Category AERODYNAMICS OF THE AIRPLANE

In this section, a brief discussion will be given of the aerodynamic forces generated by the control-surface deflection of the tail unit and their effect on the force and

3 Figure 8-39 Position of the flap neutral point for flap designs of Fig. 8-38.

moment equilibrium of the whole airplane. For the case of zero control-surface deflection, the contributions of the horizontal tail and the vertical tail, respectively, to the aerodynamic forces of the whole airplane have been given in Secs. 7-2-1 and 7-3-1.

Elevator For the contribution of the horizontal tail with deflected elevator to the pitching moment of the whole airplane, Eqs. (7-3a) and (7-3b) yield

Here, from Fig. 7-5, rH is the distance of the lift force of the horizontal tail from the moment reference axis of the airplane.

The change in the moment caused by the elevator deflection at constant angle of attack is thus obtained as

(8-29)

Here, the quantity r’H of the previous equation has been replaced by the lever arm г"#, which is the distance of the flap neutral point from the moment reference axis of the horizontal tail.

For the two-dimensional flap wing in incompressible flow, the position of the flap neutral point is given in Fig. 8-15. The change in the pitching moment caused by the elevator deflection at constant lift coefficient (zero-moment coefficient) is obtained in analogy to Eq. (7-15) by substituting —(данІдг)н)^н f°r eH as

(8-30)

Here г#ДГ is the distance of the neutral point of the elevator from the neutral point of the whole airplane (see Fig. 7-6b).

Rudder The contribution of the vertical tail with a deflected rudder to the yawing moment of the whole airplane becomes, from Eqs. (7-49a) and (7-49b),

d°iv in dav Qv Av r’y

CMzV~~dav Pv~d^VvJ <7. XT

Here, r’y from Fig. 7-36 is the distance of the side force of the vertical tail from the moment reference axis of the airplane.

The change in the yawing moment caused by the rudder deflection is thus given as

дсщгУ _ d°iv dav Чу A v r’y (8-31)

Ьтіу dcxy Э7}y q^, A s

Here the quantity r’y of the previous equation has been replaced by the lever arm r’y, which is the distance of the flap neutral point from the moment reference axis of the vertical tail.

Rudder moments Information on the rudder moments of the airfoil of infinite span for incompressible flow is found in Sec. 8-2. The control-surface moments of the elevator and rudder and also of the ailerons cannot, in general, be computed with sufficient accuracy, because for the control-surface moments the transformation, from the airfoil of infinite span (plane problem) to the wing of finite span is not possible in a reliable way. The control-surface moments for control surfaces with balance provisions of Fig. 8-2 (inner balance, outer balance, balance tabs) are particularly difficult to determine because they are greatly affected by the boundary layer as well as by’ inviscid flow problems. The control-surface moments must therefore be determined largely through wind tunnel and flight tests (see Stiess [18]). Some wind tunnel measurements on the control-surface moments of tail surfaces with inner and outer balances were reported by Schlichting and Ulrich [39].

[1]The temperature gradient dT/dH determines the stability of the stratification in the stationary atmosphere. The stratification is more stable when the temperature decrease with increasing height becomes smaller. For dT/dH = 0 when n = 1, Eq. (1-13), the atmosphere is isothermal and has a very stable stratification. For n = у — 1.405, the stratification is adiabatic (isentropic) with dT/dH — —0.98 К per 100 m. This stratification is indifferent, because an air volume moving upward for a certain distance cools off through expansion at just the same rate as the temperature drops with height. The air volume maintains the temperature of the ambient air and is, therefore, in an indifferent equilibrium at every altitude. Negative temperature gradients of a larger magnitude than 0.98 K/100 m result in unstable stratification.

[2] Translator’s note: According to the definition given by NASA, the angle of sideslip is the angle between the direction of the incident flow and the symmetry plane of the airplane. The angle of yaw is referred to a chosen direction, which may sometimes be the direction of the airflow past the body, making the angle of yaw equal to the angle of sideslip. Under some conditions, however, as in turning, a different reference direction may be used.

[3]The angle 0 has been designated here as the angle of yaw. For the difference between angle of yaw and angle of sideslip see the footnote on page 13.

[4]These quantities may be called in the text simply “thickness” and “camber” when a misunderstanding is impossible.

[5]The influence of friction on lift will be considered in Sec. 2-6.

[6]The Joukowsky mapping function, Eq. (2-21), can be given in more general form in various ways, leading to additional profile shapes that are obtained from mapping circles. For example, when in Fig. 2-14a the mapping circle does not pass through the point +a on the real axis but rather through a point located somewhat farther outside, the sharp trailing edge of the normal Joukowsky profile is replaced by a rounded edge.

[7]It is necessary to take the Cauchy principal value

X-S

[8]Note that S/c = X — j | cos and 77 = zO.

[9]Note that, according to Jaeckel [30], the folio-wing relation applies:

[10]Translator’s note: Remembex that the term “nose iadius” does not necessarily imply a circular nose. The definition of nose radius is of the kind found in Figs. 243 and 244. The curvature can, therefore, be relatively large locally on the nose, even if the radius in the above sense is not small.

[11]In addition to the geometric twist, there is an aerodynamic twist, characterized by a twist angle measured against the profile zero-lift direction instead of the profile chord.

[12]To distinguish between the coefficients of the total forces and moments, the indices of which are always expressed in capital letters, lowercase letters will be used for the indices of the coefficients of local forces and moments.

[13]If the profile coefficient c^ is known over the span, it may be replaced by Cca — C]_,ao 27Г.

form and the constant induced downwash velocity over the wing span.

[14]Note that, according to [23],

[15]Here, the prime (‘) on the summation sign indicates that the term n — v is to be omitted in the summation.

[16]2a/ is the induced downwash angle fax behind the wing, f -» °°.

[17]Kraemer [79] conducted a more detailed study into the application of the momentum law to the computation of the induced drag; see Sears [79].

[18]The suction force is considered positive when acting upstream.

[19]In what follows, the index e of forces and moments and their coefficients, which indicates the axis system used, will be omitted.

[20]Foi flight mechanical computations, the axis of rotation coincides with the lateral axis through the airplane’s center of gravity.

position; x0 = axis of rotation for vanishing lift due to pitch.

[21]The value of the Tolling moment due to sideslip of the total airplane depends on the vertical position of the wing relative to the fuselage in addition to the dihedral.

[22]The profile drag is not taken into account.

[23]The trigonometric functions will be given as sinh’3 rather than arcsinh.

[24]For Ma = 0, Eq. (4-5) reduces to the well-known relationship of incompressible flow. It is not valid for hypersonic flow; see Sec. 4-3-5.

