wing.

The equivalence of Eqs. (3-130) and (3-126) will now be shown for the not – rolled-up vortex sheet. The induced velocity field very far behind the wing with components Vo0(y, z) and wm(y, z) can be expressed through the two-dimensional velocity potential Ф(у, z) as

Неге Ф(у, z) satisfies the potential equation

Introduction of Eq. (3-131) into Eq. (3-130) and integration by parts, the first integral with respect to y, the second with respect to z, yield, with Eq. (3-132),

where surface II has been extended to infinity. Since the values of Ф, ЭФ/ду, and ЭФ/dz vanish at the boundaries y = ±°° and z = ± whereas the potential according to Eq. (348b) changes abruptly in the z direction for z=±0 by the amount Фи(у, 0) — Фг(у, 0) = Г (у), with Eq. (3-13 lb) the induced drag becomes

(3-133)

The integration limits у = ± 00 can be replaced by y = ±s, because Ф(у, 0) = 0 beyond the wing span. Now, by introducing Eq. (3-20) with w«,(y) = — 2w*(y) into Eq. (3-133), Eq. (3-126) is finally obtained, as was to be proved.

Equation (3-130).is valid for the not-rolled-up vortex sheet. Kaufmann [41] showed that the same induced drag is obtained for the rolled-up vortex sheet, where it must be assumed, however, that the cores of the two free vortices have finite velocities.

Practical computation of the induced drag From Eq. (3-126) the formula for the coefficient of induced drag is obtained with 7 = Г1Ы1Ж and щ = w-JU„ from Eq. (3-71 b), and with 7]-y/s as

(3-134)

where /1 is the aspect ratio of the wing from Eq. (34). By expressing the circulation distribution 7 by a Fourier polynomial as in Eq. (3-65<z), the result of the integration becomes, with щ from Eqs. (3-73) and (3-65b),

cDi — zc/L У!

(3-135b)

In the second relationship, the lift coefficient cL was determined from the coefficient d of Eq. (3-66a). In Eq. (3-135b) the first term represents the value for the elliptic circulation distribution [see Eq. (3-3lb)]. Since the second term is always positive, the important theorem follows that the induced-drag coefficient for elliptic circulation distribution is a minimum. This theorem is true for fixed aspect ratio and for fixed lift coefficient cL. It was proved first by Muhk [63].

A summation formula for the coefficient of induced drag can be derived in a way similar to that which led to the summation formulas for the lift-related coefficients of Table 3-1. It has the form

where the values for ain have to be computed from Eq. (3-83).

Equation (3-134) for the induced drag will now be applied to trapezoidal wings with symmetric twist. This example explains the relationship between twist and induced drag. In Sec. 3-3-2 it was shown that the circulation distribution of a symmetrically twisted wing can be put together from that of a wing without twist and a zero distribution. In the same way as the circulation distribution was split up in Eq. (3-63), the induced angle of attack of the twisted wing can be split up:

+ <*io(v) (3-137)

When the wing has no twist, the induced drag is determined just by the term C2. The wing with twist requires, in addition to this term, a term Cj that is proportional to cL, and a term C0 that is independent of <?£. Here, the first term represents a linear twist, the latter a quadratic twist. As can easily be seen by comparison with Eq. (3-3lb), the constant C2 is unity for an elliptic circulation distribution. For the wing without twist the coefficient C2 signifies physically, therefore, the ratio of the induced drag to its minimum value for elliptic circulation distribution.

As an example, the induced drag of a trapezoidal wing with twist is given in

Figure 3-57 Induced drag of symmetrically twisted tapered wing of various aspect ratios л and various tapers Л from Eq. (3-138) (lifting-surface theory), (a) Wing planform with linear twist. (b) Induced drag of wing without twist, from Eq. (3-139a). (c), (d) Twist contribution to the induced drag from Eqs. (3-1396) and (3-139c).

Fig. 3-57. It is based on symmetric linear twist with as(rf) — |tj|Qi. The corresponding circulation distribution has been computed from the lifting surface. Figure 3-51 b indicates that the induced drag of the wing without twist has a minimum for a taper X ~0.45 at all aspect ratios A. The value of this minimum is only a little different from that of the elliptic wing (C2 = 1). For delta wings (X = 0) and rectangular wings (X = 1), cDi is in many instances considerably larger than for elliptic wings. The contribution that is independent of the lift, Fig. 3-57c, is always positive. The sign of the contribution that is linearly dependent on lift, Fig. 3-51 d, depends on the value of the taper. When the taper X ~ 0.45, this contribution is zero for all aspect ratios. The reason is found in the nearly elliptic circulation distribution over wings of this taper without twist. Furthermore, by means of Eq. (3-1356), it can be shown that Cx = 0 for elliptic wings with arbitrary twist and that

M

cDio = О, = я A 2 nal (3-140)

w=2

is the contribution of the induced drag caused by the twist for zero lift.

Investigations related to the establishment of the local drag distribution along the span are compiled in [1].

3- 4-1 Drag of Wings of Finite Span

The total drag of a wing of finite span, Sec. 1-3-2, is composed of profile drag and induced drag:

D = Dp + Di (3-124)

The profile drag Dp is created by friction effects. It is almost independent of the wing aspect ratio. A procedure for the theoretical determination of the profile drag has been developed in Sec. 2-5-2. Experimentally, the profile drag can be determined through wake measurements (momentum-loss measurements). The induced drag Dz- exists only at a wing of finite span. It is created by the flow processes at the. wing tips and can be determined from the laws of inviscid flow. In Sec. 3-2-1 it has been demonstrated that the induced drag is proportional to the square of the lift.

The drag coefficient cD —D/Aq„ of a wing with elliptic lift distribution is, from Eq. (3-326):

cD=cDP+cDi (3-125 a)

= cDp + ^ (3-1256)

where A, from Eq. (3-4n), is the wing aspect ratio.

There are two methods available for the determination of the induced drag. They differ in their physical concepts. In the first method, the induced drag is found from the pressure forces that act on the wing itself. In the second method, the induced drag is obtained from energy considerations. The latter approach allows the determination of the induced drag of only the whole wing. Conversely, the first method produces, within the framework of simple lifting-line theory, the local distribution of the induced drag. Truckenbrodt [86] summarizes the state of the art of the drag of wings. Basic considerations to the drag problem stem from Jones [38]. Also, the comprehensive compilation of experimental data of the wing aerodynamics by Hoerner [30] must be mentioned.

In the previous sections of this chapter, the fluid was considered to be incompressible and inviscid when establishing the theory of lift. The wing theory based on this concept is in good agreement with measurements as long as the angle of attack is small to moderate; see, for example, Figs. 3-38, 3-39, 3-42-3-44, and

3- 49. Only in the range of large angles of attack does the effect of friction have significance for the lift. In particular, the maximum lift of a wing is not only determined by its geometry, but it is also considerably affected by friction. Determination of the maximum lift of a wing by strictly theoretical methods is not yet possible. Cooke and Brebner [11] report on flow separation from wings in general terms. Schlichting [73] presents the aerodynamic problems of maximum lift of wings in comprehensive form.

From measurements it is known that the maximum lift coefficient is strongly dependent on the geometric profile parameters (thickness, camber, nose radius) and on the Reynolds number. In Sec. 2-5-1 this relationship was discussed briefly; see, for example, Figs. 2-39 and 2-42-2-44. These previously reported results should be supplemented by the statement that the maximum lift of an unswept wing is essentially a problem of two-dimensional flow. A large aspect ratio of unswept wings of finite span cannot have an important effect on flow separation and consequently on the maximum lift because in this case the flow over the major portion of the wing deviates only a little from plane flow. Quite different are the conditions for wings of small aspect ratio. Here the flow around the wing tips reaches to the middle of the wing. For strongly swept-back wings, which includes delta wings, the flow conditions are particularly complex because the leading edge acts in a similar way as the tips of an unswept wing. For these kinds of wings, even the attached flow is much harder to assess than that for unswept wings, because the flow directions in the boundary layer may deviate from that of the outside flow (departure of the boundary layer to the wing tips, boundary-layer fence).

