Category AERODYNAMICS OF THE AIRPLANE

Velocity potential (linearization) For slender body shapes (wings) in an incident flow of velocity Uaz in the direction of the x axis (longitudinal axis), the local velocities differ only a little from (Jm in direction and magnitude. The total flow can then be separated into a basic flow and a superimposed perturbation flow by setting

U = – p и V = v W — v: (44)

where u, v, w are the perturbation velocities caused by the wing, for which it is required that

и < Ux v <: w <4

By retaining only the largest terms (linearization), the potential equation of compressible flow for such a flow problem takes the form

where Ma — Ufa is the local Mach number. This equation is valid for subsonic, transonic, and supersonic flow.[24] It is nonlinear in the velocity potential. The components of the perturbation velocities are obtained from Eq. (4-5) as

(4-6)

In analyzing the linearized Eq. (4-5) further, the first term requires particular attention because it changes sign when passing the speed of sound (Ma = 1) and thus changes the mathematical character of the differential equation. By retaining only the linear terms in ufU^, the local Mach number Ma in Eq. (4-5) may be expressed by the Mach number of the incident flow Mz« = U^ja as follows:

(4-7)

For pure subsonic and pure supersonic flow, the simplified potential equation

is obtained by replacing Ma by Ma*, in first approximation.

This differential equation for Ф is now linear. For pure subsonic flow, it is of

the elliptic type, as is the equation for incompressible flow. For pure supersonic flow, however, it is of the hyperbolic type.

When the undisturbed flow velocity is equal to the speed of sound {Mam = 1), a transonic flow results whose velocity field may include stations of Ma = 1. In this case, from Eqs. (4-5) and (4-7), it follows that

This differential equation for Ф is nonlinear. Therefore, the computation of transonic flow fields is considerably more difficult than computations of pure subsonic and pure supersonic fields. The potential equations derived above [(4-8) and (4-9)] will now be used to derive similarity rules for three-dimensional wing theory at subsonic, supersonic, and transonic flows.

Similarity rules for subsonic and supersonic flow For subsonic flows, similarity rules can be derived from Eq. (4-8) according to Prandtl [73], Glauert [27], and Gothert [28], the application of which greatly simplifies the computation of compressible potential flows. These procedures can be applied similarly to supersonic flows; see Ackeret [1]. Of the various possible derivations of these similarity rules, the so-called streamline analogy will be applied.

The similarity rules for a wing in subsonic or supersonic incident flow (Mг» ^ 1) are obtained through a transformation of the potential equation, Eq. (4-8). This transformation is such that the Mach number of the undisturbed flow no longer appears explicitly in the transformed – potential equation. To this end, a transformed reference flow is established for the given flow in a suitable way. The variables of the reference flow are designated by a prime:

x’ =X y’ = cxy z’ = cxz Ф = с2Ф’ U’ac = (4-10)

By introducing these terms into Eq. (4-8), the factor cx > 0 is determined in such a way that the Mach number is eliminated, resulting in

c, = Іl – Ma% Wa„ < 1) (4-1 la)

c, = ]/Ма%, – 1 (ifa„ > 1) (4-1 lb)

These cases can be combined to

cx = У|1 – МаІ! (4-12)

With Eq. (4-11) for the transformed reference flow, the following differential equations are obtained for the velocity potential:

(4-13)

(4-14)

The transformed equation for subsonic flow is identical to the potential equation for incompressible flow, and the transformed equation for supersonic flow is identical to the linear potential equation, Eq. (4-8), at Mach number Маж =J2. This transformation shows that computations of subsonic flows of any Mach number can be reduced to computations of flow at Mz« = 0, and computations of supersonic flows to those at Ma^ = Jl.

The transformation factor сг in Eq. (4-10) remains undetermined for the time being. Its value will be given later.

Application of the transformation formulas, Eq. (4-10), to wings of finite span will now be treated. The coordinate system x, y, z of Fig. 4-6 will be used with its X axis parallel to the incident (undisturbed) flow. Equation (4-10) describes the procedure for determining the transformed wing from a given wing of a given Mach number where the flow field about the transformed wing is to be computed, according to the above rules, for subsonic flow at Ma^ = 0 and for supersonic flow at Mdoo =/2. The transformed wing according to Eq. (4-10) is then obtained from the given wing by decreasing or increasing, respectively, the dimensions in the directions normal to the incident flow direction (y and z directions) by the factor d of Eq. (4-12).

•For the wing planform, the following relationships between the transformed

Figure 4-6 Wing geometry, (a) Wing plan – form; ctlcri, taper; л~Ь2/А, aspect ratio; A = wing planform area; ip= sweep – back angle. (&) Profile section у = const; Z(ix) = profile contour; h/c = relative cam­ber; f/c = relative thickness; a. = angle of attack.

Figuie 4-7 Application of subsonic and supersonic similarity rules to the example of a tapered, swept-back wing, (a) Given wing, to be computed for Mach numbers Maoo = 0.7, 0.9, 1.1, and 2. (b) Trans­formed wing for these Mach numbers.

From Eqs. (4-15b) and (4-15c), the remarkable relationship

A’ tan <p’ — A tan <p (4-15d)

is obtained, where it is immaterial to which of the pianformcontour lines the sweepback angle is referred, for example, the leading edge or the trading edge.

In Fig. 4-7 the transformation of the wing pianform is explained through the example of a swept-back wing. The crosshatched wing pianform in Fig. 4-7a is the shape of the given wing, the flow field of which is to be determined for the various Mach numbers Max = 0.7, 0.9, 1.1, and 2.0. The corresponding transformed wing planforms are shown in Fig. A-lb, where at Мг« <1 the transformed wings are to be computed for incompressible flow (Маж = 0), and at Маж > 1 for Ma*, = J2.

In Fig. 4-8, the wing pianform transformation as given by Eqs. (4-15a)-(4-l 5c) is explained in more detail. Here, A’/A and cotej’/cotip are plotted versus Маж. Again, the given wing pianform, which is to be computed for the various Mach numbers, has been crosshatched. The open wing planforms represent the trans-

Figure 4-8 Illustration of the applica­tion of subsonic and supersonic similar­ity rules; aspect ratio A and sweepback angle у of the transformed wing vs. Mach number.

formed wings for the corresponding Mach numbers. When the given Mach numbers are Afaoo = 0 and Mz» = y/2, respectively, the transformed and the given wings are identical. Figure 4-8 shows that, in the subsonic range, an increase of Mr» results in a decrease of the aspect ratio whereas the sweepback angle increases. For Mz« 2, the aspect ratio of the transformed wing approaches A! -> 0 and the sweepback angle ip’->90°. In the supersonic range Mr» >y/2, the aspect ratio of the transformed wing increases with Mr» whereas the sweepback angle decreases. In the limit of very large Mr», the aspect ratio of the transformed wing Л’ -*00 and the sweepback angle 0. The remarkable result is found that for large Mach numbers the three-dimensional wing flow field is converted into a two-dimensional field.

The Prandtl-Glauert-Gothert-Ackeret rule is also applicable to asymmetric incident flow (yawed wings); see Truckenbrodt [28]. For the profile cross section and angle of attack of Fig. 4-6b, Eqs. (4-10) and (4-12) lead to the following expressions:

This shows that for Mr» < /2, the transformed wing has less camber, is thinner, and has a smaller angle of attack than the given wing; conversely, for Mr» >y/2 it has more camber, is thicker, and has a larger angle of attack.

After the effect of the transformation, Eq. (4-10), on the wing geometry has been discussed, the relationship between the pressure distributions of the given and the transformed wing must be studied.

The dimensionless pressure coefficients cp = (p — p«)/(p«.i7»/2) assume, within the framework of linearization, the approximate form

(4-18 a)

(4-Ш) where the velocities of the incident flow £/«. of the given and transformed flow must be equal.