[25]rhe integral of the second equation is obtained through integration by parts.

[26]The opposite trend is found in transonic flow, in which the changes of the flow quantities are small in the lateral and strong in the longitudinal direction.

TThis formula and its comparison with measurements will be discussed in more detail in Sec. 5-3-3.

[27]In this section, the drag of the fuselage as obtained in inviscid flow (wave drag) is designated as Dp. Because of viscosity effects (friction), a contribution Dp must be added to this drag [see Eq. (5-17)].

[28]It can easily be seen that the flow pattern of the two counter-rotating vortices at у and yp and at (y + dy) and (yp + dyp), respectively, contains, as a streamline, the circle of radius R about the origin.

[29]For a blunt fuselage nose and tail, Eq. (6-6) gives finite values for dLp/dx, contrary to the exact values dLpjdx = 0.

[30]For the fuselage, a different transformation formula of the pressure coefficient was given by Eq. (5-53), where the angle of attack was transformed according to Eq. (5-52d). However, within the framework of the linear lift theory, the Eq. (6-32) for the fuselage is equivalent to Eqs. (5-52d) and (5-53).

[31] = – –00——- UK (6-31 a)

l/Malo — 1

w=—ot00lJoz (6-31 b)

The solid curves signify the mean values of the induced velocities over the circumference. They are essential for the computation of the lift distribution of the

[32]The stripe method is a procedure whereby the local lift coefficient is set proportional to the local angle of attack based on the lift slope of the plane problem, which from Eq. (4-46) is given for supersonic velocities by {dcijda)«, = 4/VMzL — 1.

[33]For simplicity it has been assumed that the ratio of the dynamic pressures qfjlq « is independent of the angle of attack a.

[34]The integral j must be evaluated after the Hadamard method of finite parts of divergent integrals.

[35]The index 0 has been omitted.

[36]The assistance of K. O. Arnold in preparing this section is gratefully acknowledged.

The flap wing of finite span in compressible flow may be treated according to the theory of the wing of finite span as discussed in Secs. 4-4 and 4-5.

Subsonic incident flow At subsonic velocities, the subsonic similarity rule (Prandtl – Glauert) of Sec. 44-1 applies. It requires the determination of a wing, to be computed for incompressible flow, that is transformed from the given geometry of the wing of finite span at compressible flow. These transformation formulas for the geometries of the wings are found as Eqs. (4-66)-(4-68). The influence of compressibility on the aerodynamic coefficients of the wing is obtained from the transformation formulas Eqs. (4-69)-{4-72). Here, the angle-of-attack distribution due to the flap deflection remains unchanged and is determined with lifting-surface theory from Eq. (8-22). Accordingly, Eqs. (8-15a) and (8-15h) give the changes of the angle of attack and of the momentum coefficient with the flap deflection. However, these equations for the incompressible reference flow now have to be evaluated for the transformed wing planform from Eq. (4-15). In Fig. 8-34, the results of sample computations for wings of finite span with deflected flaps are shown. They are the three wings discussed several times previously, namely, a trapezoidal, a swept-back, and a delta wing; see Table 34.

Supersonic incident flow The computation of the aerodynamic effect of a flap on a wing of finite span at supersonic velocities is in some respect simpler than at subsonic velocities. This becomes obvious from Fig. 8-35, which shows a rectangular

Figure 8-33 Measured rolling-moment coef­ficients of a delta wing as shown in Fig. 8-3la, with flaps extending over the entire half-span for several angles of attack a. Comparison of theory (« = 0.75) and mea­surements from Truckenbrodt and Gronau.

wing and a delta wing with flaps of constant chord extending over the entire trailing edge. When the flap is being deflected, an additive lift is generated only on this flap that is equal to the lift of a rectangular wing of span b and of the flap chord су. The lift of the wing lying before the flap is not changed by the flap deflection.

To compute the lift caused by flap deflection, the results for the rectangular wing of Sec. 4-5-4 may be recalled. From Eq. (4-112), the lift coefficient produced by the flap and referred to the total wing area A is given as

bcL _ Af 4 Эт?/ A iMa%. – 1

which is valid for b sjMaL — 1, but independent of the wing shape.

For the rectangular wing of Fig. 8-35, the change of the zero-lift angle caused by the flap deflection can easily be determined. Because Эй/Эту = — (Эс^/Згу)/ (.дсь/да), Eqs. (8-25) and (4-112) yield

ЛЁ = _ f 2Л (8-26)

dr? f 1 2Л ІМаІг – 1 – 1

where ‘hf = Cflc=AfjA is the control-surface chord ratio. In this equation, the fraction on the right-hand side, which is always greater than unity, gives the

Figure 8-35 Aerodynamics of the flap wing at supersonic incident flow, {a) Rectangular wing with flap extending over the entire trailing edge. (b) Delta wing with flap extending over the entire trailing edge.

correction of the value for the two-dimensional flap wing, as can be verified by comparison with Eq. (8-16a).

The pressure distribution on the flap of a wing in supersonic incident flow may also be established quite easily. Figure 8-36 shows a flap design in which the right-hand-side edge of the flap is an “outer edge,” the left-hand edge an “inner edge,” both of which are parallel to the incident flow direction. When the flap is deflected, Mach lines originate at either upstream edge. In the case of no intersection of these Mach lines on the flap, the pressure distribution in zone 1 is

і

Figure 8-36 Pressure distribu­tion due to flap deflection on a rectangular flap at supersonic incident flow.

that for plane flow. The resultant pressure coefficient on the upper and the lower side is, therefore, with Eq. (4-85) and Table 4-5,

(8-2 7a)

The flow in zones 2 and 3 is cone-symmetric. For zone 2, Eq. (4-111) yields

For zone 3, Tucker and Nelson [47] found the expression

cpз = — arccos (— Ocppi

In these expressions t – y/x tan ц = (y/x) sjMaL — 1, and у is measured from the upstream corners of the flap. In Fig. 8-36, the pressure distributions are shown for a section x = const. On the side of the inner edge, the flap deflection causes, within the range of the Mach cone, a lift on the undeflected wing that is equal to the lift loss at the adjacent portion of the flap.

Furthermore, Fig. 8-37 shows a flap arrangement with a swept-back outer edge of the flap such as, for example, is found in delta wings. In Fig. 8-37a, the outer, edge is a subsonic edge. If the two Mach lines originating at the two upstream flap corners do not intersect on the flap, zone 1 has again, as in Fig. 8-36, the pressure distribution of plane flow. In the case of the subsonic edge (m < 1) of Fig. 8-37a, the pressure distribution of zone 4 is of the kind given in Fig. 4-67 for a delta wing with a subsonic leading edge. In the case of the supersonic edge (Ma« > 1) of Fig. 8-37b, where the Mach cone from the right-hand upstream corner lies entirely on the flap, the pressure distributions of zones 5 and 6 are of the kind given in Fig.

