Category AERODYNAMICS OF THE AIRPLANE

Theoretical determination of sidewash Computation of the distribution of the induced sidewash velocity for a known circulation distribution can basically be done like that of the downwash, namely, with the help of the Biot-Savart law. A few qualitative considerations may be noted first. In Fig. 7-40, a symmetric and an asymmetric circulation distribution are compared. Because the circulation distribu­tions have been taken as constant, the symmetric distribution of Fig. 7-40a produces one horseshoe vortex, and the antimetric distribution of Fig. 740b two horseshoe vortices turning in opposite directions. It is immediately obvious that in

the middle plane, у = 0, a down wash velocity —w is obtained for the symmetric circulation distribution but a sidewash ±v for the antimetric distribution, having reversed signs on the upper and lower sides. The latter results essentially from the

counterclockwise-turning “double vortex,” shed in the middle. However, this highly idealized vortex model is insufficient to determine the induced sidewash quantitatively. The computation of the induced sidewash must be based on a variable circulation distribution I'(y), for example, like that for the sideslipping wing – fuselage system of Fig. 7-38. The sidewash velocity very close to the vortex sheet is obtained in analogy to Eq. (2-46л) as

(7-62)

where the upper sign applies above the vortex sheet and the lower sign below. The validity of this equation can also be checked by inspecting Fig. 7-38c and d. There, the slope of the circulation distribution is shown for у = 0, and the sign of the sidewash velocity v is indicated. The induced sidewash angle = v/U„ is obtained from Eq. (7-62) by introducing the dimensionless circulation distribution 7 = rjbUoo and the dimensionless coordinate in the span direction 17 —у/s as

(7-63)

By introducing the expression 7(17) = 7g(v) + Plp(v) for the circulation distribu­tion, where 7g is the distribution in straight flight and (3y@ the additive circulation for sideslipping flight, Eq. (7-63) yields, for the efficiency factor of the vertical tail in the vortex sheet,

(7-64)

The above derivation shows that Eqs. (7-62)-(7-64) are valid for any distance behind the wing for a not-rolled-up vortex sheet.

From Eq. (7-64) it is seen that the efficiency factor changes abruptly in the vortex sheet. The quantity dy^dr] is obtained from the circulation distribution of the sideslipping wing-fuselage system. The y@ distribution for the high-wing airplane is illustrated in Fig. 7-4ІЯ. The determination of the induced sidewash outside the vortex sheet has been studied by Jacobs and Truckenbrodt [14]. By applying the Biot-Savart law, the induced sidewash angle for a given circulation distribution 7(17) is obtained from lifting-line theory as

(7-65)

with r as in Eq. (7-29). For unswept wings and a very large distance (| -> TO), Jacobs [14] gave a simple procedure for the evaluation. The solution for arbitrary wing

planforms has been studied by Gersten [10]. For large distances behind the wing it suffices to use the values for £

In conclusion, results of a few sample computations will be reported. In Fig. 7-41 the induced sidewash field is given for a high-wing system. Figure 7-4la illustrates the geometry and the additive circulation distribution 7^ due to the sideslipping. Figure 741 b represents the streamline pattern of the induced velocity field very far behind the wing, and Fig. 7-4lc gives the distribution of the sidewash factor d&vldj3 as a function of the distance from the vortex sheet for the middle plane 7? = 0. This figure demonstrates the discontinuity of the sidewash factor at the vortex sheet f = fi and the strong drop with distance from the vortex sheet. Figure 7-41d gives the distribution of the sidewash factor in the span direction for several distances from the vortex sheet.

In Fig. 7-42 for a high-wing and for a low-wing airplane the curves of constant local efficiency factor of the vertical tail Ъ§у]Ъ$ = const are shown for the transverse plane at the location of the vertical tail. The total efficiency factor of the
vertical tail is obtained from this through integration over the vertical tail height. The field of the curves 9j3y/9j3 = const is independent of the angle of attack of the airplane. There is, however, a dependence of the efficiency factor of the vertical tail on the angle of attack because, with a change of the angle of attack, the vortex sheet is displaced relative to the vertical tail (see Fig. 7-20). This influence is quite noticeable, as may be seen by comparing the cases cL — 0 and cL = 1 in Fig. 7-42. For the system of wing, fuselage, and vertical tail of Fig. l-39a, Jacobs [14] applied this method to determine the efficiency factors theoretically (Fig. 7-39c). The agreement with measurements in Fig. 7-39b is satisfactory.

The problem area of the interaction of wing, fuselage, and vertical tail at sideslipping has been investigated by Puffert [28]. The concepts established for the induced sidewash have been translated into that for the rolling wing by Michael [23] and by Bobbitt [5].