This leads with Eq. (4-10) directly to

cp = сгс’р (4-19)

The still-unknown transformation factor c2 is determined from the kinematic flow conditions for the two wings (streamline analogy). These are, within the framework of linearized theory,

W = U°°1? w’=V°°l5 (4’20)

where w and w’ are the z components of the perturbation velocity on the profile contour zc and zc, respectively (Fig. 4-6b). Because w= дФ/dz and w’ = ЪФ’jbz, we find with Eq. (4-10):

(4-21)

The meaning of the subsonic and supersonic similarity rules can now be summarized as follows: From the given wing and the incident flow Mach number, the transformed wing is found by multiplying the dimensions of the given wing in the у and z directions and its angle of attack by the factor cx = Vl(l — Mzi)|, whereas the dimensions in the x direction remain unchanged. For subsonic velocities, the flow about the transformed wing is computed from the incom­pressible equations; for supersonic velocity, however, it is computed from the compressible equations for MzTO = /2. If the incident flow velocities are equal for both wings, the pressure coefficients are related by

With regard to practical applications, it is expedient to choose a transformation in which only the dimensions in the у direction (wing planform) are distorted, whereas the dimensions in the z direction (profile and angle of attack) remain unchanged. Such a transformation is obtained from the above version I by removing the distortion in the z direction according to Eqs. (4-16a), (4-16b), and (4-17). Thus, from Eq. (4-22), the pressure coefficient is changed, within the limits of the linearized theory, by the factor Vll — Mai, I, that is, c’p = c’p Vl I —MaL|, and the pressure coefficient becomes

This relationship is shown in Fig. 4-9. Thus, the following version is obtained for the subsonic and the supersonic similarity rule.

From the given wing and Mach number, a transformed wing is formed by multiplying the dimensions of the given wing in the у direction with the factor c — Vl(l —Mala)I, whereas the dimensions in the x and z directions remain unchanged. For the transformed wing thus obtained, the incompressible flow field is computed when the given incident flow Mach number lies in the subsonic range. When the Mach number lies in the supersonic range, however, the flow field about the transformed wing is computed from compressible equations at Mam = V2. For equal incident flow velocities £4 of given and transformed wings, the pressure coefficients are interrelated through Eq. (4-23). From the subsonic and supersonic similarity rules, the following generally valid relationships for the aerodynamic coefficients are obtained: Let the function

Cp = <5’/i h; A", cot f’-j; (4-24)

describe the dependency of the pressure coefficient on the geometric wing data at Maoo = 0 or Max = fl. Then the corresponding dependency of the geometric wing data at an arbitrary Mach number is obtained, because of Eqs. (4-15) and (4-22), in the form:

S = 7 7=- fM; A 1 – Mal ; cot^] 1 — M<&| ; f) (4-25a)

У1- Ma^ I c sl

Here 5 stands for the relative thickness tjc, the relative camber height А/с, or the angle of attack. This equation can be written in a simpler form:

cr= ,_____ І___ (4-256)

]/j 1 — Ma^o) C 4

From this formula for the pressure distribution, the lift coefficient is obtained in corresponding form by integration over the wing surface:

cL = .. 5 —ідІЛ: Л tan93; A ^ 11 — Ma%) (4-26)

Here 5 stands for the angle of attack or for the relative camber height. By going to the limiting case of the airfoil of infinite span (X= 1, ip — О, Л-»°°), the subsonic similarity rule transforms into the well-known Prandtl-Glauert rule of plane flow.

A formula analogous to Eq. (4-26) for the drag coefficient (wave drag) that is valid, however, only for supersonic flow (see the discussions of Sec. 4-5-5) is given as

cD = — A F2 (A A tail у, Л ІМа% — 1) (4-27)

№4 – і

For wings with zero angle of incidence, 5 is the wing thickness ratio tjc. In this case, the drag coefficient at zero lift cD = c^q is proportional to 52.

Figure 4-9 Illustration of the applica­tion of subsonic and supersonic similar­ity rules (version II): transformation of the pressure coefficients.

The outstanding value of the above formulas lies in their describing the Mach number effect in a simple way. They can, however, also be used to great advantage for the classification of test results.

Transonic similarity rule For flows of velocities near the speed of sound (transonic flows), a similarity rule can be derived after von Karman [103] that is related to those for subsonic and supersonic flows. For wings in a flow field of sonic incident velocity (Mao, = 1), it is obtained from the potential equation, Eq. (4-9).

Contrary to the similarity rules for subsonic and supersonic flows, for which

the dependency of aerodynamic coefficients from the geometric wing parameters

and the Mach number was investigated, only1 the dependency of the aerodynamic coefficients on the geometric parameters must now be studied, because Max = const = 1.

The problem can be posed in the following way: Given is a wing with all geometric data (planform and profile) at an angle of attack zero. What, then, is the geometry of a reference wing, also in an incident flow field of Ma„ = 1, that has an affine pressure distribution equal to that of the given wing? To answer this question, the following transformation is introduced into Eq. (4-9) [see Eq. (4-10)]:

x’ = X y’ = csij v! = c3z Ф = с4Ф’ U’cc = (4-28)

where the quantities without primes refer to the given wing, those with primes to

the reference wing.

Introducing Eq. (4-28) into Eq. (4-9) yields, with

4 = (4-29)

the following nonlinear differential equation for the velocity potential of the transformed flow:

For an additional relationship between the constants c3 and c4, the kinematic flow conditions, Eq. (4-20), for both wings have to be established.

For chord-parallel incident flow, this relationship is

where 5 = t/c is the thickness ratio of the wing profile, which has been assumed to be symmetric.

Hence, with Eq. (4-29):

The distortion of the geometric data of the wing planform is given by the factor c3 in Eq. (4-28). Hence, the following transformations are valid:

As an example for the transonic similarity rule, the transformation for a swept-back wing is presented in Fig. 4-10.

Transformation of the pressure distribution is obtained in analogy to Eqs. (4-18) and (4-19) merely by replacing c2 by c4s that is, cp=c4cp. With сл according to Eq. (4-23), it follows that

If the pressure distribution is to be related to the geometric parameters, Eq. (4-34), considering Eqs. (4-33a)-(4-33c) leads to

Cp = <32/3 / A tanЛ<51/8. (4-35)

Hence it is shown that the pressure coefficient from the transonic similarity rule is proportional to 52/3, whereas it is proportional to 5 according to the subsonic and supersonic similarity rules of Eq. (4-25).

From Eq. (4-35) the following expression is found for the drag coefficient,

cD = (55/3 F (/, A taпер, .Id1 /3) (4-36)

showing that the drag coefficient is proportional to 5S/3, whereas it is proportional to 52 according to Eq. (4-27).

Figure 4-10 Application of the transonic similarity rule for sonic incident flow to the example of a trapezoidal swept-back wing. (a) Thickness ratio 8 = t/c — 0.05. (b) Thick­ness ratio 5′ = t’/c’ = 0.10.

The formulas for the airfoil of infinite span (X— 1, <p= 0, A -»«>) wiH be given in Sec. 4-3-4 in extended form (Mz„ « 1 instead of Max = 1).

In Sec. 2-5-2 the friction drag of wing profiles in incompressible flow was discussed. Particularly, in Fig. 2-48 the influence of the Reynolds number on the drag of a flat plate in chord-parallel incident flow was demonstrated. The insight gained then will now be extended to the case of compressible flow.

Wall flow The compressible boundary layer is decisively affected by the heat transfer between the wall and the streaming fluid. Here, the case of the wall without heat transfer (adiabatic wall) is of particular importance.