Figure 8-38 Lift due to flap deflec­tion at supersonic incident flow. Curve 1, inner flap. Curve 2, tip flap. Curve 3, full-span flap.

4-69 for a delta wing with a supersonic leading edge. The pressure in zone 6 is constant, Eq. (4-89):

CP6 — —- cppi (8-28)

with m = tan 7/tan д. The pressure distributions in zones 4 and 5 have been determined by Tucker and Nelson [47].

Finally, a few data will be given, in the following two figures on the lift produced by the flap deflection and on the position of its center of application. Figure 8-38 gives the total lift of three rectangular flaps. Flap 1 has two inner edges (inner flap), flap 2 an inner and an outer edge (tip flap), and flap 3 two outer edges (full-span flap). Shown in this figure is the ratio of the total lift produced by the flap to the lift of the two-dimensional flap wing as a function of the quantity bfS/Malc — Ijcf. Flap 1 does not cause any lift loss compared with the two-dimensional flap wing; Eq. (8-25) applies to flap 3. The lift of flap 2 is the arithmetic mean of those of flaps 1 and 3. Figure 8-39 shows the position of the lift force of the flap (flap neutral point). Here, xy is the distance of the flap neutral point from the axis of rotation. For flap 1, the flap neutral point lies at the flap half-chord. It shifts forward for flaps 2 and 3.

The rolling moment due to aileron deflection can be computed very easily by realizing that the lift force at antimetrically deflected flaps acts, in very good approximation, on the half-span of the flap.

Further information on rectangular flaps is found in Schulz [47]. Flaps on rectangular, delta, and swept-back wings have been investigated by Lagerstrom and Graham [47]. Flaps with outer (horn) balances have been studied by Naylor [47].

8-3-1 Flaps on the Wing in Incompressible Flow

Computational methods The aerodynamics of the flap wing of infinite span (plane problem) has been discussed in the previous section. Now the effect of a flap (control surface) on a wing of finite span will be treated. A further geometric parameter, the span of the flap, is added (see Figs. 7-1, 7-3, and 8-4a). Furthermore, in many cases the flap chord ratio varies over the flap span (see Fig. 8-1). To determine the lift distribution, a wing with a deflected flap is equivalent to a wing with an additional angle-of-attack distribution over the span (twist). For a flap covering only a portion of the span, this additional angle-of-attack distribution is discontinuous. The angle-of-attack distribution that is equivalent to a given flap deflection is obtained from the theory of the flap wing of infinite span as

(8-21>

where doijdrif is the local flap effectiveness from Eqs. (8-8a) and (8-9a) and from Figs. S-la and 8-9a. If the flap chord ratio f varies over the span, it is a function of the span coordinate 17 =y/s.

According to the procedure for the computation of the lift distribution on wings of Sec. 3-3, the additive circulation distribution caused by the flap deflection can be determined for such an angle-of-attack distribution. Special attention should be paid to the station of discontinuity in the angle of attack.

The case of a symmetric angle-of-attack distribution corresponds to a landing flap at the wing or an elevator at an all-wing airplane as shown in Fig. 7-3. The antimetric angle-of-attack distribution corresponds to the ailerons (Figs. 7-1 and

7- 3).

Following simple lifting-line theory (Sec. 3-3-3), Multhopp (Chap. 3, [60]) developed a method for handling the discontinuity in the angle-of-attack curve. In Fig. 8-29, a result of this method for a trapezoidal wing of aspect ratio A = 2.75 and taper A =0.5 is shown as curve 1. The station of discontinuity in the angle-of-attack distribution ay lies at щ = 0.5. In Fig. 8-29a it is symmetric, in Fig.

8- 29b it is antimetric. According to Fig. 8-29, the symmetric flap deflection at the wing outside generates a considerable lift, even in the wing middle section. The circulation distributions according to extended lifting-line theory (three-quarter – point method, Sec. 3-3-4) are also shown in Fig. 8-29 as curves 2. As should be expected, extended lifting-line theory gives a smaller lift than simple lifting-line

theory. A computational method for the lift distribution on wings with flaps, based on lifting-surface theory (Sec. 3-3-5), is given in [46]. This method requires the availability of the angle-of-attack distributions caused by the flap deflection on the cj4 line (I/) and on the trailing edge (%r). They are, considering Eq. (8-21),

where the coefficients da/drjf and dcmjdr]f from Eqs. (8-8д) and (8-8b) and from Eqs. (8-9a) and (8-9b), respectively, are known from the profile theory of the flap wing and depend only on the control-surface chord ratio.* An improved method for describing the effect of the angle-of-attack discontinuity has been given by Hummel [46]. Lift distributions of wings with deflected flaps (angle-of-attack distribution with a break) have been computed by Bausch [5] from simple lifting-line theory for a wing of elliptic planform. For a wing with a trapezoidal planform, corresponding computations have been published by Richter [5]. A large number of computations
have been conducted by de Young [10], who applied extended lifting-line theory; however, he did not exclude the station of discontinuity in his computations. Investigations, applying lifting-surface theory, have been conducted by Truckenbrodt and Gronau [46] on delta wings with deflected flaps.

A summary of American tests on wings of finite span with flaps that extend only over a portion of the span is given in [14]. It includes the separation characteristics of such wings; compare the publications [31, 54].

Results of a few sample computations of the lift distribution of wings with flap and control-surface deflections will be given in the following section.

Landing flaps, elevators For the wing of elliptic planform, the change in the mean zero-lift angle caused by the flap deflection is obtained according to Sec. 3-3-3. For a sectionwise-constant, symmetric angle-of-attack distribution, Eq. (3-81) yields, after integration,

Here the flap (control surface), having a constant flap chord ratio, extends from —7?0 to + rjo. The relationship between aу and the flap angle i? y is given by the theory of the two-dimensional flap wings of Eq. (8-21). The coefficient Эа/Эоу is. shown in Fig. 8-30 as a function of the flap span. This result is obtained by both simple and extended lifting-line theories.

A further example, in which Truckenbrodt and Gronau [46] applied lifting – surface theory, is shown in Fig. 8-31. It deals with a delta wing of aspect ratio Л* = 2b*/cr = 2 equipped with a flap that is symmetrically deflected. The flap chord ratio = crfc, however, varies between X/= | at the wing root and Л/ = 1 at the wing tips. The local flap effectiveness was obtained by introducing Eqs. (8-9a) and (8-9b) into Eqs. (8-22a) and (8-22b). The changes of the mean zero-lift angle bajbrf and of the mean zero-moment coefficient bc^fbrf were computed first.