Contrary to unswept wings, the flow over strongly swept-back wings without twist separates locally first at the wing tips because the lift load has its maximum there (see Fig. 347).

When the angle of attack increases, the separated region expands inward in span direction. This behavior is discussed in more detail in [26]. A very comprehensive compilation of material on this behavior of swept-back wings at large angles of attack and high Reynolds numbers has been given by Furlong and McHugh [16].

The effect of the aspect ratio and the sweepback angle on the maximum lift coefficient will now be examined using some test results.

In Fig. 3-53, results are plotted for the maximum lift coefficient of rectangular wings and swept-back wings of constant chord (</? = 45°). The Reynolds numbers of these measurements are Re ^ 106. Figure 3-53c confirms that the maximum lift coefficient for A > 2 is almost independent of the angle of attack. For very small aspect ratios, cLmax is somewhat larger than for large aspect ratios. Particularly noteworthy in Fig. 3-53b is, for aspect ratios Л < 2, the strong increase to values of а 30° in the angle of attack for which the maximum lift coefficient is obtained.

In Fig. 3-54 curves are given for the lift coefficients of a series of delta wings plotted against the angle of attack. When the aspect ratio A decreases, the lift slope becomes considerably smaller, while the maximum lift coefficient and the corresponding angles of attack increase. The lift slopes dcLjda of these wings have been presented earlier in Fig. 3-38. Maximum lift coefficients cLmzx for these and additional delta wings are plotted in Fig. 3-55 against the aspect ratio. Comparison

Figure 3-53 Maximum lift coefficients of rec­tangular wings = 0) and swept-back wings of constant chord <> #0), Reynolds number Re » 106. (a) Maximum lift coefficient C£,max vs. aspect ratio л. (b) angle of attack a for max vs. aspect ratio A. Curve 1, = 0°; profile NACA

0015, from Bussmann and Kopfermann [25]. Curve 2, = 45°; profile NACA 0012, from

Truckenbrodt [85]. Curve 3, ip = 0°; 8 « 0.10, mean values of various measurements. Curve 4, ^=35°; 6 « 0.10, mean values of various mea­surements.

A=0.83 1.61 2.38

Figure 3-54 Lift coefficients ci vs, angle of attack a. for delta wings of various aspect ratios – t; taper = 2> thickness ratio 5 = 0,12, Reynolds number Re ^ 7 * 10s, from Truckenbrodt [85].

with Fig. 3-53д shows that the increase in cLraaK for small aspect ratios is considerably larger than for rectangular and swept-back wings. Also, Fig. 3-556 shows a strong increase of &cLmax at small aspect ratios in agreement with Fig. 3-53b. Experimental studies on the separation characteristic of delta wings have been carried out by Truckenbrodt and Feindt [85] by means of simple wake measurements.

Figure 3-55 Maximum lift coef­ficients of delta wings, Reynolds number Re « 106. (a) Maximum lift coefficient C£max vs. aspect ratio Л. (b) Angle of attack a for C£,max vs – aspect ratio A. Curve 1, delta wing; A = 0; pro­file NACA 0012, from Lange and Wacke [25]. Curve 2, delta wing; A = profile NACA 0012, from Truckenbrodt [85]. Curve 3, mean values of various mea­surements.

The wing theory treated so far establishes a linear correlation between lift coefficient and angle of attack. It is designated, therefore, linear wing theory. It is known from experimental investigation that for wings of very small aspect ratio, A < 1, lift coefficients Ci are considerably larger than those obtained from linear theory when plotted against the angle of attack. Figure 349 illustrates this behavior for rectangular wings of aspect ratios Л =0.2, 0.5, 1.0, and 5.0 as compiled by Gersten [21]. The dashed theoretical curves represent linear theory as discussed earlier. Although linear theory produces the right lift slope {dcLlda)a~0 even for small aspect ratios, strong deviations of the measurements from linear behavior are already obvious for small angles of attack.

All wing theories discussed so far are based on the concept that bound and free vortices lie in the same plane. A linear relation between lift and angle of attack is the necessary consequence.

This much simplified vortex model must be abandoned for a theoretical explanation of the nonlinear relation between lift and angle of attack. A first trial in this direction was made by Bollay [8]. He used a vortex model similar to Fig.

3- 50tf in which the free vortices no longer lie in one plane but rather are shed in the downstream direction from the wing tips under the angle a/2 with the wing plane. Bollay assumes that the bound vortices are constant over the span. Gersten [21]

Figure 3-49 Measured lift coefficients vs, angle of attack a for rectangular wings of aspect ratios A = 0.2, 0.5, 1.0, and 5.0. Curve 1, linear theory of Scholz. Curve 2, nonlinear theory of Gersten.

refined this vortex model by prescribing a variable circulation distribution over the span (Fig. 3-50b). The cL(a) curves based on this theory are given in Fig. 349 as solid lines. They are in very good agreement with this theory (see Winter [102]). By the same theory, pitching moment, induced drag, and lift distribution along the span have also been determined. Agreement between tests and theory is good in these cases, too. Furthermore, the nonlinear theory has been extended by Gersten to arbitrary wing shapes. It represents an extension of the lifting-surface theory of Sec. 3-3-5 to the nonlinear angle-of-attack range. The cL(a) curves as determined

from this theory and the comparison with test data are shown in Fig. 3-51 for a swept-back and a delta wing.

It is known from test results that the aerodynamic coefficients of wings of small aspect ratio are strong functions of the wing leading-edge design. This is true particularly for swept-back and delta wings with sharp leading edges which, even at very small angles of attack (a — 3°), promote flow separation from the leading edge of the kind shown in Fig. 3-52. Starting at the wing tips, two vortex sheets form on the two leading edges that roE up into free vortices when floating downstream.

This process was first discussed by Legendre [55] and has been treated in

Figure 3-51 Lift coefficient of swept – back wings with sharp leading edge and small aspect ratio vs. angle of attack.

(——– ) Linear theory from Eq.

(3-lOli). (——— ) Nonlinear theory of

Gersten. (o) Measurements, (a) Swept – back wing.1 = 1, =1, (/3 = 45°. (b) Delta wing л = 0.78, К =

Figure 3-52 Bursting of the free vortices of a delta wing according to Hummel. Aspect ratio /і = 0.78, taper = 0.125. (a) Vortex formation shown schematically, (b) a = 20°, /3 = 0°, no bursting. (c) a= 30°, (3 = 0°, bursting of the vortices at large angles of attack, (d) a = 20°, (3 = —10°, bursting of one vortex of yawed wing. (e) a = 20°, (3 = 0°, bursting of one vortex through an artificial pressure rise.

numerous other publications [4, 10, 33, 55, 98]. The roll-up of vortex sheets has been studied theoretically by Roy [71] and by Mangier and Smith [58, 80]. Roy established details through numerous flow-pattern photographs. Under certain circumstances, a striking change in the structure of the rolled-up vortex sheets can be observed that can be termed bursting of the vortices. Figure 3-52b-e shows smoke pictures of this phenomenon from Hummel [33]. The bursting of vortices occurs (1) at large angles of attack in symmetric incident flow (Fig. 3-52c), (2) at the vortex of the upstream-turned side of yawed wings (Fig. 3-52d), and (3) when an obstruction is placed into the vortex flow (Fig. 3-52e). Naturally, the bursting of vortices has a strong effect on the aerodynamic properties of the delta wing; compare [4, 33]. These processes affect lift and pitching moment as well as drag.

Further investigations of nonlinear effects on wings of small aspect ratios, especially on delta wings, are reported in [19, 32, 57, 67]. A very recent survey of the aerodynamic properties of slender wings with a «harp leading edge has been given by Parker [66].

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].