The laminar boundary layer of compressible flow can be treated theoretically, but theoretical studies dealing with the turbulent compressible boundary layers are still limited to semiempirical theories of the type of the Prandtl mixing-length hypothesis, in which, however, additional assumptions must be made. Drag coefficients of the flat plate at zero incidence over Reynolds and Mach numbers are given in Fig. 4-4 in comparison with measurements. Agreement between computa­tion and measurement is not satisfactory in all cases. However, some uncertainty of measurement at high Mach numbers should be taken into account. Also, in Fig. 4-5,

Figure 4-3 Heating of a solid wall through friction; W — velocity boundary layer; T = temperature boundary layer.

the ratio of the drag coefficients at compressible and incompressible flow are presented against the Mach number up to very high Mach numbers. The decrease of friction drag is very pronounced at high Mach numbers. Curve 1 of the two theoretical curves is valid for the adiabatic wall, curve 2 for the wall with heat transfer. Measurements of several authors are in good agreement with theory. For completeness, the friction coefficients of the flat plate at zero incidence are also given for compressible laminar flow.

4- 2-1 Temperature Effects in Compressible Flow

It is a peculiarity of all compressible flows that their aerodynamic processes are always coupled with thermodynamic processes. The pressure changes in the flow, in general, are connected to temperature changes that may be determined from the equation of state [Eq. (1-la)].

Stagnation flow An outstanding station in the flow about a body is, according to Fig. 4-1, the front (upstream) stagnation point, at which the velocity is zero. The flow quantities at the stagnation point will be designated by the index 0. The pressure in the undisturbed flow of velocity is p„, the density £«,, and the temperature Too. At the stagnation point, the velocity is vv0 = 0, and the pressure, density, and temperature are p0, Q0, and T0, respectively. A pressure increase Ap = Pq — p„ takes place on the streamline incident on the stagnation point, which causes a temperature increase AT=T0—T00. The pressure coefficient at the stagnation point is obtained with steady, that is, isentropic compression as

The dependence of the pressure coefficient at the stagnation point cp0 on Маж is shown in Fig. 4-2a. For moderately high Mach numbers of the incident flow,

Figure 4-1 Temperature rise through com­pression.

particularly in the subsonic range, Eq. (4-1) is reduced by binomial expansion to cpо ~ 1 + Маїо, as also shown in Fig. 4-2. Agreement of the approximation with the exact formula, Eq. (4-1), is quite good up to Max = 1. For Ma„ 0, Eq.(4-1) becomes the well-known formula for the stagnation pressure of incompressible flow, Po – P oo = (@=o/2)w«, which is the basis for velocity measurements with the Prandtl impact-pressure tube (pitot tube). Such a tube measures the pressure difference (pо —poo) in compressible flow as well.

At an incident flow of supersonic velocity, the pressure changes from p„ to pQ discontinuously through a shock wave located somewhat upstream of the stagnation point (Fig. 4-2д). The pressure change can be determined in this case by first computing the pressure jump across the shock wave from the equations of the normal shock. In the subsonic flow behind the shock wave, the pressure change is isentropic. The result of this computation is

At very high Mach numbers, Ma„ -» °°, the pressure coefficient approaches a finite value, which for air of 7= 1.405 is cp0 max = 1-84. It can be seen from Fig. 4-2я that for supersonic incident flow, the pressure increase at the stagnation point with unsteady compression (which describes the physical reality) is considerably smaller than that obtained from the computation of steady compression.

Temperature The pressure increase of the stagnation point is always tied to a temperature increase. It is obtained from the energy equation as

This temperature increase at the stagnation point is shown in Fig. 4-26. It is equally valid for steady and for unsteady compression. It increases with the square of the velocity, and therefore, reaches appreciable values in the supersonic incident flow range. Note that a temperature increase according to Eq. (4-3) occurs not only at the stagnation point and its vicinity but, approximately, everywhere along a solid wall. In a thin layer (friction or boundary layer) close to the wall, the kinetic energy of the moving gas is transformed into heat through viscous effects (see Fig.

4- 3). This results in heating of the wall by an amount AT= T — Тж, which can be represented approximately by a relationship similar to Eq. (4-3), it can be realized, therefore, that a “heat cushion” is found over the entire surface of a body immersed in a flow of high velocity. In the immediate vicinity of the stagnation point this heating is produced by compression, and on the remaining portion of the surface by friction.

4- 1 INTRODUCTION

The theory of the wing in incompressible flow was discussed in Chap. 2 for the two-dimensional problem (infinite span) and in Chap. 3 for the three-dimensional problem (finite span). In this chapter the wing in compressible flow will be treated. Both subsonic and supersonic flows will be considered, depending on whether the flow velocities are lower or higher than the speed of sound, respectively. The connection between these two kinds of flow is formed by the transonic flow. Flows with very high supersonic velocities, so-called hypersonic velocities, are designated as hypersonic flows.

The influence of compressibility must be taken into account at Mach numbers higher than Ma «0.3. Compressible flow is of great significance for flow about wings, because the Mach numbers in aeronautics are, in general, considerably higher than 0.3.

The discussions of this chapter will be organized similarly to those of Chaps. 2 and 3, in that first the airfoil of infinite span in compressible flow (profile theory) and then the wing of finite span in compressible flow will be treated. From gas dynamics it is known that the compressible flows of subsonic and supersonic velocities are basically different; the same is true, naturally, for wing flows (see Fig. 1-9). For the theoretical treatment of compressible wing flow, the so-called linear theories will be applied predominantly, because they yield results that can be interpreted easily and thus aEow the establishment of general validity and of practical conclusions. For the theoretical considerations, mainly inviscid flow will be assumed as in Chaps. 2 and 3.

Besides the textbooks on gas dynamics listed in Section ЇІ of the Bibliography,
basic questions of compressible wing flow are treated by Taylor [94], Prandtl [73, 74], von Karman [101], Howarth [34], Robinson and Laurmann [78], Sears [83], Kuo and Sears [47], Heaslet and Lomax [30], Jones and Cohen [39], and Garrick [26]. Furthermore, more recent results and understanding of the theory of the aerodynamics of wings in compressible flow are presented in progress reports for certain time intervals by, among others, von Karman [102, 104], Ashley et al. [3], Ktichemann [45], Schlichting [81], Landahl and Stark [48], and Hummel [35]. Problems of experimental wing aerodynamics are treated by Frick [24]. In this connection, the comprehensive compilations of experimental data on the aero­dynamics of lift and drag by Hoerner and Borst [32], and Hoerner [31 ] should be mentioned.

Rectangular wing of finite span In the simple case of a rectangular wing c(y) — c of constant profile over the span, the contour is represented by z(x, у) = z{x) for —s<y <s. Hence, introduction into Eqs. (3-176) and (3-175д) and integration over у yield

with

It is easily verified that the quantity Ли is negative in general. This means that the perturbation velocities on the contour of a finite wing are reduced in comparison with those on the infinitely long wing. For the wing middle (root) section, у — 0, the perturbation velocity becomes

(3-181)

For Л — 0 this reduces to и = 0.

On the parabola profile Z — 25X(l —X) of thickness ratio 5 = tjc, the

maximum (perturbation) velocity at station x = c/2 of the middle section у = 0 is, from Eqs. (3-179) and (3-180),

^ = -5/lsmh-‘ () (3-182)[23]

Uoo 77 J

Thus, for the plane case, Д-*°°, it follows that umzx pl/£/oo =48/л, in agreement with Eq. (2-97).

Results for the rectangular and the parabolic profiles are given in Fig. 3-71. Figure 3-7lor shows the maximum perturbation velocity, which lies at X= 0.5, as a function of the aspect ratio for the two sections 77= 0 and 77= 0.5. In Fig. 3-7lb the maximum perturbation velocities are depicted for various aspect ratios over the span coordinate. In conclusion, it should be stated that the maximum perturbation velocity on the wing of finite span becomes noticeably smaller than on the wing of infinite span only for aspect ratios A < 2.