Figure 8-30 Change of the mean zero-lift angle due to flap deflection for an elliptic wing with various forms of the flap, from Bausch.

Figure 8-31 Measured aerody­namic coefficients of a delta wing with symmetrically deflected flap extending over the entire trailing edge. Aspect ratio л* = 2, profile NACA 0012; comparison of the­ory (y. = 0.75) and experiment, from Truckenbrodt and Gronau. (a) Geometry, (b) Lift coefficient vs. angle of attack, (c) Lift coeffi­cient vs. pitching-moment coeffi­cient.

Ailerons In Fig. 8-32, the rolling-moment coefficients are given for a wing of elliptic planform and antimetric control-surface deflection. Figure 8-32a gives the

Figure 8-32 Rolling-moment coeffi­cient vs. flap deflection for an elliptic wing, from Bausch. (a) Flap extending over the entire half-span; curve 1, extended lifting-line theory; curve 2, simple lifting-line theory, (b) Effect of the flap span.

rolling moment of the ailerons plotted against the aspect ratio with each aileron extending over the entire half-span. The extended lifting-surface theory of Eq. (3-100) yields

^cMx _____ 1 —- (8-24z)

d(Hf У&2 – j~ 4 – f – 2

where k – TtAlc’Loo *=» A/2. For comparison, this coefficient according to simple lifting-line theory is added. The rolling moment of the ailerons for the case of an aileron extending over only a part of the wing half-span is shown in Fig. 8-32b. In this case, Eq. (3-100) yields

Цш = (&мЛ N/rr^3 (8-244)

where (dcMxjdaf)Vo=0 is given by Eq. (8-2Ad) and Fig. 8-32<z. For the delta wing of Fig. 8-3 a, the theoretical coefficients of the aileron rolling moment of antimetricaily deflected ailerons extending over the entire half-span are compared in Fig. 8-33 with measurements. Agreement between theory and experiment is good for small and moderate angles of attack.

Aileron investigations and comprehensive experimental results are summarized in [12, 45].

General remarks As has been mentioned in Sec. 8-1, the take-off and landing devices on the wing serve to increase the maximum lift coefficient. A great variety of arrangements are utilized to increase the maximum lift. The older kinds of take-off and landing devices consist of flaps and balance tabs attached to the wing trailing edge or the wing nose (Fig. 8-3). More recently, devices have frequently been used that increase the lift through boundary-layer control by suction or ejection. A brief account of this method has been given in Sec. 2-5-3. A comprehensive survey of the various methods for the increase in maximum lift is included in Lachmann [28].

The effect of take-off and landing devices on the lift characteristic c^(a) of a

wing is presented schematically in Fig. 8-16. Curve 1 gives the values without flap deflection. Curve la shows the increase in the coefficient cLmax by boundary-layer control at the wing nose. Curve 2 gives the values with flap deflection, and curve 2а again the increased values of cLmax through boundary-layer control at the nose. Curves 3 and За give the corresponding data when, in addition, the boundary layer at the flap nose is controlled as well. The summary report about theoretical and experimental studies on boundary-layer control by Carriere et al. [8] should be mentioned. Earlier, a paper on the properties of flap wings was given by Young [55].

Flaps The simplest method of increasing C£,max is the deflection of a cambered flap as shown in Fig. 8-1 la. This effect is obtained because the flap deflection increases the effective camber of the wing, resulting in a lift augmentation that may be considerable. As an example, Fig. 8-1 la shows cL against the angle of attack for several flap deflections. The increase in С£тах depends on the flap chord ratio Xf, the highest values are usually obtained for Xf = 0.20-0.25 [7].

A quite simple landing device in terms of design is the split flap as shown in Fig. 8-3e. This is a flat plate lying against the lower side of the wing and turning about its forward edge. The lift curves cL(a) of Fig. 8-17b for several flap angles rjf are similar to those of the cambered flap (compare Fig. 8-17a). The effectiveness of the split flap is, according to Gruschwitz and Schrenk [19], due not only to an increased camber but also to a reduction of the static pressure on the suction side of the profile. In Fig. 8-18, the pressure distribution is shown for a wing with

Figure 8-16 Effect of flap deflection and boundary-layer control on the lift of a flap wing (schematic). Explana­tions in the text.

Figure 8-18 Pressure distribution on a wing with deflected split flap, from [19]. Curve 1, without flap deflection. Curve 2, with flap deflection.

deflected split flap. Because of the flow around the sharp trailing edge of the deflected plate, a strong low-pressure range is formed in the wake of the flap, having an effect up to the upper side of the wing.

Basically, the cLmax value increases with Reynolds number. In Fig. 8-19, the results on the effect of the Reynolds number on the value of cLmax are given, both for a wing without a flap and one with a 60° deflection of a split flap. Young [54] reports on the separation characteristics of flap wings. Flaps extending over only a portion of the wing span will be treated in Sec. 8-3.

The effectiveness of the simple cambered flap is limited by the flow separation occurring at large deflection щ right behind the flap nose. By boundary-layer control at the station of greatest danger. of separation, the lift-increasing effect of the cambered flap can be improved, as shown schematically in Fig. 8-16. Boundary-layer control by suction or ejection requires a considerable design and construction effort and will be discussed later in more detail. On the other hand, the slotted flap as shown in Fig. 8-3&, first suggested by Betz [6] and by Lachmann [27], represents a simple design for natural boundary-layer control. The slotted flap functions in such a way that the air, flowing through the slot from the lower to the upper side, carries the boundary layer, formed on the wing, into the free flow before separation can occur. Starting at the flap nose, a new boundary layer forms that can again grow over a larger distance before separation.

The maximum lift coefficient cLmax depends on the separation processes at the main wing in front of the flap as discussed in detail in Sec. 2-5-1. The most unfavorable flow conditions occur shortly behind the profile nose of the wing and at large angles of attack, а « a(c£,max). Here, the pressure increase that follows the

Figure 8-19 Change of maximum lift coeffi­cient with Reynolds number for a wing without and with a split flap. Flap chord ratio Kf =0.20, flap angle 17f= 60°, from [7].

suction peak usually leads to boundary-layer separation at the wing leading edge (see Fig. 2-44). By boundary-layer control, similar to that of the trailing-edge flap, separation can be shifted to larger angles of attack. The extension of the linear range of the <?x(a) curve of Fig. 8-16 leads to a considerable additional lift gain.