Basic formulas The local lift coefficient c{(y) of a wing section у is obtained through integration of the pressure distribution over the wing chord in analogy to Eq. (2-54) as

*>•09

Ф )~<^Г)/ ДСЛХ>У)&Х (3-53)

Xf(y)

The total lift coefficient cL=L! Aq00 of the wing is thus obtained with qoc =qXJIo!2 as

cl—JJ Acpdxdy

(A) s

= 2 ФШ dy

Compare also with Eq. (3-13). By using Eq. (3-54#), the total lift is obtained through integration of the pressure distribution over the wing chord. With Eq. (3-15), it may also be obtained from the Kutta-Joukowsky theorem. Here the circulation distribution has to be taken from Eq. (3-39), the distribution of vortex density from Eq. (344). Below, a further expression for the total lift will be derived by applying the momentum law. As in Fig. 3-21, a cylindrical control surface is arranged about the wing. The axis of the cylinder runs in the direction of the incident flow velocity Ux. The two base surfaces I and ЇІ of the cylindrical control surface are assumed to be very far upstream and downstream of the wing, respectively. The diameter of the control cylinder is chosen large enough to make pressure and velocity on the cylindrical surface equal to the values px and of the undisturbed flow on surface I, respectively. In computing the lift from the

Figure 3-21 Computation of lift by means of the momentum law and of the induced drag by the energy law.

momentum law, it can be assumed that the free vortex sheet is parallel to the incident flow direction far downstream of the wing.*

The fluid mass permeating an area element dy dz of surface II per unit time is QUoodydz. It produces, together with the velocity w«, induced by the wing, a momentum component in the z direction of magnitude qU^Woo dy dz. Since the induced velocity on surface I is zero, the integral of the momentum over the surface II represents the force exerted normal to the incident flow direction to the wing, that is, the lift

(3-55)

(3-560 (3-56*)

Now, the identity of Eqs. (3-55) and (3-15) will be shown for the not-rolled-up vortex sheet. The field of the induced velocities very far downstream of the wing can be described by means of a two-dimensional velocity potential Ф(у, z) [see Eq. (3-49b)], where Woo = Ъф/dz. By introducing this expression into Eq. (3-55), integration over z yields

— $

On the boundaries у = ±°° and z = ±°°, the values Ф vanish, whereas in the vortex sheet, at z — ±0, the potential in the z direction from Eq. (3-48b) changes abruptly by the amount ДФ(у, 0)= Фи(у, 0) — Ф^(у, 0)=Г(у). The integration limits y = ±°° may be replaced by у = ±s = ±bj2 because ЛФ(у, 0) = 0 outside of the wing span. Introduction into Eq. (3-56<z) yields Eq. (3-56b), in agreement with Eq. (3-15). The total lift thus depends only on the circulation distribution over the wing span. It is thus immaterial whether the circulation distribution is created by wing

”Kxaemer [79] points out the decisive significance of the inclination of the free vortex sheet for computation of the induced drag by means of the momentum law.

planform (aspect ratio, sweepback, taper), wing twist, or camber of the wing surface.

Certainly, Eq. (3-55) is also valid for the rolled-up vortex sheet as in Fig. 3-8. Now let b0 = 2sо be the distance between the two free vortices of circulation strength Г0, whereby the circulation distribution along the span is symmetric (Fig.

3- 22). The induced velocity w*. at a point of the yz lateral plane very far behind the wing (.x -* °°) becomes, from the Biot-Savart law,

so +У, Sp ~y

(sQ+y)2+z^(so-y?+z2

Introducing this expression into Eq. (3-55) and integrating twice yield

L=QUcorQb0 (3-57)

By taking into account the Kutta-Joukowsky lift theorem, this formula states that the lift of a wing of span b = 2s and of variable circulation distribution Г (у) is equal to the lift of a wing of span b0 and over the span constant circulation distribution Г0. Comparison of Eqs. (3-566) and (3-57) yields the distance between the two free vortices:

s

Ъо=^1Г[у)йу (3’58)

0

This relationship can also be interpreted as a statement that the vortex moment about the longitudinal axis (x axis) remains constant during roll-up. For the right

oc(?]) = oiA’i1) + *An) y(v) = УI (v) + yz(v)

zero. Consequently, the circulation distribution of the twisted wing at given angle of attack a = const is given by

y(v) = + Yoiv) (3-63)

The zero distribution 70(h) is obtained from

To (V) = 7g(v) + <*0 T U(V) (3-64)

Through procedures similar to those applied for the lift, integration over 17 for a known circulation distribution 7(17) produces other simple relationships for the lateral distance of the lift center of a wing-half, for the lift force of a wing-half, and also for the rolling moment about the x axis. They are summarized in Table 3-1.

Introduction of a Fourier series Computation of the integrals for the coefficients of lift and rolling moment turns out to be particularly simple when the circulation distribution is expressed as a Fourier polynomial of the form

M

у (#) = 2 £ aft sin, a# (3-65я)

/і=і

This procedure was first introduced by Trefftz [69] and Glauert [23]. The first term in Eg. (3-65a) represents the elliptic circulation distribution 7 = 2аг sin d – = 2*iVl — V2 as treated previously in Sec. 3-2-1. After execution of the integrations over — 1 <7? < 1 and 0< # <7r, respectively, the coefficients of lift and rolling moment are obtained with ch? = — sin d-dd – as

(3-67*)

obtained with jj. = 1 and /і = 2 for the lift coefficient cL and the rolling-moment coefficient cMx (Table 3-1). In addition,-quadrature formulas are also given for the lateral distance of the lift center of a wing-half riL=yLjs and for the lift coefficient of a wing-half c*. Table 3-2 contains the coefficients for the formulas for practical application of the last column of Table 3-1.

«(У) = *«(У) – r »іІУ)

(3-70*)

the effective or, respectively, the induced angle of attack becomes*

*e(v) = f(v)r(v)

dy dr/’

2 n J dr]’ 1] — 7]’

-l

The formula for аг(7?) can also be written, through integration by parts, as

By introducing Eqs. (3-71<z) and (3-71Z?) into Eq. (3-69), the Prandtl integral equation for the dimensionless circulation distribution 7(17) is obtained in the form

*(n) = f(>i)rM+£f

-1

*See footnote on page 139.

Figure 3-23 Wing section y: a(y), angle of attack against zero-lift direction: a. g(y), geometric angle of attack against wing chord; a0 (y), zero-lift angle,

a = dig — a0.

In abbreviated form, the integral equation of the simple lifting-line theory can be written:

z(r/) = «;(>?) H – /(?;) y(y)

Schmidt et al. [76] deal with the mathematical formulation of the simple lifting-line theory and present comprehensive results.

Solution with Fourier polynomials A convenient method of solving Eq. (3-72) for the circulation distribution consists of expressing the circulation distribution as a Fourier polynomial such as Eq. (3-65) (M = n). By introducing Eq. (3-65a) into Eq. (3-71 6), first the induced angle of attack is obtained[14]:

(3-73)

After introduction of Eqs. (3-65a) and (3-73) into Eq. (3-726), the following equation is obtained, defining the Fourier coefficients an

M

<x($) sin$ — a„ [2/(#) sin# – f w] sinw#

Here the distribution of the angle of attack a(&) and the wing planform f(&) are given beforehand. The coefficients ax, a2,. -. , aM are determined by satisfying Eq. (3-74) at M points d-M along the span. This results in a system of M

linear equations for ax to aM. Lotz [56] simplified this procedure by introducing Fourier polynomials for the functions a($) sin # and f(6) sin &. After the Fourier coefficients an have been determined, the circulation distribution is obtained from Eq. (3-65<z) and the distribution of the local lift coefficients from Eq. (3-596). Weinig [93] suggests that the theory of the lifting line be solved by comparison with the corresponding grid flow.

Wing of elliptic planform The elliptic wing has been treated in Sec. 3-2-1. There it was shown that an elliptic wing without twist has an elliptic circulation distribution over the span. The elliptic wing with twist may be computed from the above formulas very easily as, among others, Schmidt [69] has shown.