Elliptic wing The theory of Sec. 3-6-2 for the computation of the perturbation velocities and the above example for the rectangular wing were based on approximate methods valid for small thickness ratios 5. One example of an exact solution will now be given.

A wing of elliptic planform and elliptic profile as in Fig. 3-72 is a general ellipsoid of which the two axial ratios are very different. Let au b, сг be the three semiaxes of the ellipsoid. Then

б = – r = г – (3-183Д)

Cr О і

Л = ¥- = -— (3-1836)

А л a. x

Figure 3-71 Maximum perturbation velocities on rectangular wings with parabolic profile at zero lift. For infinite spantrmaXp] = (4/-7г)5 £/«, at station x/c = X = 0.5. (c) Dependency on aspect ratio.1. (b) Dependency on span coordinate ц =yjs.

Figure 3-72 The geometry of the elliptic wing with elliptic profile (general ellipsoid).

For a general ellipsoid, the velocity distribution on the contour can be determined in closed form. The pressure distribution on the surface of the ellipsoid in a flow parallel to the x axis is, from Maruhn, Chap. 5 [40],

where cp = (p — p «>)/(р/2)£/£ is the dimensionless pressure coefficient and A — Афіїаі, Ci(bi), a quantity that depends on the two axis ratios of the ellipsoid.

Equation (3-184) demonstrates that the pressure minimum and thus the velocity maximum lie at x = 0. This velocity maximum is constant along the у axis. From Eq. (3-184), Cpmjn = 1 A2 = 1 > with Umzx — Um "fwmax

being the maximum velocity on the contour. Hence, the maximum perturbation velocity becomes

^=A(8,A)-i (3-185)

h oc

In Fig. 3-73, the ratio wmax/wmaxpl is plotted against the aspect ratio A for various thickness ratios 5, where umaxpl =5(7*. The curve 5 ->0 represents linear

theory. The curves for the other values of 5 show the deviations of the exact solutions from linear theory.

Swept-back wing Another example is the swept-back wing of constant chord. The wing of infinite span (see Fig. 3-74) is considered first. Its sweep-back angle is ip and the profile z(x, у) = z(xr) is constant over the span, where xr is the x coordinate of the middle (root) section. The wing sections at large distance from the plane of symmetry are always in quasi-two-dimensional flow. Its velocity distribution can be determined by assuming an incident flow normal to the leading edge of the magnitude СЛ» cos </>. The result is a perturbation velocity in the x direction that, for the swept-back wing, is smaller by the factor cos than that for the unswept wing (plane problem):

Uy(y °°) = Mpi cos y

Now the velocity distribution on the middle section is to be computed. From

The integration requires special caution [see footnote to Eq. (3-175)]. The result for the middle section is

(3-187)

This relationship was first published by Neumark [65]. The first term represents the velocity distribution on a wing section far away from the wing plane of symmetry

Figure 3-74 Geometry of the swept-back wing of infinite scan.

as in Eq. (3-186). The second term represents the change in velocity distribution caused by wing folding. In the case of backward sweepback (tp > 0), the perturbation velocity in the front part of the middle section is reduced, and in the rear part of the middle section it is increased.

The above equation has been extended to generalized parabolic profiles. The result is presented in Fig. 3-75 for profiles with relative thickness positions Xt = xtjc = 0.2, 0.3, and 0.5. The curves for the sweepback angles v? = —45°, 0°, and +45° show a very considerable influence of sweepback on the velocity distribution over the middle section. The maximum perturbation velocities are shown once more separately in Fig. 3-76 over the sweepback angle.

Figure 3-76 Maximum perturbation velocity at mid­dle (root) section of swept-back wings of constant chord and infinite span vs. sweepback angle <p; see Fig. 3-75.

Figure 3-77 Velocity distribution of swept-back wing of constant chord, with aspect ratio -1 = 2.0 and sweepback angle y? = 53° at zero lift for several sections along the span, according to Neuxnark. Wing profile: parabolic profile Xt = 0.5. ы^тахСV = “) = (4/tt)(6 cos = maximum perturbation velocity of swept-back wing of infinite span at section у — «.

For a swept-back wing of constant chord and finite span, corresponding computations have been made by Neumark [65]. The velocity distribution м of a wing of aspect ratio /1= 2 and sweepback angle у = 53° is illustrated in Fig. 3-77 for various sections along the span. It is related to the maximum perturbation velocity of the swept-back wing of infinite span at a section far away from the wing root [Eq. (3-186)]. For the same wing, the lines of constant velocity (isobars) are drawn on the wing planform in Fig. 3-78. This figure demonstrates particularly well that, as a result of the sweepback, the maximum perturbation velocity increases

Figure 3-78 Isobars of a swept-back wing of aspect ratio л = 2, with sweepback angle |*э= 530 at zero lift, from [65]. Curves u{x, y)/u<Pttaax(y= «) = const; see Fig. 3-77.

considerably in the vicinity of the wing middle (root) section, and that the velocity maximum at this middle section has shifted far back.

Investigations on the pressure distribution over the middle section of a lifting swept-back wing of infinite span have been conducted by Kiichemann and Weber [48], and those on a swept-back wing with an arbitrary symmetric profile by Weber [923.

3-6-1 Displacement Problem of the Wing

The theory of the wing of infinite span as discussed in the previous sections of this chapter was based on the assumption of a very thin profile (skeleton). For the theory of the wing of finite span, the extension from the skeleton theory to the theory of the inclined wing of finite thickness (profile teardrop) has long been available (Sec. 244). A similar extension for the wing of finite span and finite thickness is still lacking. However, for the wing of finite span and finite thickness of the wing profiles (symmetric profiles), there does exist a computational method that allows the determination of the displacement effect of the wing and thus of the pressure distribution on the surface of such wings, provided that the lift is zero. It represents, therefore, a teardrop theory for wings of finite span; note the publications of Keune [42] and Neumark [65] and compare also [43]. The method of singularities is used in which the body within the flow field is replaced by a system of sources and sinks. The fundamentals of this method have been established in Sec. 2-4-3 for two-dimensional flow and applied to the problem of an airfoil in plane flow.

For the assessment of the effect of compressibility in both two-dimensional and, in particular, three-dimensional flow, it is important to know the maximum perturbation velocity on the wing. The computational procedures, treated in the following sections, for the velocity disturbance on wings of finite span and finite thickness are of significance, therefore, for the aerodynamics of the wing of high subsonic velocities.

3-6-2 Method of Source-Sink Distribution

Source system of the wing of finite span For the computation of the three – dimensional flow field about a slender body resembling a wing of finite span and finite thickness, a distribution of three-dimensional sources and sinks is established in the plane of the surface A (wing planform plane). An area element dx dy carries the source strength

у) = q(x, у) dxdy (3-171)

when q(x, y) designates the source strength per unit area. The source strength q(x, y) must satisfy the so-called closure condition

fj q{x, y)dxdy = 0 (3-172)

U)

in order to form a closed body shape. Compare also the corresponding expressions for the plane case, Eq. (2-92).

Velocity distribution on the wing contour Superposition of the velocity field, produced by the source distribution, with a translational flow of velocity £/«, the direction of which lies in the source plane (Fig. 3-70), produces a closed stream surface that can be interpreted as the contour of the wing of finite thickness; compare again Sec. 24-3. Let u, v, and w be the velocities induced by the source distribution (perturbation velocities) and let z^x, y) = z(x, y) be the shape of the wing contour, symmetric to the xy plane. Then the condition for tangency of the velocity resultant on the entire contour is

» = + «)-J*-+ »(З-ПЗ*)

= A (3-1736)

ax

This is the kinematic flow condition. Since, for slender bodies, the velocities и and v are small compared to the incident flow velocity except for the immediate

vicinity of the leading edge and the wing tips, it is sufficient to work with the simplified form, Eq. (3-173Z?). This is, formally, the same relationship as in the plane case. Both for the kinematic flow condition and for the computation of the pressure distribution on the contour, the velocity components u, v, w are required on the contour. For slender bodies, it is sufficient, however, to compute the velocity components in the wing plane, as in the teardrop theory of plane flow. This simplifies the problem, considerably.