Another effective arrangement for the increase of the maximum lift is the slat (flap before the wing leading edge) as shown in Fig. 8-3/, whose characteristics have already been discussed in Sec. 2-5-3. A polar curve of it is given in Fig. 2-53. Figure

8- 20 shows the lift coefficient plotted against the angle of attack for a wing without and with a slat. In agreement with profile theory, the slat does not generate a noticeable change of the profile camber, because this would cause a parallel shift of the cL(ct) curves without and with slat. Because of natural boundary-layer control, the maximum lift coefficient of a wing with a slat is reached at very large angles of attack.

An effect similar to that of the slat is produced by the so-called nose flap, first proposed by Kruger [44]. Here, the increase of a(cLmax) results from a different effect, namely, the shape of the profile nose, responsible for the separation process, which is changed favorably by the flap deflection (see also Fig. 2-44).

In addition to the conventional landing devices on the trailing edge discussed so far, the double-section wing as shown in Fig. 8-3c and the Fowler flap as shown in Fig. 8-3<i must be mentioned. The former is a simpler design of the slotted flap. The latter consists of a flap that is driven out rearward and deflected. A simultaneous camber and area increase is thus accomplished.

Frequently, several landing devices are utilized in combination to establish a maximum lift that is as large as possible. As an example, Fig. 8-21 gives the lift coefficient of the profile Go 819 with a slat and a double-section flap against the angle of attack. The favorable effect on the boundary layer of the flow through the slot between the slat and the main wing is clearly indicated by comparison with the measurement when the nose slot is closed. In this latter case, the cLmayi values for

Figure 8-20 Lift coefficient Ci(a) of a wing with slat, from [48]. Profile Clark Y, Reynolds number Re — 6 • 10s. Curve 1, without slat. Curve 2, with slat.

Figure 8-21 Lift coefficient c^(q) for the profile Go 819 slat and double-slot flap, from Wuest [53].

all measured flap angles are lower by Acimax ~ 0.6; also, the flow separation leads to a larger lift drop than for the open nose slot.

Comprehensive data on the maximum lift coefficient of wings with and without landing devices are given in [32, 33, 46].

Suction In an effort to increase further the maximum lift of wings, suction was studied quite early (see Betz [4]).

The suction intensity is defined by a dimensionless suction coefficient as

(8-19)

Here Q is the volume removed per unit time, A is the wing area, and Ux is the incident flow velocity. The maximum lift can be increased considerably by slot suction. Comprehensive tests on this method were conducted by Schrenk [4]. The most effective method, particularly for thick profiles, was found to be slot suction with a flap wing. Lift coefficients up to about cL = 4 may be obtained, as shown in Fig. 8-22 for a thick profile with flap and suction. Here the coefficients of suction are about Cq = 0.01-0.03 and the suction pressures cp =(p —p^/q™ — ~2 to —4, where Q stands for the total flow volume removed, p for the pressure in the suction slot, and qm = (p00/2)£/’« for the dynamic pressure of the incident flow. The effect of suction lies in its keeping the flow essentially attached to the flap. The greatest danger of separation is near the flap nose. If the decelerated boundary layer at this

Figure 8-22 Lift coefficients of flap wings with slot suction, from Schrenk.

station is removed strongly enough by suction, the flow over the entire trailing-edge flap may be kept attached. After favorable wind tunnel results had been obtained, for flap profiles with suction, the Aerodynamische Versuchsanstalt Gottingen (AVA) conducted the first flight tests of the suction effect in the early 1930s. The possible gain in lift for fully attached flap flow {cq = Cq£) over the lift of uncontrolled •flow (cq = 0) may be seen in Fig. 8-23. This diagram shows cL as a function of flap deflection at several angles of attack of the wing. Note that the lift for potential flow is reached when the suction is just strong enough for complete prevention of separation. Arnold [4] studied the computation of the required amount cql. More recently, both slot suction and continuous suction through perforated walls have been applied, the latter at the trailing-edge flap as well as at the wing nose. Further developments of suction procedures have been summarized by Regenscheit [36] and Schlichting [36].

The continuously distributed suction has been studied theoretically by Schlichting and Pechau [38]. Flight tests by Schwarz [38] and by Schwarz and Wuest [38] confirm the feasibility of nose suction.

Ejection The boundary layer may be controlled by ejection as well as by suction for increased maximum lift. This method has been applied most successfully to the wing with a trailing-edge flap. By tangential ejection of a thin jet of high velocity at the nose of the deflected flap, flow separation from the flap can be prevented and the lift can be increased. Critical for the effectiveness of ejection is, according to Williams [51], the dimensionless momentum coefficient

= QjOjVj (8-20)

Q CO A

where the index / refers to the conditions in the jet and the index •» to those of the incident flow.

Comprehensive studies on the lift increase of flap wings with ejection have been conducted by Thomas [43]. In Fig. 8-24, a typical result of these measurements is given, namely, the gain in the lift coefficient AcL against the momentum coefficient Cj for several flap angles rif. The curves AcL versus Cj clearly show two ranges: first, a very steep increase at small momentum coefficients; and second, a considerably smaller increase at large momentum coefficients. The first range is that of boundary-layer control. It extends to the momentum coefficient that just suffices to produce complete flow attachment back to the flap trailing edge, thus completely preventing separation. The second range of considerably smaller lift gain with the momentum coefficient is the range of supercirculation. Here, the “hard jet” (of very high momentum) acts similarly to an extended mechanical flap.

In Fig. 8-25, the lift coefficient of a wing at fixed flap deflection is plotted against the angle of attack for several momentum coefficients су. The ejection has a similar effect as an increased camber (flap deflection). Flow separation sets in at smaller angles of attack, however, than without ejection. Inspecting Fig. 8-16 shows that an additional lift gain can be generated by combination with a boundary-layer

Figure 8-23 Lift increase due to slot suction at the trailing-edge flap for completely attached flap flow, from Arnold. (——————– ) Measurements with­out suction. ( ) Measurements with

suction.

Figure 8-24 Flap wing with ejection, lift increase леї vs – momentum coefficient Cj for various flap angles г?/ at constant angle of attack a — —5°, from Thomas.

Figure 8-25 Lift coefficient of a wing with ejection over the trailing-edge flap, from Williams, profile f/c = 0.08. Flap deflection rj/ = 45°, flap chord ratio hf = 0.25.

control at the wing nose, either by suction or by ejection (see Gersten [15]). Even when the flow is completely attached, a further increase in lift may be accomplished by stronger ejection on the flap. This is the result of supercirculation and the jet reaction force. This problem area has been summarized by Poisson – Quinton [34] and by Williams [51]; see also [28]. Levinsky and Schappelle [29] developed a method aimed at maintaining potential flow through tangential ejection on flap wings.