For the elliptic wing c = crs/ —r2 — cr sin $ and A = 46/ясу [Eq. (3-9)], and thus from Eq. (3-70),

2 sin # / (#) = — ^L°°

(3-756)

Hence, Eq. (3-74) for the wing with elliptic planform becomes

M

a(&) sin&= £ (k n) ansmn& (3-76)

n— 1

and the coefficients an can be computed directly through a Fourier analysis as

П

an = —^—— f л (г?) sin?? sinnfi d& (3-77)

к – f – П 71 J о

This solution will now be discussed for a few particularly simple angle-of-attack distributions. Setting

a sinm# rn hq

<*W = rm – (3-78)

sm»

the corresponding circulation distribution is obtained with an = 0 for пФт and with am = rml(k + m) for n = m as

y(&) = 2 ——— sin « (3-79)

‘ к -f – m к – f – m

For a wing with the aspect ratio A = 6, that is, к – 3, the results for m = 1, 2, and 3 are presented in Fig. 3-24. Here m = 1 gives the constant angle-of-attack

Figure 3-24 Lift distribution according to the simple lifting-line theory of an elliptic wing at various twists [Eq. (3-78)]; aspect ratio Л = 6. (a) Wing planform. (b) m — 1: wing without twist. (c) m~ 2: linear angle-of-attack distribution, (d) m = 3: parabolic angle-of-attack distribution.

Figure 3-25 Lift slope of elliptic wings vs. aspect ratio; = 2тг. (1) Simple lifting-line theory, Eq. (3-80&). (2) Ex­tended lifting-line theory, Eq. (3-98). (- о -) Exact solutions according to Kin – ner [44] and Krienes [47].

distribution (wing without twist), m — 2 gives the linear angle-of-attack distribution such as, for instance, is encountered in a rolling motion, and m — 3 gives the parabolic angle-of-attack distribution (symmetric twist, cL — 0).

Circulation distribution and coefficients of lift and rolling moment of the elliptic wing with twist are obtained from Eq. (3-66a) and also from Eq. (3-665) by introducing the corresponding coefficients an according to Eq. (3-77). The lift coefficient thus becomes

ГС CL = ——— Га($) sin2# cZ# fc-M я) (3-8 Од) 0 dcj пЛ dot к + 1 (3-805)

For the wing without twist, a = const, the coefficient of lift slope is obtained in agreement with Eq. (3-34#). It is presented in Fig. 3-25 as a function of the aspect ratio /I. Also shown are the results based on the extended lifting-line theory that will be treated further in Sec. 3-34, and the exact solution for the elliptic wing.*

From Eqs. (3-80s) and (3-805) the zero-lift angle is obtained with Eq. (1-23) as

-f-1

Q! o = — ~ J*«(??) І і — rf ch] (3-81)

-i

For approximate computations, the relationships of the elliptic wing can be applied to other wing shapes.

Quadrature method of Multhopp The simplest and most used method for the computation of the lift distribution of unswept wings according to the simple

"Here, the wing planform is an exact ellipse, which, e. g., becomes a circular disk for Л = 4/ir.

lifting-line theory is that of Multhopp [60]. This method will be briefly sketched now: Starting from the expressions for the circulation distribution [Eq. (3-65)] and for the Fourier coefficients [Eq. (3-67)] in connection with Eq. (3-68), the summation expression, Eq. (3-67b), is introduced into Eq. (3-73). The induced angle of attack at the discrete stations

is then obtained in the form

(3-83)[15]

M

о*і„ b, vyt ^, bvnyn (v 1,2,…, ilf)

n= 1

with the universal coefficients

M+ 1 w 4 sin $v

_!-(-! y-n

ш 2(M +1) (cos &v — cos &n)2

By introducing expression (3-83) for the induced angle of attack into the integral equation for the circulation distribution Eq. (3-72b), the following system of equations is obtained for the values of jv:

M

(bw +fv)7v = ‘ «V + 2’ bvn yn (v = 1, 2, . . . , M) (3-85)

П= 1

This is a system of M linear equations for the M circulation values jv = 7(17*,) with v = 1, 2,. .., M. In Eq. (3-85), the following relationships apply:

&v = a(r? v) fv = ~T~ with cv = c(rjv) (3-86)

C l<x> Cp

For M~1 and M— 15, the universal coefficients are compiled in Tables 3-3 and 34. The values bm for ^—«1=2, 4,.. . are equal to zero. For the numerical solution of the system of equations, it is significant that the system of M equations can be split up into two systems of (M + l)/2 or (M —1)/2 equations, respectively, which can be solved conveniently by iteration. By splitting an arbitrary angle-of – attack distribution into its symmetric and antisymmetric contributions, the procedure of the numerical solution can be further simplified. For a continuous behavior of wing chord c{r) and angle of attack a(rf), usually M = 15 points along the span are sufficient for all practical purposes. For discontinuous angle-of-attack distributions, as found for flap deflections, Multhopp recommends that one split off

Table 3-3 Universal coefficients and bvn for the computation of circulation distributions, forM= 7, according to Eq. (3-84)a V 1 (7) 3 (5) 2 (6) 4 Vv 0.9239 0.3827 0.7071 0.0000 bvv 5.2262 2.1648 2.8284 2.0000 n n 2(6) 1.8810 0.8398 1(7) 1.0180 0.0560 2. 4(4) 0.1464 0.8536 3(5) 1.0972 0.7887 6(2) 0.0332 0.0744 5(3) 0.0973 0.7887 — — — 7(1) 0.0180 0.0560 aAfter Multhopp [60].

the discontinuity stations before applying the above computational procedure; see Chap. 8.

Equations (3-85) are valid for unswept but otherwise arbitrary wings of sufficiently large aspect ratio (/1 > 3) and also for arbitrary angle-of-attack distributions.

Further results of the simple lifting-line theory In Fig. 3-25 the dependence of the lift slope on the aspect ratio is shown for a wing of elliptic planform. This result is approximately valid also for wings of different—for instance, trapezoidal—planforms.

To demonstrate the effect of the aspect ratio on the lift distribution, the circulation distributions over the span were computed for three rectangular wings with c = const and aspect ratios A — 6, 9, and 12. When A increases, the circulation distribution approaches more and more a rectangular distribution. Figure 3-26 demonstrates this fact. Illustrated are the local lift coefficients ct with reference to the total lift coefficient cL along the span. For A 00 (plane problem), Ci/cL = 1, and for very small aspect ratios {A -*0) the lift distribution is elliptic. This can easily be seen from Eq. (3-12b), which for /l-»0 goes to a(rj) = a2-(v) because f(rj) = 0. Hence, for a= const, cq = const, meaning, from Sec. 3-2-1, that the circulation distribution is elliptic.

To show the effect of wing taper on the lift distribution, Fig. 3-27 illustrates the circulation distribution for four trapezoidal wings without twist of aspect ratio A=6 and tapers X=ctfcr = 0, and 1. The taper has a strong effect on the distribution of the local lift coefficients along the span. This can be seen in Fig.

3- 28 in which the curves Ci/cL are shown. The strongly tapered wings have, near the wing tip, local lift coefficients that are considerably larger than the total lift coefficient cL. This fact is significant for the flow-separation characteristics of such

Table 3-4 Universal coefficients bw and bvn for the computation of circulation distribution, for M = 15, according to Eq. (3-84)a

я After Multhopp [60].

Figure 3-26 Lift distribution ciJci of rectangular wings without twist of aspect ratios A — 6, 9, 12; also limiting curves for Л -* 0 and Л = 2rr.

wing shapes at high lift coefficients. With increasing angle of attack, separation begins approximately at the station of maximum local lift coefficient, hence on strongly tapered wings close to the wing tips, but on rectangular wings in the middle of the wing.