The source distribution of Eq. (3-171) constitutes a three-dimensional source. Thus, the velocity potential of the source distribution q(x’,y’) at a point x, y, z is obtained as

(3-174)

where the integration is performed over the wing area A covered by sources. The corresponding velocity components are found from Eq. (3-45). At a point x, у of

ji,

the wing plane z — 0, they become

u{x, y, 0) v{x, y, 0) ■W (z, y, 0)

The upper sign is valid for z >0, the lower for z <0. Hence, the induced velocities normal to the xy plane are discontinuous across the source layer (wing plane).

Introduction of Eq. (3-175c) into the kinematic flow condition Eq. (3-1736) yields

q(x, y)^2V^~- (3-176)

dX

Consequently, the source strength is proportional to the slope of the contour in the zx plane [see also Eq. (2-906)]. The formulas obtained by properly introducing Eq. (3-176) into Eqs. (3-175a) and (3-175Z?) describe the velocities added at the location of the wing by flow displacement (profile teardrop) of the wing. Presentation of these formulas is omitted. For the wing of infinite span (plane problem), Eq. (2-94) yields

^Because of the singular points of the integrands in Eqs. (3-175a)-(3-175c), integration of Eq. (3-175д) must be conducted first over x! and then over y’. For Eq. (3-175Z?) the reverse order of integration is necessary. If the integration is to be performed in a different order, however, the Cauchy principal value must be taken for the second integration in either equation.

Also, in this case ypl = 0.

It should be stated here that the velocity differentials (perturbation velocities) of wings of finite span in Eqs. (3-175a), (3-1756), and (3-176) are proportional to the profile thickness ratio 5 = tjc, in analogy to the two-dimensional profile theory [Eq. (3-177)]. The above linear theory is sufficiently accurate for all practical purposes up to about 5 = .

The resultant velocity on the contour is

(3-178)

when quadratic terms in и and v are neglected.

Yawed flight During steady yawed flight, the incident flow condition is determined by the sideslip angle /? (Fig. 3-61 b) in addition to the angle of attack a. Because of the asymmetric incident flow, in addition to lift, drag, and pitching moment, additive forces and moments are created, namely, the side force due to sideslip Y, the rolling moment due to sideslip Mx, and the yawing moment due to sideslip Mz (see Fig. 1-6). They vary linearly with /3 for small angles of sideslip. The derivatives of the dimensionless coefficients with respect to the sideslip angle are, therefore, independent of the sideslip angle. They are termed stability coefficients of lateral motion. All three coefficients for a wing are strongly dependent on the sweepback angle and the dihedral angle.

First, the wing without dihedral will be treated, followed by a discussion of the effect of the dihedral angle.

A fundamental treatment of the yawed wing was first given by Weissinger [94]. The resulting theory can be designated as simple lifting-line theory in the sense of Sec. 3-3-3. In this theory it is assumed that free vortices are shed only from the trailing edge and that these are parallel to the incident-flow direction. The inclination of the free vortex strips against the wing axis of symmetry is of secondary effect on the results of the Weissinger theory. In this theory, Weissinger [94] introduced a correction factor taking into account the effect of the wing end flaps on the rolling moment due to sideslip. Later Gronau [25] made compre­hensive computations of the rolling moment due to sideslip and the yawing moment due to sideslip, mainly for swept-back and delta wings, using the method of the extended lifting-line theory (Sec. 3-34). Here, too, the effect of the free vortex strip inclination has only approximately been taken into account.

Test results from various sources for the rolling moment due to sideslip of rectangular, swept-back, and delta wings against the aspect ratio are shown in Fig. 3-64. For comparison, theoretical curves from [25] are added. Agreement between measurements and theory is good. With decreasing aspect ratio, the rolling moment due to sideslip decreases strongly. This presentation reveals further that sweepback causes a strong increase in the rolling moment due to sideslip. This means that the rolling moments due to sideslip of swept-back and delta wings, in particular, are strongly dependent on the lift coefficient (see also Kohlman [45]). Figure 3-65 gives the corresponding plots for the yawing moment due to sideslip.

A wing in asymmetric flow (yawed wing) corresponds aero dynamically to a wing of asymmetric planform (see Fig. 3-17). Based on this concept, its circulation distribution can be computed from the extended lifting-line theory (Sec. 3-3-4), or the lifting-surfaces theory (Sec. 3-3-5). However, the required computation effort is considerably greater than for symmetric incident flow because of the asymmetry of the wing planform.* The result of such a computation is the circulation distribution over the span, measured normal to the incident flow direction. With it, the total lift and the neutral-point positions are obtained from the formulas of Sec. 3-3-2. In this way the rolling moment about the experimental x axis is also determined.

The rolling moment due to sideslip, accordingly, is proportional to the total lift. The circulation distribution for the three wings without twist examined earlier has been computed by this method at three different angles of sideslip. In Fig. 3-66, the circulation distributions for (3 — 0° and /3=10° have been presented over the span coordinate, measured normal to the direction of the incident flow. For all three wings, the circulation distribution changes very little with the angle of sideslip.

It should be mentioned that this behavior is typical for wings without dihedral. The locations of the neutral points for three angles of sideslip are inscribed into the wing planforms. The coefficients of the rolling moment due to sideslip and the coordinates of the neutral points are compiled in Table 3-7. The yawing moment due to sideslip is caused by the difference in drag of the two wing-halves. It consists of a contribution from the profile drag and one from the induced drag. The latter

Figure 3-66 Circulation distribution of three yawed wings without twist in sideslipping, based on the lifting-surface theory of Truckenbrodt. Angle of attack a = 1, measured in the section parallel to the incident flow direction, j = rjVb. Geometric data of the wings from Table 3-5. (a) Trapezoidal wing: f = 0a; л = 2.75; Л = 0.5. (b) Swept-back wing: <*г = 50°; л = 2.75; = 0.5. (c) Delta wing: ^ = 52.4°; л = 2.31; = 0.

contribution is proportional to the square of the lift, precisely like the induced drag. The side force due to sideslip of a wing without dihedral can be determined approximately by considering that the profile drag of a yawed wing acts parallel to the direction of the incident flow, but the induced drag acts in the direction of the wing axis of symmetry. Consequently, in asymmetric incident flow, only the component of the profile drag су — cDp sin /3 acts in the direction of the wing-fixed lateral axis. Hence the side force slope is

(3-153)

Wing with dihedral The dihedral of a wing is understood to be the inclination of the left and the right wing-halves relative to the xy plane (Fig. 3-6lb). The dihedral angle is designated as v; in the general case v may vary along the span. The stability coefficients of yawed flight 3cy/0j3, dcMx/dj3, and dcMzld(3 of the wing are strong functions of the dihedral. For the total airplane, the contributions of the wing to the side force due to sideslip 3cy/0j3 and to the yawing moment due to sideslip are relatively small. Conversely, the contribution of the wing to the rolling moment due to sideslip of the total airplane is of decisive significance. Selection of the wing dihedral is governed exclusively by the requirement of a flight mechanically favorable value of the rolling moment due to sideslip.[21]

The aerodynamic effect of the dihedral in yawed flight is due to the angle of attack, increased by the amount da of the leading half-wing, and the angle of attack decreased by A a of the trailing half-wing. This angle Л a can be determined
as fohows: From Fig. 3-67a and Ъ, the lateral component of the incident flow Vy = V sin 13 produces on either half-wing a component normal to the wing of amount

V„ = j-Fysinv

Together with the component Vx = V cos (3 of the incident flow, the additive angle-of-attack change becomes

(3-154д)

(3-1546)

The second relationship is valid for small angles of sideslip and small dihedral angles. The exact establishment of the dihedral angle from a given wing geometry must be based on Eq. (3-11).