Jet flaps Effects very similar to those generated by a solid trailing-edge flap are obtained by ejecting a high-speed jet under a certain angle V/ near the wing trailing edge. This method, illustrated in Fig. 8-26, is termed a jet flap. The vertical component of the reaction force of the jet is supplemented by an induced lift that may be many times larger than the jet reaction (supercirculation). This effect has been studied by many experiments [34, 52].

In Fig. 8-26, the theories of Spence [42] and Jacobs [9] are compared with experiments on a symmetric profile with jet flap. The figure shows the dependence of the lift slopes dcLfda and dcLJdrjj on the momentum coefficient Cj as defined by Eq. (8-20). Here the momentum coefficients Cj are much larger than in Fig. 8-25. Up to values of about c;- = 0.1, the jet acts on the boundary layer; for larger values of Cj it essentially causes the circulation to increase (supercirculation). Either lift, slope increases strongly with increasing Cj. For Cj = 4, the lift slope dci/da has about twice the value of that without ejection (cy = 0). The agreement of theory

Figure 8-26 Profile with jet flap, com­parison of theory and experiment for lift slopes bcj^jda and Эс^/Эру. Theory from Spence and Jacobs. Measurements from Dimmok [52]. (®) rjy = 31°. O) py = 58°.

and experiment is good. Helmbold [20] studied the theory of the wing of finite span with jet flap.

A comprehensive wing theory for the wing of finite span with jet flap has been developed by Das [9]. An example of this theory and a comparison with experiments is given in Fig. 8-27 for a swept-back wing with a jet flap spanning the entire trailing edge. Agreement between theory and experiment is good. Murphy and Malmuth [9] report on the computation of the aerodynamics of the jet flap wing in transonic flow. The jet flap wing near the ground has been studied by Lohr [30]. The aerodynamic problems of the maximum lift have been summarized by Schlichting [37]. Questions of the practical application of the jet effect to the generation of high lift on wings with and without flap are discussed in the summarizing paper of Korbacher [25].

Air brakes, spoilers The aerodynamic effect of air brakes has been investigated repeatedly (see Arnold [3]). In particular, various positions of the brakes on the lower and upper sides of the wing have been studied. Figure 8-28 shows the result of three-component measurements for a wing with air brakes over the entire span. The polar curves illustrate the very large drag increase. Compared with the wing alone, the drag coefficient is about 20 times larger.

Devices of a similar kind mounted only on the upper side of the wing are also, termed spoilers. By extending them on only one side of the wing, they can be used

Figure 8-27 Lift coefficient of a swept-back wing with jet flap; com­parison of theory and measurement from Das [9]. Aspect ratio.1 = 3.5, sweepback angle уз = 45°, jet angle 7}/ = 30°.

Figure 8-28 Three-component measure­ments on a rectangular wing with air brake, from Reller [3]. Aspect ratio л = 5.1; flaps extend over the entire span. WO, wing without flap; S, flap on suction side; P, flap on pressure side.

for control about the vertical and longitudinal axes. The flow separation from the wing caused by the spoiler leads to a strong, one-sided lift loss and thus to a rolling moment. Wing tunnel test results on spoilers and a few computations on the effect of the spoiler are found in [11, 21, 23, 50].

Lift and moment The theory of the flap wing of infinite span in compressible flow may be derived approximately from the profile theory of compressible flow as given in Sec. 4-3. There solutions were obtained for subsonic incident flow using the subsonic similarity rule (Prandtl, Glauert) and for the supersonic incident flow using the supersonic similarity rule (Ackeret). The following formulas apply for fixed flap chord ratios = Afinc.

For subsonic incident flow (.Ma« < 1),

УМа^ – 1

In Fig. 8-14, the changes in zero-lift angle and zero moment caused by the flap deflection are given as a function of the flap chord ratio.

By using the above coefficients, the position of the flap neutral point can be computed with Eq. (8-6), where it has to be considered that dcLlda= lit! Vl —Mala for Maa° < 1 and dcLjdot= AjjMaL — 1 for Ma„ > 1. The position of the flap neutral point is given in Fig. 8-15 against the flap chord ratio fy. Here the relationships xNf = c/4 + (AxN)f applies for Маж < 1 and xNу = cj2 4- (AxN)f for Afecc > 1.

At supersonic velocities the flap neutral point lies much farther back than at subsonic velocities, as should be expected. The following expressions are obtained for the coefficients of the flap moment (control-surface moment) at subsonic incident flow (Mr*, < 1):

dCm f ( dCmf dcL dcL /inc

Figure 8-15 Position of the flap neutral point vs. the flap chord ratio for compres­sible flow (subsonic and supersonic veloci­ties).

dcmf______ 1___ /Эст A

dVf ~ Vl-Md 9i?/ Anc

Again, the coefficients marked inc are those of incompressible flow from Eq. (8-14) and Fig. 8-13. Corresponding relationships are found for the coefficients of flap loading.

For supersonic velocities (Mzoo > 1), the Ackeret rule yields (see Sec. 4-3-3)

dcmf____ 3_

dcL ~ 2

The coefficients of flap loading are determined immediately as cXf = 2cmf by realizing that the pressure distribution over the flap chord is constant.

8- 2-1 The Flap Wing in Incompressible Flow

The flap wing as a bent plate The fundamentals of the theory of the flap wing of infinite span in incompressible flow have been given in Sec. 2-4-2. In its simplest form, the wing with a deflected flap is replaced by a bent plate as shown in Fig. 2-24, on the chord of which, according to Glauert [16], a vortex distribution is arranged.

In Fig. 8-7 these theoretical coefficients have been given against the flap chord ratio f. The problem of the single-bent plate has been solved by Keune [24] with the method of conformal mapping. The most important result of this study is the confirmation of Glauert’s approximate solution for small flap angles. For larger flap

angles, the deviations are more pronounced. In Fig. 8-7 these results are added to the results of comprehensive test series on wings of various flap shapes. The measured coefficients have been taken from test series for small flap angles. The coefficients thus obtained have been designated as Эа/Эт? уе and dcM/drife. Comparison of theory and experiment shows that the measured values are smaller than the theoretical ones for both the change in the angle of attack and the change in the moment. The curve for the wing with a split flap (spreader flap) shows the largest deviation from the theoretical curve. For larger flap deflections, the flap effectiveness declines. This trend is shown in Fig. 8-8 by assigning an effective flap angle rye to each geometric flap angle rf. This coordination applies approximately to the moment change as well.

The differences between the theoretical curves and the measurements in Fig.