3- 3-1 Methods of Wing Theory

The theoretical basis for this section was laid in Sec. 3-2. For practical applications, the computational methods discussed below (simple and extended lifting-line theories, lifting-surface theory) proved to be particularly convenient and may be characterized as follows: The simple lifting-line theory applies only to wings with straight cl4 lines in symmetric flow, that is, to unswept wings. It gives good results for larger aspect ratios (Л>3) and allows the determination of lift distributions over the span from which total lift, rolling moment, and induced drag, but not pitching moment, may be computed. The extended lifting-line theory (three-quarter – point method) applies to wings of any planform and aspect ratio. Thus, it applies to swept-back and yawed wings. It gives the lift distribution over the span from which total lift, rolling moment, induced drag, and, approximately, pitching moment are obtained. The lifting-surface theory, like the extended lifting-line theory, applies to any wing and aspect ratio, but gives lift distributions over the span and over the chord from which total lift, rolling moment, induced drag, and also pitching moment, and thus the neutral-point position of the wing, are found. Accurate knowledge of the neutral-point position is particularly important for swept-back wings.

Summaries and detailed presentations on the methods of wing theory in incompressible flow are given by Betz [6], von Karman and Burgers [88], Robinson and Laurmann [70], Thwaites [82], Weissinger [96], von Karman [89], Flax [15], Hess and Smith [28], and Landahl and Stark [52]. The development of the lifting-line theory as a “singular perturbation problem” is due to van Dyke [87]; see also the references on page 111. Extensions of wing theory to include nonlinear angle-of-attack effects and the behavior of wings near the ground (ground effects) are found, for example, in [8, 19, 21, 40] and [2, 81, 100], respectively. Although it is not possible in this book to treat the questions of nonsteady flow that are important for airplane aerodynamics, the references [2, 50, 52, 53] shall be mentioned in this connection. Problems of flexible wings are discussed in [22].

Studies on design aerodynamics have been prompted by Kiichemann and accomplished for swept-back wings in particular [3].

The lifting-surface theory of Sec. 3-2-2 can be transformed into a simpler theory of the kind given in Sec. 3-2-1 by replacing the continuously distributed circulation along the
chord by a vortex line, arranged at a suitably chosen station on the local chord (lifting-line theory). Let x’c = xc(y’) be the location of this lifting-vortex line which, from the results of Sec. 2-3-2 for the inclined flat plate, is expediently placed on the quarter-point line (Fig. 2-37). Then the function G(x, y; y’) of Eq. (342д) becomes

0(x, У У’) = Пу’) 11 + -7====^====-} (3-50e)

i{x – xcy2-f – (y – у)“ /

Here Г(у’) is the total circulation around the wing section у. Furthermore, for у — у and x > xc this function becomes

0(x, y y) = 2Г{у) (3-50b)

The kinematic flow condition [Eq. (3-40)] can be satisfied in this case at one point of the chord only. This control point has the coordinate xp(y): Expediently, it is placed on the three-quarter-chord station, measured from the leading edge (three-quarter point, theorem of Pistolesi), see Sec. 24-5. Hence, the expression in parentheses on the left-hand side of Eq. (343) becomes

where a(y) is the measured angle of attack relative to the zero-lift direction (Fig.

3- 18).

By introducing Eqs. (3-51) and (3-50) into Eq. (343), the integral equation for the circulation distribution from the extended lifting-line theory i§ obtained as

Compared with the simple lifting-line theory discussed in Sec. 3-2-1, Eq, (3-52) has the great advantage that it is also applicable to yawed and swept-back wings. This extended lifting-line theory is also called the three-quarter-point method. It was developed in detail and applied particularly by Weissinger [95]. Also Reissner [95] was engaged in the establishment of a solid foundation for this lifting-line theory.

For the swept-back wing a vortex arrangement as in Fig. 3-20 is obtained. In Fig. 3-20a the replacement of the wing by a system of elementary wings and in Fig.

3- 20b the equivalent vortex system according to PrandtTs concept (Fig. 3-9) are demonstrated.

Figure 3-20 Vortex system of a swept-back wing (lifting-line theory), (a) Substitution of the wing by elementary wings, (b) Bound and free vortices according to Prandtl (see Fig. 3-9).

In Prandtl’s lifting-line theory and in the three-quarter-point method described above, the wing is replaced by just one lifting line. Wieghardt [101] proposed the arrangement of several lifting lines in series. This method can be designated as a multiple-points method. Scholz [77] developed this method in more detail and applied it especially to the cambered rectangular wing.

Vortex system of the lifting surface To simplify the problem, it was assumed in Sec. 3-2-1 that the circulation representing the wing was concentrated on one line (lifting-line theory); see Fig. 3-7. This concept is a fairly good approximation for a real wing only when its chord is much smaller than its span (wing of large aspect ratio). When the chord is no longer much smaller than the span, it is necessary to replace the concept of a lifting line by that of a distribution of lifting vortices over the wing chord. Such a continuous vortex distribution over the wing chord was the basis for the skeleton theory (Sec. 24-2). In the preceding section, the free vortices were assumed to be distributed on the surface. By applying this concept of a continuous circulation distribution logically to the wing of finite span, a vortex distribution on the surface results that varies in chord and span direction (lifting surface). An outline of this lifting-surface theory will now be derived. This theory is of practical importance particularly for wings of small aspect ratio, for swept-back and delta wings, and for yawed wings. This vortex distribution on the surface can be taken to be a distribution of singularities in the sense of Sec. 24-2. During the further development of wing theory, instead of vortex distributions, dipole distributions will be used occasionally; see, for example, Prandtl [69 (1936)].

After the fundamental publication of Prandtl on wing theory using vortex distributions, Blenk [69] further developed this theory by extending the two – dimensional Bimbaum-Ackermann theory, Chap. 2 [8], to three dimensions.

The distribution of vortex strength over a given surface can be accomplished in various ways. Let the wing surface have an arbitrary shape, and let a rectangular wing-fixed coordinate system be chosen whose у axis is normal to the incident flow direction.

A first possible approach to the replacement of the wing by a vortex distribution is to cover this surface with two areal vortex distributions kx(x, у) and ky(x, y), as in Fig. 3-15. The former distribution consists of vortex lines parallel to the x axis, the latter of those parallel to the у axis. The ky vortices are of the kind that was previously applied to the two-dimensional wing theory (see Fig. 2-20); the kx vortices, however, resemble the free vortices in the vortex sheet behind the wing (see Fig.- 3-9). Only the ky vortices contribute to the lift of the wing when the incident flow is in the x direction. The vortex distributions kx(x, y) and ky(x, y)

Direction of incident flow

Figure 3-15 Wing with areal vortex distribution. kx — vortex density of vortex lines in the x direction, ky — vortex density of vortex lines in the у direction.

cannot be chosen arbitrarily; rather, they must produce velocities induced by the vortex sheet that satisfy the condition of irrotationality dufdy — dvfdx = 0.

According to Eq. (2-46л), in the vicinity of the vortex sheet (z -> 0) the perturbation velocities are

(3-37)

where the upper sign is valid above, the lower sign below the vortex sheet. Hence

This relationship is called the condition of source-free vortex distribution.

The connection between circulation and rotation (Stokes’s theorem) yields kx ~ cox and ку~Ыу, which is another formulation of the spatial vortex conservation law.