The lift distribution of a wing with dihedral during yawed motion may thus be determined by adding the geometric angle-of-attack distribution of the wing without dihedral to the antimetric* twist from Eq. (3-154) (see also Fig. 3-67). As in Fig. 3-67, the lift (Z/2 L-AL/2) acts on the leading wing-half, the lift (Z/2 —JZ/2) on the trailing wing-half. Z is the lift for symmetric incident flow and A LI2 is the additive lift of one wing-half in yawed motion.

For the determination of the aerodynamic forces of the two wing-halves, it has to be realized that, as Fig. 3-67 demonstrates, the resultant incident flow direction is deflected up by the angle A a on the leading wing-half but deflected down by the same angle da on the trading wing-half. These angle-of-attack changes are relative to the angles of symmetric incident flow. The resultant aerodynamic forces on the two wing-halves undergo the same direction changes. The exact determination of the side force due to sideslip, of the rolling moment due to sideslip, and of the yawing moment due to sideslip requires computation of the lift distribution on the given wing for the antimetric angle-of-attack distribution in Eq. (3-154).

Approximate expressions for the aerodynamic quantities of the yawed wing with dihedral giving an explicit account of their dependence on the dihedral angle and the total lift coefficient can be gained, however, through the following estimations: The side force due to sideslip resulting from the dihedral is, from Fig. (3-67Й),

being the additive lift of one wing-half, where, from Eq. (3-154b), Aoc=v(3. Consequently, the coefficient of the side force due to sideslip becomes

(3-155)

The coefficient (dcLjdot)v of this equation can be determined exactly only by computation of the lift distribution on a wing with antimetric twist as in Fig. 3-67c.

Translator’s note: The word antimetric, found repeatedly in the text, has been coined by the

authors to avoid an inconvenient expression like “acting or pointing in opposite directions but being of equal magnitude.”

Figure 3-67 Aerodynamics of a wing with dihedral in sideslipping, (<z) Wing planform, xу plane. (b) Dihedral yz plane, (c) Additive antimetric angle-of – attack distribution due to the dihedral a a = ± v(3, (d) Incident flow resultant and aerodynamic-forces re­sultant of the two wing-halves.

As an approximation, however, it may be assumed that this coefficient is equal to that of a wing without twist of aspect ratio Л/2. Equation (3-155) reveals, then, that the coefficient of the side force is proportional to the square of the dihedral angle and independent of the total lift coefficient. Introduction into Eq. (3-155) of the lift-slope value for an aspect ratio Л/2 from the extended lifting-line theory of Eq. (3-98) yields, for the unswept wing,
where к — яЛ/с’і oo.

Measurements that confirm the above formula are reported in the summary account [72].

The rolling moment due to the dihedral is (see Fig. 3-61b)

w л A L

Mx = 2 —yL

where у і designates the distance of the center of the additive lift of the half-wing, AL/2, from the wing root. For the rolling-moment coefficient cMx =MxfqAs results, corresponding to the above discussion,

(3-157)

Here, (vl)v —Уь!$ is the dimensionless distance of the center of the additive lift of the half-wing from the wing root. This equation demonstrates that the coefficient of the rolling moment due to yaw as a result of the dihedral is proportional to the dihedral angle and independent of the total lift coefficient. To a good approxima­tion, (t}l)v can be set equal to fir = 0.424. With this value, the following approximate relationship for the unswept wing is obtained. Here (dcifda)v from Eq. (3-98) for one-half of the aspect ratio A /2 is again introduced.*

The additive yawing moment due to sideslip resulting from the dihedral is very small in general. Its sign is such that it tends to turn the leading half-wing further upstream. This comes about because, as shown in Fig. 3-67(7, the resultant aerodynamic force at the leading half-wing is being turned toward the front and at the trailing half-wing toward the rear. Measurements are given in [72].

Rolling motion A linearly variable vertical velocity Vz = coxy is obtained when the wing executes a rotary motion about the longitudinal axis as in Fig, 3-68<z (see also Fig. 3-61c). Superposition with the incident flow velocity V results, from Fig. 3-68b and c, in an additive antimetric angle-of-attack distribution

Atx{ri) — r]Qx

where Qx = coxs/V is the dimensionless angular rolling velocity. This angle-of-attack distribution produces an antimetric lift distribution along the span and consequently a moment about the x axis that always tends to inhibit the rotary motion. This moment is designated rolling moment due to roll rate or roll damping. The

^Through evaluation of Eq. (3-100) with Eq. (3-154), Eq. (3-158) may be established as solution for the elliptic wing.

asymmetric force distribution along the span furthermore produces a yawing moment, the so-called yawing moment due to roll rate. These two moments are proportional to the dimensionless rolling angular velocity Qx, making their coefficients Ьсмх№&х and dcMz/bQx independent of Qx. In determining the aerodynamic force of the two wing-halves from Fig. 3-68c, it should be noted that relative to the symmetric incident flow direction, the resultant incident flow direction of the downward-turned wing-half is deflected upward by the angle A a and that of the upward-turned wing-half deflected by the same angle A a downward. Consequently, the local aerodynamic forces on the two wing-halves undergo the same directional changes.

For the determination of the roll damping of a given wing, the antimetric circulation distribution та(т?) over the span has to be established following a procedure for the computation of the lift distribution of Sec. 3-3. Hence the roll damping is, from Table 3-1,

(3-160 b)

Accordingly, the roll-damping coefficient bcMxjdQx is independent of the total lift coefficient of the wing. The roll-damping coefficients of the three wings (trape­zoidal, swept-back, delta) examined earlier are found in Table 3-5.

A simple approximate formula for the roll damping of unswept wings is obtained by setting a = p in Eq. (3-100):

Use of this approximate formula is not recommended for wings of strong sweepback. A more accurate computation should be made. Schlottmann [75] demonstrates the theoretical determination of the roll damping of slender wings by a nonlinear theory and experimental confirmation of the computed results.

The yawing moment due to roll rate tends to turn the downward-turning wing-half forward. This behavior can be understood as follows: On the downward – moving wing-half, the resultant incident flow direction is turned upward and consequently the resultant aerodynamic force turns forward. On the upward-moving wing-half, the resultant aerodynamic force consequently turns rearward. On a section у of an unswept wing, the force dD’ = dDt — dLAa = dL(oci — da) is thus obtained in direction of the undisturbed incident flow.[22] Here dDt — dL&i from Eq. (3-17). Integration produces the induced yawing moment due to roll rate:

With dL= gVTdy from Eq. (3-14) and у = FjbV and A a from Eq. (3-159), the coefficient of the yawing moment is determined as

The total circulation у is composed of the contribution of the wing in symmetric incident flow ys and of the contribution ya created by the rotary motion for aa=7] that is, y= ys + Qxya. Correspondingly, the induced angle of attack becomes щ – ais +Qxaia. Introduction of these relationships into the above equation yields

(3-162)

For wings without twist of elliptic circulation distribution, a simpler evaluation of the integral is possible. The circulation distribution is obtained from Eq. (3-65), specifically, ys with p = 1 and ya with д= 2, З,…, M. Correspondingly, the induced angles of attack are found from Eq. (3-73), ot. is with n= 1 and aia with n = 2, 3,, M. By taking into account Eq. (3-65b) with r] = cos # and dr} = — sin в d&, the integration over G <n yields the relationship

(3-163)

where ax — cL/-nA and a2 = —(2/тгА)(дсМхІд@х) from Eqs. (3-66a) and (3-666).

By introducing Eq. (3-161), the following approximate formula for the coefficient of the yawing moment due to Toll rate is finally found:

= _ J_ y^3 +4-1

8QX 4 |/£2 – f – 4 – f 2

Thus the coefficient of the yawing moment due to roll rate is proportional to the total lift coefficient.