8- 7д and b cannot be fully explained by the influence of the profile thickness. They should essentially be due to friction effects. For theoretical studies of the flap wing, it is advisable to apply empirical corrections to the coefficients of the flap wing as obtained from profile theory. This is accomplished simply by multiplying the effect of the camber on the coefficients Эа/Эру and Эсд^/Эру with an empirical factor «. Then the adjusted coefficients assume the form

(8-9 b)

Here the terms with the index к = 1 are the theoretical values from Eqs. (8-8a) and (8-8b). In Fig. 8-9, these coefficients for x = 0.75 are also shown; they agree satisfactorily with the measurements of Fig. 8-7.

In Fig. 8-10, the theoretical values for the position of the flap neutral point from Eq. (8-6) are plotted against the flap chord ratio with bc^jda = 2n. In this figure, the distance between the flap neutral point and the leading edge,

Figure 8-8 Correlation between the effective flap deflection туг and the geometric flap deflection ту for sev­eral flap designs (see Fig. 8-7).

Figure 8-9 Reduction of flap effectiveness from Eqs. (8-9a) and (8-96). (a) Change of angle of attack due to flap deflection. (6) Change of pitching moment due to flap deflection.

xNf= c/4 + (AxN)fb is given, where cf4 is the position of the wing neutral point. It is noteworthy that, for small flap chords, the flap neutral point lies at c/2. This is in consequence of the fact that the deflection of even a small flap strongly affects the pressure distribution on the front portion of the wing.

Computation of the flap loading (control-surface loading) and of the flap moment (control-surface moment) requires that the pressure distribution on the deflected flap be known. The theoretical pressure distribution on a bent plate is illustrated in Fig. 2-28, whereas Fig. 8-11 gives the experimentally determined pressure distribution on a wing with split flap from Schrenk ‘[40] (see also Seiferth [41]).

The aerodynamic force on the flap (flap loading), the knowledge of which is important for computation of the structural strength of the flap, is obtained from the pressure distribution on the flap as

L’f=bf I (pi~ pu) dx = Cifbfcfqoo

(fif)

Figure 8-10 Position of the flap neutral point vs. the flap chord ratio for incompressible flow.

In Fig. 8-12 the two coefficients have been plotted against the flap chord ratio f.

The flap moment (control-surface moment) of a wing portion of width bf, referred to the control-surface axis of rotation, is

Mf——bf f (Pi-Pu)(x-Xf) dx=cmfbfc}qOB (8-13)

(cf)

Figure 8-11 Pressure distribution on a wing with a slot flap, from Schrehk.

Figure 8-12 Flap loading; theory from Glauert. Curve 1, change of the coefficient of flap loading 0 with lift coefficient. Curve 2, change of the coefficient of flap loading with flap angle.

where Xf is the position of the axis of rotation as shown in Fig. 8-1 and су is the flap chord. The theory of the flap wing (bent flat plate, Sec. 2-4-2) yields the. following relationships for the control-surface moment coefficient cmf.

= " ЪЦ 1(3 _ 2Xf) ‘/V(1-V) – (3 – %) “resin V051 (8-14*)

In Fig. 8-13, these coefficients are plotted against the flap chord ratio Ay. Test results for simple cambered flaps are also shown. They lie considerably below the theoretical curves. These differences are caused by the influences of the profile thickness and, particularly, of the friction.

To reduce the control-surface moment Afy, several forms of control-surface balance arrangements have already been shown in Fig. 8-2. Of these, only the inner balance and the balance tab can be considered two-dimensional problems. At the inner balance, the control-surface moment is decreased by moving the axis of rotation rearward. Then, in deflecting the control surface, a control-surface “nose” protrudes from the profile, forming a contour that is hardly accessible to computation. To determine the aerodynamic coefficients of the flap wing with inner balance, mainly experimental studies have to be applied, such as, for example, those published by Gothert [18]. The aerodynamic coefficients of a flap wing with balance tab were first treated by Perring [16] using the theory of the multiple-bend plate. A comparison of his theoretical results with measurements is given by Gothert [18]. In this case the effect of friction is particularly strong.

The flap wing as a wing profile Several investigators have studied theoretically not only the flap wing of finite thickness but also the effect of flap arrangement and

Figure 8-13 Coefficient of control-surface moment vs. flap chord ratio Xf, theory from Gothert. (a) Change of the coefficient of control-surface moment with lift coeffi­cient. (b) Change of the coefficient of the control-surface moment with flap deflec­tion.

flap shape and, particularly, that of a slot between the fixed airfoil and the movable flap. In particular, the publications of Allen [2], Fliigge-Lotz and Ginzel [13], Keune [24], and Jacob and Riegels [22] should be pointed out. The results of these studies have been presented systematically by Gothert [18] within the framework of an experimental study. Furthermore, comprehensive test results on flap wings have been reported by Wenzinger [49] and by Keune [24]. Summary accounts of these studies are found in [7] and [45]; compare also [1, 35]. Theoretical investigations on the behavior of the boundary layer of flap wings and comparisons with measurements have been conducted by Goradia and Colwell [17].

The following aerodynamic coefficients are introduced for the wing with control surface:

Here the lift coefficient and the pitching-moment coefficient are referred to the geometric quantities of the wing, as in the case of the wing without control surface [see Eq. (1-21)]; The control-surface moment Mf (flap moment, hinge moment) is referred to the axis of rotation of the control surface; its sign can be seen from Fig.

8- 1. The coefficient of the control-surface moment cmf is referred to the geometric
quantities of the control surface. These three aerodynamic coefficients depend on the angle of attack a and the control-surface angle rjf.

As an example of measurements, the lift coefficient Ci of a simple flap wing is plotted against the angle of attack in Fig. 8-5a for several flap angles rjf. The flap deflection T]f causes, corresponding to Fig. 2-24, an additive camber and thus, at constant angle of attack, an increase in lift. The curves cL(a) for several angles rjf are parallel to each other. The dependence of the lift coefficient on a and rjf for small angles may be expressed as

(84a)

(8 ЛЬ) where da/drif indicates the change in the zero-lift direction of the wing because of the flap deflection (flap effectiveness) [see Eq. (7-3&)]. The coefficient da/drjf depends strongly on the control-surface chord ratio. Data on this effect have been given in Fig. 2-25a for a flap wing of infinite span.

In Fig. 8-5b, the lift coefficient cL is plotted against the moment coefficient cM for several flap angles rjf- The flap angle causes a parallel shift of the moment curves. The dependence of the moment coefficient cM on cL and rjf for small values of these parameters may be expressed as

486 AERODYNAMICS OF THE STABILIZERS AND CONTROL SURFACES

Here, dcMldr)f gives the change of the zero moment with the flap deflection. This coefficient depends strongly on the flap chord ratio. Data on the wing of infinite span have been given in Fig. 2-25b.