A second possible way to represent a wing by a vortex distribution consists, as suggested by Glauert [23], of replacing the wing by so-called elementary wings of infinitesimal span dy and of chord c(y) (Fig. 3-16). Each elementary wing occupies its special location within the wing boundaries as defined by the wing geometry. The vortex system of each elementary wing consists of a number of vortex lines, one behind the other, parallel to the у axis, which is equivalent to a series arrangement of horseshoe vortices as introduced in Sec. 3-2-1. This representation was given for arbitrary wing planforms by Truckenbrodt [84], among others. In Fig. 3-17, this concept is again demonstrated by the example of a yawed swept-back wing. Note that the free vortices of the individual horseshoe vortices have been drawn separately in this picture, but only for clarity; actually, all of them are located on two parallel lines of distance dy. The circulation distribution density of

Figure 3-16 Substitution of a lifting wing by an elementary wing of span dy and chord c(y).

the elementary wing in direction of the chord (x direction) is k(x) per unit length. In the terminology of Fig. 3-15, к corresponds to ky of this figure. From Eq. (244), it follows that the circulation of a surface element of the elementary wing with span dy and chord dx is

tІГ{х, у) = Jc{x, у) сіх (3-38)

and the total circulation of the bound vortex of the elementary wing at the wing section у becomes

xr

Г(у) = J k(x, y) dx (3-39)

xf

where Xf(y) and xr(y) designate the x coordinates of the front and rear edges of the section, respectively. The same circulation is found in the two free vortices originating at the trailing edge of the elementary wing.

Figuie 3-17 Vortex system of a yawed wing, from [84].

Within the framework of linear wing theory, that is, limitation to small profile camber of the individual wing sections and to small angles of attack, it can be assumed that bound and free vortices of all elementary wings lie in the same plane (,xy plane). This assumption was also made for the profile theory in Sec. 2-4-2.

Equation for the determination of the circulation distribution To establish an equation for the computation of the circulation distribution, an expression first must be developed for the requirement that the lifting surface carrying the vortices is a stream surface, that is, that the normal component of the resultant velocity is equal to zero on this surface. This is the so-called kinematic flow condition. In Fig.

3- 18 a wing cross section у of the lifting surface (skeleton surface) z^(x, y) = z(x, y) is sketched. It is located in a flow field of incident flow velocity [/„ that forms the geometric angle of attack ag(y) = clf + z(y) with the chord.* Here aF is the angle of attack, measured from the x axis, and e(y) is the twist angle.

The kinematic flow condition becomes, in analogy to Eq. (249),

(340)

where w(x, y) is the velocity in the z direction induced by the total vortex system at the point x, у of the xy plane (w > 0 in the direction of the positive z axis). The brackets contain the term describing the angle between the incident flow direction and the skeleton tangent. Equation (340) must be satisfied in all points x, у of the lifting surface.

Furthermore, as a next step, the induced velocity w(x, y) on the lifting surface must be determined from the given vortex distribution k(x, y). To simplify the problem, the induced velocity is computed, however, at the projection of the lifting surface on the xy plane that is identical with the vortex sheet. The induced velocity w(x, y) at an arbitrary point of the xy plane is obtained by first determining the contribution of one horseshoe vortex of one elementary wing (Fig. 3-19). The total induced velocity w(x, y) is then the result of integrating first over one elementary wing in the x direction and consecutively in the у direction over the total number

^Contrary to Sec. 3-2-1, the incident flow velocity is designated now by U„ instead of V.

Figure 3-18 Illustration of the kinematic flow condition of wing theory.

Free vortex pair.

Control point

Figure 3-19 Explanation of the determination of the induced velocity w(x, y) of the horseshoe vortex of an elementary wing.

– <7{x, у; у) – =£ G'<*• V-V"> dy’ £ J ІУ — У )“

(3-41)*

of elementary wings. Execution of this integration yields the following result, as shown in detail in [84]:

with the kernel function

xr(y’)

G{x, у: у’) = Г к(х’, у’) (1 + —=^=) dx’ (ЗА2а)

J У(® — О2 + (у – уп

х/(у’)

X

У, У) — 2 jk(x’, у) dp’ (3-42Ъ)

xf(y)

For the derivation of Eqs. (3-41) and (342), the Biot-Savart theorem must be applied in such a way that the point in which the induced velocity w{x, y) is to be computed is first positioned outside of the vortex sheet (z Ф 0). It is then shifted into the vortex sheet (z-* 0). On the right-hand side of Eq. (341), the first term represents the self-induction of the elementary wing in the section у =y (downwash), whereas the second term represents the external induction of all other elementary wings in the sections у Фу (upwash). Finally, introducing Eq. (341) into the kinematic flow condition Eq. (340) yields

G(x, yy’) is related to k(x, y) through Eq. (342). Equation (343) is an integral equation for the circulation distribution k(x, y) of the lifting surface in which the angle of attack aF and the wing shape zix, y) are given quantities. To satisfy the

Kutta condition for the wing, the vortex density k(x, у) at the trailing edge x = xr(y) must disappear [see Eq. (2-51)]. After having determined the vortex density k(x, y) from Eq. (343), the resultant of the pressure distribution of lower and upper surface at the point x, y, from Eq. (2-53), takes the form

(344)

Here, <7oo = gU’Lfl is the dynamic pressure of the incident flow.

As in the case of the Prandtl wing theory (Sec. 3-2-1), the wing geometry (twist and camber) can be established with Eq. (343) when the wing area and vortex distribution k(x, y) are given quantities. The indirect problem requires quadratures as in Eqs. (342) and (343). When the wing geometry (planform and angle of attack) is given, Eq. (343) produces the vortex distribution on the wing surface. This direct problem leads to an integral equation for the vortex distribution k(x, y), the solution of which poses considerable mathematical difficulties. Approximation methods need to be applied, therefore, which can be laid out in various ways.

A first possibility for obtaining an approximate solution is given by imposing beforehand the vortex distribution k(x, y) in the direction of the wing span y. By selecting for k(y) an expression of m terms, the first of which may, for instance, represent the elliptic distribution, the integral equation Eq. (343) can no longer be satisfied on the whole lifting surface, but only on m sections in the chord direction.

A second possibility for the establishment of approximate solutions consists of imposing beforehand the vortex distribution k(x, y) in the direction of the wing chord x, for example, using the Birnbaum normal distribution of Eq. (2-61). If one selects for k(x) an expression of n terms, then the integral equation can be satisfied only on n lines along the span. Such procedures have been established for n = 1 (first normal distribution) by Weissinger [95], for n — 2 (first and second normal distributions) by Multhopp [62] and Truckenbrodt [84], and for n = 5 by Wagner [91] and also by Kulakowski and Haskell [12].

A third possibility consists of imposing beforehand distributions with m terms over the span and simultaneously distributions with n terms over the chord. In this case the integral equation can be satisfied at (m • n) points suitably distributed over span and chord. Such a procedure was applied by Blenk [69]. More recently, the so-called panel procedure was developed [46] (see Sec. 6-3-1).

Previously, Falkner [14] presented a procedure in which discrete vortices were arranged in both the chord and span directions. Also, the work of Jones [39] and Lan [51] must be mentioned.

Velocity potential The induced velocity field of the vortex system of a wing can also be defined by means of a spatial velocity potential Ф(х, у, z). Here the velocity components induced by the vortex system are

According to Truckenbrodt [84], the potential is

with Cr{x, y, z; y’) = f k(x’, y’) (1 4 -……………… ж ~ – dx’ (347)

J 1 {z – *’)2 -f (y — y’f 4 з2 /

xf(y’)

This expression for the velocity potentials of a lifting surface had been presented earlier in similar form by von Karman [89] and Burgers [69]; see also [84].

The potential is discontinuous at the lifting vortex sheet and in the free vortex sheet behind the wing. Closer investigation shows that it changes abruptly when crossing the. vortex sheet from the upper to the lower surface. This step of the potential above (index u) and below (index Ї) the vortex sheet is given at the lifting surface [xf(y) <x <л:г(у)] by

Фы(4 у) — Ф/ (4 у) = J к(х’, у) dx’ (348а)

Xf(.V)

and the free vortex sheet [x >xr(y)] by

xr(y)

фи(х> У) — фі (4 у) — S к(х’> У) dx’ — г(у) (348Ь)

х/(у)

Very far upstream and very far downstream of the wing, the function Ф, in terms of Г from Eq. (3-39) becomes

Ф ( — oo, y, z) = 0 (349a)

S

(349b)

—s

Equation (349b) represents the two-dimensional potential of the induced velocity field in the yz plane far behind the wing (potential in the Trefftz plane [69]).