Fundamentals As has been shown in Sec. 7-2-2, the effect of the wing on the horizontal tail at symmetric incident flow lies essentially in the downwash of the

wing. The fuselage and the relative position of wing and fuselage (wing high position) contribute little to the interference. In all cases, however, the effectiveness of the horizontal tail is reduced by the wing and fuselage.

Considerably different conditions prevail for the effect of the wing and fuselage on vertical tads at asymmetric incident flow. Schlichting and Frenz [35, 36] showed that vertical tails are markedly affected only by a combination of wing and fuselage. This interference results in an increase or in a reduction of the effectiveness of the vertical tail, depending on the high position of the wing. This influence on the vertical tail is caused physically by the quite asymmetric circulation distribution over the span of wing-fuselage systems. This asymmetry, explained by Fig. 6-6, causes a rolling moment due to sideslip. This fact has been discussed in Sec. 6-2-3. In Fig. 7-38, this antimetric circulation distribution along the span is illustrated for a sideslipping high-wing airplane. The lift increase of the leading wing-half and the lift decrease of the trailing wing-half generate a pressure drop on the upper side of the wing toward the advancing wing-half. This pressure drop leads to an induced flow, as explained in Fig. 7-38, which revolves around the wing. This velocity induced at the wing is effective at the vertical tail as an induced

Figure 7-38 Evolution of induced side-wash of a wing-fuselage system in yawed flight, (a), (b) Geometry (high-wing airplane), (c) r(y) = circula­tion distribution, Tg(y) = circulation distribution at symmetric incident flow, (d) Induced velocity field at the location of the vertical tail.

lateral velocity of about the same magnitude. Figure 7-38<7 shows immediately that, for conventional positions of the vertical tail, the incident flow angle of the vertical tail is decreased by the lateral velocity v, that is, that the effectiveness of the vertical tail is reduced. As in Fig. 6-6d, the sign of the induced lateral velocity is reversed for the low-wing airplane from that of the high-wing airplane. This results, for the same relative positions of fuselage and vertical tail, in an increased effectiveness of the vertical tail. In consequence of its evolution, the lateral velocity induced by the fuselage-wing interference is proportional to the sideslip angle 0 and independent of the angle of attack a. Thus the resultant velocity in the у direction at the location of the vertical tail is

Vy — {3 Uc0 4- vg – f Ug

where j3C/co is the lateral velocity due to the sideslip angle, vg is the induced lateral velocity at symmetric incident flow, and fivp is the additional induced lateral velocity due to sideslip as in Fig. 1-3M. The effective sideslip angle of the vertical tail is

Hence the efficiency factor of the vertical tail is

(7-61)

because vg is independent of j3. Because = d(3v/d@, Eq. (7-61) is identical to

required to determine the efficiency factor of the vertical tail.

For an experimental confirmation of the above considerations, a few test results on the efficiency factor are plotted in Fig. 7-39b for the wing-fuselage-vertical tail system of Fig. 7-39a. From measurements of the yawing moment due to sideslip with (yvV) and without (oV) vertical tail, a mean efficiency factor of the vertical tail has been established by Jacobs [14] in the form

where (dcMzld@)y is the contribution of the vertical tail to the yawing moment.* This experimentally determined efficiency factor is given in Fig. 7-39b as a function of the high position of the vertical tail. The result is

for the low-wing airplane:

1 (stabilizing)

*This has been determined as the difference of the measurements on fuselage and vertical tad and of the fuselage alone.

and for the high-wing airplane:

< 1 (destabilizing) dp

To evaluate the above equations, the lift slope dciyjday must be known for the interference-free vertical tail. Basically, this can be computed with the methods of three-dimensional wing theory. Since the shapes of the vertical tails are in most cases quite asymmetric, this task is particularly complicated. Hence, wind tunnel measurements are indispensable for the acquisition of these aerodynamic quantities of the vertical tail. An attempt has been made in Fig. 7-37 to represent the measured lift slopes of single-fin assemblies with partial fuselages as a function of a uniquely defined aspect ratio Ay = b2v/Av. The meaning of Ay and by is obvious

Figure 7-37 Measured lift slopes of an interference-free vertical tail with partial fuselages from DYL measure­ments and Koloska [13].

from the sketch in Fig. 7-37. The aspect ratios Ay he between 1 and 2. Fuselages of round and rectangular cross sections and with horizontal and vertical tail edges were investigated as well as systems with and without horizontal tails. The ratio of the fuselage height hp to the span by of the vertical tail was limited by hplby = 0.35 and 0.5. Curve 1 of Fig. 7-37 shows the theoretical trend for the lift slope as in Fig. 3-32. It represents approximately the test points for vertical tails with circular fuselages and with horizontal tails. For such vertical tails, the lift slope follows the relationship

=——- 2яЛу— (7-58)

da. v 4 + 2

Curve 2, lying considerably lower than curve 1, represents a vertical tail with fuselages of rectangular cross section and without horizontal tail surfaces. Between these curves Me, as curve 3, the results for systems of circular fuselages without horizontal tails and those of rectangular fuselages with horizontal tails. Additional measurements for vertical tail assemblies with two fins are given in [35]. Theoretical studies of the lift slope of a vertical tail with a horizontal tail have been conducted by Rotta [31].