Frequently it is advantageous to specify the location of the aerodynamic center of the additional forces generated by the flap deflection. This point is termed the flap neutral point. The distance of the flap neutral point from the neutral point of the wing without flap deflection (= neutral-point displacement) is obtained from Eqs. (8-5) and (84) as

(A xN)f _ _ bcMjbrif c dcLfbrif

where 3cLbrf may be taken from Eq. (8-4a).

In Fig. 8-6, cL is shown as a function of the control-surface moment coefficient cmf for several values of rjy. Here, too, a linear relationship applies of the form

The dependence of this coefficient on the flap chord ratio will be discussed in Sec.

8- 2. The condition cmf= 0 determines a certain coordination of щ and cL and thus also of 7у and a for self-setting of the free control surface.

For the aerodynamics of the wing with control surface (flap), the most important geometric parameters as shown in Fig. 8-1 are as follows:

Control-surface angle (flap angle): r? y Control-surface chord ratio (flap chord ratio): f—Cfjc

These quantities have already been given for the whole wing with control surface (flap wing) in Sec. 2-4-2 and in Fig. 2-24. If the control surface does not extend over the whole span, as in, for example, the aileron in Fig. 8-4*2, the span of the control surface bA = 2sA becomes another important geometric quantity. On the horizontal tail plane and the fm, the control surface usually extends over the whole span of the horizontal tail bH and the height of the vertical tail h v,

Figure 84 Geometry of the control surface, (a) Ailerons. Cb) Elevator, (c) Rudder.

respectively (Fig. 84b and c). In many cases the control-surface chord ratio is varied along the span. In this case it is preferable to use the control-surface area ratio A ft A’ instead of the control-surface chord ratio Cf/c, where Af is the control-surface area and A’ is the wing area within the span range of the control surface.

7- 1 INTRODUCTION

7- 1-1 Function of the Flaps and Control Surfaces

As has been explained in Sec. 7-1, the tail surfaces of an airplane serve a twofold purpose, namely, to stabilize and to control the airplane. In general, the tail surfaces consist of a fixed part, the stabilizer, termed a fin on the vertical tail and a (horizontal) tail plane on the horizontal tail, and a movable part, the control surface, termed an elevator on the horizontal tail and a rudder on the vertical tail. There is another set of control surfaces attached to the wing, termed ailerons; see Figs. 7-1 and 7-3. The tail surfaces, with the control surfaces fixed, serve to stabilize the airplane. The corresponding aerodynamic problems have been discussed in detail in Chap. 7. The airplane is controlled by deflection of the control surfaces. Control about the lateral axis is accomplished with the elevator, that about the vertical axis and the longitudinal axis with the rudder and the ailerons.

The geometry of the tail surfaces and of the ailerons is that of an airfoil with a flap (flap-wing) as shown in Fig. 8-1 (see also Fig. 2-24). The aerodynamic effect of the control surfaces consists of an additive lift produced by their deflection. This lift, acting on the tail surfaces or the wing, respectively, controls the airplane. The aerodynamic forces acting on the control surfaces generate a moment that, referred to the control-surface axis of rotation, is termed control-surface moment or hinge moment. The effect of the control surface should be strong enough to generate an additive lift that, for a given control-surface deflection, is as large as possible. At the

same time, however, the hinge moment should be as small as possible so that the forces needed for the operation of the control surfaces also remain small. A control surface in the form of a simple flap as shown in Fig. 8-1 has relatively large hinge moments. Efforts have therefore been made to reduce the moments required to move the control surfaces. This has been accomplished by means of so-called control-surface balances, as shown in Fig. 8-2. The most important types of aerodynamic control-surface balances are the inner balance (nose balance) as shown in Fig. 8-2д, the balance tab as shown in Fig. 8-2b, and the outer balance (horn balance) as shown in Fig. 8-2c. In all cases of control-surface balance, it is important that the lift increase caused by the control-surface deflection (control – surface effectiveness) should, if possible, not be reduced by the control-surface balancing.

The airfoil with control surface of Fig. 8-1 may serve two purposes: first, to control the airplane, and second, to be used as a landing device. In the latter case, its effect is to increase the maximum lift of the airplane, thus holding down the landing speed. This lift increase is usually accompanied by a drag increase. In Fig.

7- 3, several designs of such landing flaps are shown. In the arrangements of Fig.

8- 3a-e, the flaps are attached to the rear end of the wing, whereas in Fig. 8-3/ and g, flaps are shown in front of the wing (slat, nose flap). Some of these arrangements are also employed as take-off assistance to reduce take-off distance.

Finally, a few more forms of flaps may be mentioned, namely, the system of a

Figure 8-2 Various forms of aerodynamic control-surface balances, (a) Inner balance (nose balance). (b) Balance tab. (c) Outer balance (horn balance).

Figure 8-3 Several control-surfaces and flaps, (a) Cambered flap. (b) Slot flap, (c) Double-section wing. (d) Fowler flap, (e) Split (spreader) flap, if) Slat, (g) Nose flap.

brake flap (air brake) on the upper and lower sides of the wing (see Fig. 8-28). They have the shape of a rectangular plate and are set normal to the flight direction. It is the function of the air brakes in their extended position to increase strongly the drag of the airplane, thus reducing considerably the speed and generating a steeper glide angle (brake effect).

The flow conditions at the vertical and horizontal tails are affected not only by the fuselage and wing but also considerably by their mutual interaction. Of special

interest here are the conditions at the tail unit at sideslipping and rolling. A tail unit at which the middle section of the horizontal tail lies over the root of the vertical tail will be considered to demonstrate this fact.

On a vertical tail in an incident flow of sideslip angle j3, a circulation distribution is generated that does not drop to zero at the root section but rather has a finite value because of the end-plate effect of the horizontal tail. A circulation discontinuity results now in the shedding of a single vortex that turns in a direction opposite to that of the rest of the free vortices. This vortex in turn induces at the horizontal tail a downwash exceeding the counteracting induction effect of the continuous free vortex sheet. The resulting circulation distribution at the horizontal tail has, as shown in Fig. 7-43b, a discontinuity in the middle of the horizontal tail; it is antimetric and generates a rolling moment due to sideslip that is reversed from that of the vertical tail (see Fig. 7-43, from Laschka [19]).

To reduce the load induced on the horizontal tail by the sideslipping vertical tail, a positive dihedral may be provided. This increases, however, the total rolling moment due to sideslip. On the other hand, the rolling moment due to sidesUp of the tail unit may be reduced by providing the horizontal tail with a negative dihedral.

By extending and applying a suitable panel method as described in Sec. 6-3-1 for the wing-fuselage system, the pressure distributions, and thus the acting forces and moments, can also be determined for the whole airplane; compare, for example, [15].