Acceleration potential For the treatment of the problem of the lifting surface by means of the Laplace potential equation there is available, besides the method of the velocity potential just discussed, the method of the acceleration potential. This was first published by Prandtl [69 (1936)].

The method of the acceleration potential has been applied to the circular plate by Kinner [44] and to the elliptic plate by Krienes [47].

3- 2-1 Fundamentals of Prandtl Wing Theory

The creation of lift of a wing is tied to the existence of a lifting (bound) vortex within the wing (Fig. 3-7). This fact has been explained in Sec. 2-2 by means of Fig. 2-4. The position of the bound vortex on the wing planform is described in Sec. 2-3-2 for the inclined flat plate. Accordingly, it is expedient to position the vortex on the one-quarter point of the local wing chord. An unswept wing in symmetric incident flow is therefore represented by a bound vortex line normal to the incident flow direction.

Figure 3-6 Distribution of local lift coefficients for a rectangular wing of aspect ratio л = 5 and profile Go 420. Reynolds number Re — 4.2 • 10s; Mach number Ma = 0.12.

Figure 3-7 Vortex system of a wing of finite span.

Since the pressure differences between lower – and upper-wing surfaces decrease to zero toward the wing tips, producing a circulation around the wing, the flow field of a wing of finite span is three-dimensional. This pressure equalization at the wing tips, shown schematically in Fig. 3-86, causes an inward deflection of the streamlines above the wing and an outward deflection below the wing (Fig. 3-8(2). In this way, streamlines that converge behind the wing have different directions. They form a so-called surface of discontinuity with inward flow on the upper surface, outward flow on the lower surface (Fig. 3-8c). The discontinuity surface tends to roll up farther downstream (Fig. 3-8<f), forming two distinct vortices of opposite

sense of rotation. Their axes coincide approximately with the direction of the incident flow (Fig. 3-8e and f). These two vortices have a circulation strength Г. Thus, behind the wing there are two so-called free vortices that originate at the wing tips (Fig. 3-7). Far downstream, these two vortices are connected by the starting vortex, the evolution of which was explained in Sec, 2-2-2. The bound vortex in the wing, the two free vortices, originating at the wing tips, and the starting vortex together form a closed vortex line in agreement with the Helmholtz vortex theorem. The vortices produce additional velocities in the vicinity of the wing, the so-called induced velocities. They are, as a result of the sense of rotation of the vortices, directed downward behind the wing. They play an important role in the theory of lift.

The starting vortex need not be taken into account in steady flow for treatment of the flow field in the vicinity of the wing. This is understandable when it is realized that the wing has already moved over a long distance from its start from rest. In this case the vortex system consists only of the bound vortex in the wing and the two infinitely long, free vortices. These form again an infinitely long vortex line shaped like a horseshoe, open in the downstream direction. This vortex is called a horseshoe vortex.

The very simplified vortex model of Fig. 3-7, having one bound vortex of constant circulation, is still insufficient for quantitative determination of the aerodynamics of the wing of finite span. A further refinement of the two simple free vortices originating at the wing is necessary. The above-mentioned pressure equalization at the wing tips causes the lift, and consequently the circulation, to be reduced more near the wing tips than in the center section of the wing. At the very wing tips even complete pressure equalization occurs between upper and lower surfaces. The circulation drops to zero. The actual circulation distribution is. similar to that shown in Fig. 3-9; it varies with the span coordinate, Г = Г(у). The variable circulation distribution Г(у) in Fig. 3-9 can be thought of as being replaced by a step distribution. At each step a free vortex of strength А Г is shed in the downstream direction. In the limiting case of refining the steps to a continuous circulation distribution, the free vortices assume an areal distribution (vortex sheet). A strip of this vortex sheet of width dy has the circulation strength dF = (dr[dy)dy. Thus the slope of the circulation distribution F(y) of the bound vortices determines the distribution of the vortex strength in the free vortex sheet.

It was Prandtl [69] who for the first time gave quantitative information on the three-dimensional flow processes about lifting wings based on the above discussed mental picture. Earlier, Lanchester had investigated this problem qualitatively (see von Karman [90]).

Lift and induced drag From the Kutta-Ioukowsky theorem [see Eq. (2-15)], the lift dL of a wing section of width dy and its circulation Г (у) are related by

dL = 5 УГ[у) dy

Figure 3-9 Wing with variable circulation dis­tribution over the span.

The total lift is obtained by integration as

L — qV f Г (у) dy (3-15)

•f-s

As the most important consequence of the formation of free vortices, the airfoil of finite span undergoes a drag even in frictionless flow (induced drag), contrary to the airfoil of infinite span. Physically, the induced drag can be explained by the roll-up of the discontinuity sheet into the two free vortices: During every time increment a portion of the two free vortices has to be newly formed. To this end, work must be done continually; this work appears as the kinetic energy of the vortex plaits. The equivalent of this work is expended in overcoming the drag during forward motion of the wing.

On the other hand, the formation of induced drag may also be understood by means of the Kutta-Joukowsky theorem as follows: The downstream-drifting free vortices produce a downwash velocity w£ behind and at the wing, after Biot-Savart. At the wing the incident flow velocity of the wing profile is therefore the resultant of the incident flow velocity V and this induced downwash velocity w£. Accordingly, the resultant incident flow direction at the wing is inclined downward by the angle а£ against the undisturbed incident flow direction, with

«< = У (3-16)

In general, w£ < V and hence a£ sin щ ~ tan a£.

From the Kutta-Joukowsky theorem (Sec. 2-2-1), the resultant dR of the aerodynamic forces at the wing cross section у (Fig. 3-10) stands normal to the resultant incident flow direction. Hence, normal to the undisturbed flow direction there is a lift component dL = dR cos « dR and parallel to the undisturbed flow direction a drag component dDi — dR sin щ «=» dRat. The latter is the induced drag of the wing cross section y, which, with Eq. (3-16), becomes

Wi

dDt = aidL = dL —

Hence, the total induced drag is obtained through integration over the wing span from у = — s toy = +s, and by noting Eq. (3-14), as

(3-18)

where W/(y) is the distribution of the induced downwash velocity that is variable in

the general case.

The distribution of the induced downwash velocity along the span is obtained by applying the Biot-Savart law to the semi-infinitely long free vortex behind the wing. The contribution of the vortex strip dy at station у to the downwash velocity at the location of the lifting line у (Fig. 3-9) is with iy—y) being the distance of the point under consideration (control point) у from the location у’ of the free vortex line. From this, the induced velocity at the wing is found by integration over the area of the free vortices as*

(3-19)

From this equation, the induced downwash velocity W/ at the location of the lifting line can be computed when the circulation distribution Г(у) is known. Finally, the induced drag can be determined from Eq. (3-18).

It should be mentioned here that the induced downwash velocity very far behind the wing has twice the value of the downwash velocity wt at the wing from

*At station y’ =y, the integrand has a singularity. The analysis shows that the integral has to be evaluated through the Cauchy principal value. Hence, the range у—є<у'<у + є must be excluded during integration and the limit operation

Ihu і f ■ ■ ■ dy’ + f. .. dy’

> О 1 —f. .*»_!_* must be conducted.

Eq. (3-19). This is obvious from the fact that far downstream the free vortices can be taken as being infinitely long vortex lines, leading to

Woo(y) — — 2и>((у) (3-20)

The velocity w0о is taken as positive in the direction of the positive z axis (see Fig. 3-21).

Prandtl’s integral equation of the circulation distribution The above considerations will now be applied to the derivation of an equation for the determination of the spanwise circulation distribution for a given wing of finite span.

The change of the incident flow direction that results from the downwash velocity induced by the free vortices was explained in Fig. 3-10. This change of flow incidence, at equal geometric angles of attack a, is responsible for the reduced lift at the cross section у of a finite-span wing in comparison with the lift at the same cross section of an infinitely long wing.