Yawing motion Motion of the airplane about the vertical axis produces additive longitudinal velocities, of reversed signs on the two wing-halves (Fig. 3-69; see also Fig. 3-6 le). An asymmetric lift distribution over the span results, creating a yawing moment and a rolling moment. This yawing moment counteracts the rotary motion and is termed, therefore, yaw-damping or turn-damping of the wing. It is very small compared with that of the whole airplane, and therefore its computation is omitted.

The rolling moment created by the yawing motion is termed rolling moment due to yaw rate or turning rolling moment. The turning rolling moment tends to turn the forward-moving wing-half upward.

The turning rolling moment can be computed in the following way: Through the rotary motion with angular velocity coz from Fig. 3-69b, a linear distribution of the longitudinal velocity is generated along the span:

7х(у) = У-щу (3-165)

To ensure that the wing is a streamlayer of this inhomogeneous flow field, the kinematic flow condition

(3-166a)

ay + y = 0 (3-1666)

must be satisfied at each point of the wing surface. Here Vn is the component of the longitudinal velocity Vx normal to the wing chord; thus, Vn — aVx (Fig. 3-69c).

In a homogeneous flow field, Vx = V, the kinematic flow condition becomes w/T+a=0. Comparison with Eq. (3-166b) demonstrates that the inhomogeneous flow is equivalent to a homogeneous flow with the mathematical angle of attack

«t = «-^ = *(l-S,4) (3-167)

where Qz = cozsjV. Consequently, the circulation distribution for inhomogeneous flow can be computed by using the computation procedures of Sec. 3-3, but by applying an angle-of-attack distribution as in Eq. (3-167). The resulting circulation distribution is Гъ = b V7b. A wing strip of width dy thus produces a lift dL – q VxTb dy = qV( 1 —Огг])Гъ dy, and the rolling moment becomes

S S

Mx = — J ydL = — 0 f Vxrbydy

~s – s

And further, the coefficient of the rolling moment cMx =Mx/qAs is found as

і

Смх= —A/(1 — Qzri) ybri drt

-i

The circulation distribution yb at the angle of attack ab, from Eq. (3-167), may be composed as follows:

П = ayu — cc Qzya (3-168)

Here yu is the circulation distribution for a = 1, and ya that for aa — r] [see Eq. (3-160)]. For the sake of simplicity, let a wing without twist a = const, be considered. Introduction of Eq. (3-168) into the equation for the rolling-moment coefficient yields

= i^A f yur]2 drj Л f yarj dr^j ci (3-169)

This equation demonstrates that the coefficient of the rolling moment due to yaw rate is proportional to the angular velocity Qz and the total lift coefficient cL. For a wing without twist of elliptic circulation distribution, the following approximation formula is obtained with Eq. (3-98) and after evaluation of the integrals similar to those of Eqs. (3-162) and (3-163):

dcMx_ j_ L, VFTt + i

8QZ 4 "И у в +4-f2 / L

This expression is nearly independent of the aspect ratio. For the three wings that have been examined (trapezoidal, swept-back, and delta, Table 3-5), the coefficients of the rolling moment due to sideslip are listed in Table 3-8.

For an accurate computation of the rolling moment due to yaw rate, it must be realized that the rotary motion of the wing causes the free vortices to be shed into a lateral flow. Hence the portions of the free vortices that lie on the wing produce an addition to the lift and thus to the rolling moment due to yaw rate. A detailed computation reveals that the coefficient of the rolling moment due to yaw rate for wings of small aspect ratio (/1< 3) depends considerably on the position of the axis of rotation.

Straight flight For longitudinal motion, the resultant aerodynamic force may be represented by lift, drag, and pitching moment. Their dependence on the angle of attack (see Fig. 3-61a) has been discussed in the previous section. The two most important coefficients are lift slope dcLjda and pitching-moment slope dcMjdcL. The latter determines the position of the aerodynamic neutral point of the wing by Eq. (1-29). The lift slope dcLjda is presented for various wing, shapes in Figs. 3-25,3-38, 3-42, 3-44, 346, and Table 3-5. The neutral-point positions of various wing forms can be obtained from Figs. 3-37, 3-38, 3-43, 344, and Table 3-5. The flight mechanical computations for the neutral-point position require great accuracy. The neutral-point position depends strongly on the individual planform. In general, therefore, it is required that for its determination the lift distribution should be computed by using the lifting-surface theory (see Sec. 3-3-5).

Pitching motion Pitching motion is actually a nonsteady motion. In general it proceeds slowly enough, however, that it can be treated as “quasi-steady.” When the wing

Figure 3-61 The motion modes of the wing, (a) Straight flight, (b) Yawed flight, (c)-(e) Rotary motions: rolling, pitching, yawing.

(Fig. 3-62) performs a rotary motion with angular velocity coy about the lateral axis through xs (Fig. 3-6Id), a vertical additive velocity Vz = озу(х — х$) is produced that varies linearly over the wing chord. Together with the incident flow velocity V, the rotary motion in chord direction produces an additive angle-of-attack distribu­tion a(x) = VzjV of magnitude

«(®) = ^ (s — XS) (3-147)

This angle-of-attack distribution produces an additive lift distribution, the integra­tion of which leads to an additive lift and an additive pitching moment. These quantities are designated lift due to pitch rate and pitch damping. Both depend linearly on cOy. It is expedient, therefore, to introduce the coefficients dcLjdQy as lift due to pitch rate and bcMjbQy as pitch damping, where Qy = tuyC^jV is the dimensionless pitching angular velocity and is the wing reference chord, introduced earlier by Eq. (3-5b). These coefficients depend only on the wing geometry and the position of the axis of rotation.

Now it wfll be explained how these two quantities can be determined and, in

Figure 3-62 Explanatory sketch for aerodynamic coefficients of the pitching wing.

particular, how their values change with the position of the axis of rotation xs[20] It is evident that there is an axis of rotation xQ for which the lift due to pitch rate is zero. For a rectangular wing, this axis of rotation lies at a distance § c from the leading edge, according to the Pistolesi’s theorem (see Sec. 2-4-5). The pitch damping, however, cannot be zero for any position of the axis of rotation. For the computation of the lift due to pitch rate, the angle-of-attack distribution of Eq. (3-147) is rewritten in the form

oc{x) = ^ (x – xQ) – F ^ (a?0 – x8) (3-148)

By setting xs = x0 in this equation, the second term becomes zero, whereas, by definition, the first term produces zero lift due to pitch rate, (dcL/d£>y) = 0. The contribution of the first term to the pitch damping will be expressed by (dcM/dQy). The second term in Eq. (3-148) represents a constant angle of attack and gives the total lift due to pitch rate as

bcL _ dcL х0 – xs

9 Qy J s d? os cy.

Here dcLjda. is the lift slope of the wing. Equation (3-149) shows that the lift due to pitch rate is a linear function of the position of the axis of rotation (see Fig. 3-63a). The moment of the pitching motion is obtained from Eq. (1-28) as

which leads with Eq. (3-149) to the pitch damping:

&cm /деm жіу — xs xo — :cs

dQjs 8®Jo Сц Cm doc

This equation shows that the pitch damping depends parabolically on xs. In particular, it is immediately obvious that for Xg = xN and for x$ = x0 the pitch damping has the same value, namely, (bcM/dQy)0, as in Fig. 3-633.

To be able to compute the pitch damping from Eq. (3-150) for an arbitrary position of the axis of rotation xs, the determination of (bcMjbQy)0 and of x0/cM is required, whereas the coefficients dcLjda and are known from earlier

discussions.