For a span element dy of a finite-span wing, Eq. (3-12) yields for the lift:

dL = cfy) j V2c(y) dy (3-2 la)

= c’i„ae{y) V2c(y) dy (3-216)

Here c(y) is the wing chord at station у (Fig. 3-9) and q(y) = c’laoote(y) is the local lift coefficient of the area element dA = c(y) dy; ae(y) is termed the effective angle of attack (Fig. 3-10) and = (dci(da)« is called the lift slope for the airfoil of infinite span. The latter value is close to 2тг, from the theory of thin profiles (see Chap. 2). For the inclined flat plate, cj« is exactly equal to 2rr. Equation (3-21) is based on the concept that a profile cross section of a wing of finite span behaves like that of a wing of infinite span (plane flow) at an angle of incidence ae.

The geometric angle of attack о(у), measured from the zero-lift position, the effective angle ae(y), and the induced angle az-(y) [Eq. (3-16)] are related by

qO) = ae0) + af(y)

as shown in Fig. 3-10. The effective angle of attack ae is obtained from Eq. (3-216) with the help of Eq. (3-14), and the induced angle of attack from Eq. (3-19) with ai= wil V as

2rQ)

Vc{y)ct

Introducing Eq. (3-23) into Eq. (3-22) yields the following basic equation for the determination of the circulation distribution:

2Г(у) f dr dy’

VcfyWio[13] + v J dy’ У —у’

-s

This is Prandtl’s integral equation for the circulation distribution of a wing of finite span as first published by Prandtl in 1918 [69]. It is a linear integral equation for the circulation distribution Г(у), where Г depends linearly on the angle of attack a. The profile coefficient ctaa is known from profile theory (Chap. 2).*

With given wing geometry [chord distribution c(y) and angle-of-attack distribu­tion a(y)], the circulation distribution can be determined from Eq. (3-24). This is the so-called direct problem of wing theory. Conversely, if the circulation distribution Г(у) is known, either the angle-of-attack distribution (twist angle) ct(y) can be computed from Eq. (3-24) when the chord distribution c(y) is given, or the chord distribution c(y) when the angle-of-attack distribution a(y) is given. This is the so-called indirect problem of wing theory. In either case, from the circulation distribution Г(у) the lift is obtained from Eq. (3-15) and the induced drag from Eq. (3-18).

From a mathematical viewpoint, the direct problem is considerably more difficult than the indirect problem, because in the former case an integral equation has to be solved while in the latter case only a quadrature has to be performed.

Elliptic circulation distribution A particularly simple solution of Eq. (3-24) that is of great practical importance is found for the elliptic circulation distribution along the span. In this case the circulation becomes

where Г0 is the circulation at the wing center y— 0 (Fig. 3-11). From Eq. (3-15), the lift becomes

L =^obVrQ

4 u

The induced down wash-velocity is obtained from Eq. (3-19). Execution of the integral yields for points within the span, jyj <b/2,

(3-21 a)

(3-276)

This remarkable result shows that, for elliptic circulation distribution, the induced downwash velocity w,-, and consequently the induced angle of attack a,-, are constant over the span (Fig. 3-11).

By introducing Eqs. (3-25) and (3-21 a) into Eq. (3-18), the induced drag is obtained with T0 from Eq. (3-26) as

(3-2Sa)

(3-286)

Here, q — (o/2)V2 is the dynamic pressure resulting from the velocity V. The induced drag is proportional to the square of the lift and inversely proportional to the dynamic pressure and the square of the span. Comparison of Eqs. (3-28b) and (3-27b) confirms the relationship Dt = cqL, given in Eq. (3-17).

The geometry of the corresponding wing is obtained in a particularly simple way when starting from the wing without twist, a(y) = a = const. Since, from Eq. (3-276), the induced angle of attack щ(у) = const, Eq. (3-22) shows that the effective angle of attack along the span must also be constant: ae(y) = const.

Figure 3-11 Elliptic circulation distribution with the corresponding elliptic wing plan-

From Eqs. (3-23a) and (3-25), it follows that the chord is distributed elliptically over the span:

Ф) = cr yj 1 – (3-29)

The elliptic wing planform is shown in Fig. 3-11.* Thus it has been demonstrated that an elliptic wing without twist has an elliptic circulation distribution. From Eq. (3-21), it also has a constant local lift coefficient ct(y) over the span.

Coefficients Finally, the most important results for the induced angle of attack [Eq. (3-27&)] and for the induced drag [Eq. (3-28b)] will also be expressed through the dimensionless coefficients of lift and induced drag. They are defined as follows:

L = cLqA (3-30a)

Di = cm<lA (3-30 b)

with A being the wing planform area. Consequently, Eqs. (3-27b) and (3-28b) yield

(3-3 la) (3-3 lb)

Here A — b2/A is the aspect ratio of the wing from Eq. (3-9). The important result for the coefficient of the induced drag of Eq. (3-31 b) is compared in Fig. 3-12 with test results for a wing of aspect ratio A = 5. The theoretical curve for the induced drag agrees quite well over the whole cL range with the polar curve of the measured data. The difference between the two curves is about constant over the whole cL range. It is caused by the effect of friction that has been neglected in the above theory. Figure 3-12 suggests splitting up the drag coefficient into a component that is nearly independent of the lift coefficient and a component that is dependent on the lift coefficient. The former is called the coefficient of profile drag cDp, the latter the coefficient of induced drag cDi. They are related by

C-D cDp cDi	(3-32 a)
cl ~CDp 1 їїЛ	(3-32 b)
For the geometric angle of attack, Eqs. (3-22) and (3-3 Ід) yield
, CL a = ae -1— e TtA	(З-ЗЗд)
cl, cL ~ j „ .1	(3-33b)

*The elliptic wing consists of two ellipse halves, the large axis of which is the cj4 line.

From Eq. (3-21), ae = cL/c’Loo because the constant local lift coefficient сг(у) and the total lift coefficient Ci are equal in this case. The latter equation allows one to determine the lift slope of the wing of finite span as a function of the aspect ratio. From Eq. (3-33b) it follows:

with c’L=dcLldot and = 2тг. Equation (3-34h) expresses the degree of reduction of lift slope and consequently also of lift because of the finite aspect ratio when the angles of attack are equal. In Fig. 3-13 this ratio of lift slopes is presented as a function of the aspect ratio.

As will be shown later in more detail, the formulas for induced drag and lift slope found here. for the elliptic wing are valid for other wing shapes in good approximation. This is true particularly for the rectangular wing, as shown by Betz [5]; see Figs. 3-32 and 3-57.

Prandtl’s transformation formulas The above-derived results on the effect of aspect ratio on lift and drag have been checked experimentally by Betz and Wieselsberger [99]. For comparison of the polar curves of two wings of aspect ratios Аг and Л2 at equal angles of attack, Eq. (3-32b) with cDp2 = CDpl yields

Figure 3-13 Ratio of the lift slope of wings of finite and infinite aspect ratios vs. aspect ratio, c’icо = 27г.

In Fig. 3-14a the measured polar curves are plotted for a number of rectangular wings with aspect ratios Ax = 1, 2, , 7. Figure 3-146 shows the result of the

transformation of these polars to the aspect ratio Л2 = 5 from Eq, (3-35). The transformed curves fall well on one curve, confirming experimentally the validity of Eq. (3-35). In Fig. 3-146 the theoretical polar curve of the reduced drag for A — 5 is also included. On the other hand, comparison of the lift curves cl(ol) of two wings of aspect ratios A and Аг of equal lift coefficient yields, with Eq. (3-336),

(3-36)

For the wings of Fig. 3-14, the lift curves were converted to the aspect ratio Лг = 5. Again, the converted curves fall together, confirming experimentally the validity of Eq. (3-36).

The two equations (3-35) and (3-36) can therefore be used for the transformation of measured drag polars cD{cL) and lift curves cL(a) at aspect ratio Лі to those of a wing with a different aspect ratio Л2 if both wings have the same profile. These equations are called, therefore, transformation formulas of the wing of finite span.