For xs – xN, Eqs. (3-149) and (3-150) yield

(3-1513)

Thus the problem of determining the lift due to pitch rate and the pitch damping for an arbitrary position of the axis of rotation has been reduced to the computation of the two coefficients (3cijdDy)N and {bcMjbQy)N for the position of the axis of rotation in the neutral point. These latter two coefficients are obtained from the lift and pitching moments as determined from lifting-surface wing theory for the angle-of-attack distribution corresponding to Eq. (3-147):

(3-152)

In Table 3-6, numerical data on the positions of the axis of rotation for zero lift due to pitch rate and. of the corresponding pitch damping are compiled for a trapezoidal wing, a swept-back wing, and a delta wing (Table 3-5). Compare also Garner [62] and Gothert and Otto [24].

In the case of airplanes with a separate horizontal tail, the contribution of the wing to the lift due to pitch rate is small compared with that of the tail surface. An

4

Figure 3-63 Lift due to pitch rate (a) and pitch damping (b) vs. position of axis of rotation Ху. хдг = neutral-point

accurate computation is therefore not required. On the other hand, in the case of all-wing airplanes, whose total pitch damping is almost completely produced by the wing, a more accurate computation may be required, depending on the specific case.

3-5-1 Contributions of the Wing to Stability

The methods for the computation of the aerodynamic forces on a wing have been discussed in detail in Secs. 3-2-3-4. This section will show how these methods can be applied to the determination of the flight mechanical coefficients of the wing. A survey on these coefficients has been given previously in Sec. 1-3-3.

The flight mechanical coefficients are determined by the motion of the wing

Figure 3-60 Drag polars of circular wings of various degrees of leading-edge rounding, accord­ing to measurements of Hansen [4-4]. Theory according to Kinner [44]. The drag at = 0 has been subtracted from the measured values. Disks I and II: cDp= 0.012; disk III: cDp = 0.008. Curve 1, with suction force from Eq. (3-125); CD = cDp + 0.274cjr. Curve 2, without suction force from Eq. (3-1446); Cjq = cjjp + Q,55c.

and the wing geometry. In the following discussions, only those coefficients will be considered that are significant for airplane stability. The coefficients that determine maneuverability will be treated later in Chap. 8.

In addition to the wing, the other parts of the airplane (fuselage, empennage) contribute, sometimes considerably, to these flight mechanical coefficients. These contributions will be discussed later, too. In the present section, only the contributions of the wing will be discussed.

The flight mechanical coefficients of the wing depend on numerous geometric parameters of the wing, such as wing planform (aspect ratio, taper, sweepback), twist, and dihedral (see Sec. 3-1-1). The dependence of the flight mechanical coefficients on wing geometry is too varied to attempt a complete description of all these interrelations. In some cases the contribution of the wing to the stability coefficients of the whole airplane is small. Further investigations will be restricted to the cases in which the wing makes an essential contribution. Reference will be made to the summary reports of Betz [6], Schlichting [72], and Multhopp [61].

Of the two axis systems of Fig. 1-6, the experimental system will be used.[19] The coefficients are defined in Eq. (1-21).

The motion of the airplane can be divided into a longitudinal motion and a lateral motion, as has been explained in Sec. 1-3-3. During longitudinal motion, the position of the plane of symmetry of the airplane does not change. This motion is characterized by three parameters: flight velocity V, angle of attack a, and pitching angular velocity cov (Fig. 3-61). The lateral motion is defined by sideslip angle (3, rolling angular velocity со*, and yawing angular velocity coz (Fig. 3-61). The stability coefficients are understood to be the changes of force and momentum coefficients with the above motion parameters.

Tangential force Earlier, in Sec. 1-3-2, the wing-fixed components of the aero­dynamic force, normal force, and tangential force were introduced together with the flow-field-fixed components of lift and drag. For small angles of attack, the normal force is almost equal to the lift, whereas the tangential force deviates considerably from the drag, even for small angles of attack. The tangential force T is taken as positive in the direction from the wing leading edge to the trailing edge. When limiting the angle of attack to small or moderately large values (see Fig. 1-7д), the tangential force coefficient cT = TfAq becomes, with T — —X,

cT = cD – cLa (3-141)

where єis the coefficient of total drag as composed from profile drag and induced drag from Eq. (3-125). By introducing cD from Eq. (3-125b) and a = (daldcL)cL into Eq. (3-141), the tangential force coefficient for elliptic circulation distribution becomes

(3’142)

With dcLjda from Eq. (3-8Ob) for the simple lifting-line theory and from Eq. (3-98) for the extended lifting-line theory, Eq. (3-142) yields

cl

CT = cDp–r – (3-143fl)

-f-~1 2 і

ct = cdp—————————— ——Сд (3-143b)

where к = uA/c’Loa. In Fig. 3-58, the difference (cDp — cT)lc2L from Eq, (3-143) is shown against the aspect ratio. Accordingly, for large aspect ratios this difference, and thus the tangential force, are independent of the aspect ratio.

Figure’3-59 illustrates the dependence of the tangential force coefficient on the lift coefficient for wings of various aspect ratios A. The profile drag coefficient had been taken to be cDp « 0.05. It is remarkable that the coefficient of the tangential force assumes negative values when the lift coefficient cL> 0.5. In this case the

Figure 3-58 Tangential force coefficient cj vs. aspect ratio a, (1) Based on the simple lifting-line theory, Eq. (3-143a). (2) Based on the extended lifting-line theory, Eq, (3-143b). Сдр = coefficient of profile drag; = 2ж.

Figure 3-59 Lift coefficient vs. tangential force coefficient Cj for wings of various aspect ratios a, from Eq. (3-1436). For comparison, the drag polars C£>(c^) are also shown.

tangential component of the resultant of the aerodynamic forces is directed upstream along the wing chord. The drag polar curves cD(cL) are also included in Fig. 3-59.

Suction force The discussions about the drag of wings of infinite span of Sec. 24-2 have shown that the flow around the leading edge of an inclined profile produces a suction force in an inviscid fluid (Fig. 2-12a). This is the result of the strong underpressure in the vicinity of the leading edge. Now, the suction force on wings of finite span will be examined. The suction force is a part of the induced drag [Eq. (3-124)], with the total drag being split into profile drag and induced drag. Equation (3-125) is therefore the expression for the drag coefficient with suction force. It has been pointed out in Sec. 24-2 that no suction force exists for very sharp leading edges. In this case, the flow around the leading edge causes local separation, eliminating the strong underpressure that results in a suction force. Rather, the resultant force of the pressure distribution over sharp-edged noses acts normal to the wing surface and, therefore, has the component La in the incident flow direction. Thus the drag coefficients with and without suction force are, respectively,

CD — c Dp + cDi (3-144<z)

cD ~ cDp (3-1446)

The difference of the drag coefficients of Eqs. (3-144<z) and (3-1446) yields the suction force coefficient cs — S/Aqm :

CS ~ cLa~cDi [18]

Comparison with Eq. (3-142) yields

cS = cDp~cT (3-146)

Consequently, the quantity of Fig. 3-58 is a direct measure of the suction force. In particular, cs = CjJ2tt for wings of very large span. This result is in agreement with Eq. (2-77), remembering that the lift coefficient for smooth leading-edge flow cLS is zero for symmetric profiles.

In conclusion, a few experimental results on wings of small aspect ratio according to Hansen [44] will be presented, confirming the above considerations. Figure 3-60 shows polar curves for a slender circular disk. To show the effect of the suction force, the leading edge of the disk was formed in several ways, as can be seen from Fig. 3-60. The suction force increases with the leading-edge nose radius. The theoretical curves for the drag coefficient with and without suction force from Eqs. (3-144л). and (3-144Z?) are added in this figure. The tests show the expected result, namely, that the measured drag coincides with the theoretical curve, including suction forces, when the leading edge is well rounded. When the leading edge is very sharp, however, the measured drag lies close to the theoretical curve without suction force. All measurements with differently formed leading edges lie between the two theoretical curves.