It is almost impossible to give generally valid data for the aerodynamic coefficients of vertical tails at compressible flow.

7- 3-1 Contribution of the Vertical Tail to the Aerodynamics of the Whole Airplane

The airplane in sideslipping flight The function and the geometry of the vertical tail have already been described in Sec. 7-1. As shown in Fig. 7-36, the vertical tail at asymmetric incident flow of the airplane of sideslip angle /1 is subject to a side force Yy. Because of its large lever arm, this side force generates the predominant portion of the yawing moment due to sideslip of the whole airplane. Moreover, the vertical tail also contributes to the side force due to sideslip and the rolling moment due to sideslip of the airplane. The contribution of the vertical tail to the yawing moment. due to sideslip of the airplane is

MzV=-rvYv (7-47)

where, from Fig. 7-36, r’v is the distance of the side force vector of the vertical tail from the moment reference axis that generally coincides with the vertical axis through the airplane center of gravity.

In analogy to Eqs. (7-2h) and {l-2b) for the horizontal tail, dimensionless coefficients may be introduced for the side force Yy and the yawing moment Mzy of the vertical tail by

Figure 7-36 Incident flow direction of the vertical tail. C. G. — center of gravity of the airplane.

Here qv is the dynamic pressure at the location of the vertical tail, which is generally smaller than the dynamic pressure of the undisturbed flow q« because of the interference of wing and fuselage with the vertical tail. The coefficient of the yawing moment of the vertical tail, referred to the wing quantities, is obtained from Eqs. (7-47M7-48&) as

The lift coefficient of the vertical tail CjV depends on the angle of attack (angle of sideslip Pv) and the rudder deflection qv of the vertical tail, in addition to its geometric data. The term dciy/dav stands for the lift slope of the interference-free vertical tail and (davldVy)Vv stands for the change in the zero-lift direction of the vertical tail caused by the rudder deflection.

In some cases the incident flow direction of the fin pv is considerably different from that of the airplane P because of the interference of wing and fuselage with the vertical tail. The two incident flow angles differ, as shown in Fig. 7-36, by the sidewash angle ft, = v/Uoo induced by the wing and fuselage at the location of the vertical tail:

(3y — (3 + (3V

Hence, for a rudder deflection of zero, the contribution of the vertical tail to the yawing moment is given as

It follows, then, that the change in yawing moment with the angle of sideslip (contribution of the vertical tail to the directional stability, Sec. 1-3-3) becomes

dCmzV___ dciy / f 3ft, qy A у r’y

3j3 dav ~r dp qx A s

The quantity

d&v _ і, Sft

dp dp

is designated as the efficiency factor of the vertical tail. From Eq. (7-52) it follows that the contribution of the vertical tail to the directional stability is proportional to the efficiency factor.

To establish the contribution of the vertical tail to the side force of the whole airplane, the coefficient of side force of the vertical tail is defined, in analogy to Eq. (7-9) for the horizontal tail, as

Yy — CyyAqc

In analogy to Eq. (7*52), the contribution of the vertical tail to the side force due to sideslip becomes

Hence, the contribution of the vertical tail to the side force due to sideslip, too, is proportional to the efficiency factor of the fin. Generally, the vertical tail also contributes to the rolling moment due to sideslip because the point of application of the vertical tail side force lies, in most cases, considerably above the airplane’s longitudinal axis.

The airplane in yawing motion Besides the sideslipping considered so far, the rotary motion of the airplane about the vertical axis (yawing motion) is also of great importance to the aerodynamics of the vertical tail. A rotary motion about the vertical axis with angular velocity coz generates a sideslip angle at the vertical tail

as the dimensionless angular sidesUp velocity. By introducing this expression for $v into Eq. (7-51) considering Eq. (7-50), the change in the coefficient of the yawing moment with the angular sideslip velocity becomes

This coefficient is termed the contribution of the vertical tail to the sideslip or yaw damping. Comparison of this formula with Eq. (7-52) shows that the contribution of the vertical tail to the directional stability is, in terms of the geometric quantities, proportional to (A y/A)(r’v/s) and that to the yaw damping is proportional to (A vJA)(r’y/s)2. The following discussions will be limited to incompressible flow.

quantitative assessment of the qualitative findings about the downwash at supersonic flow, first the simple case of a wing with constant circulation distribution over the span will be investigated. In this case, for supersonic flow the effect of the wing on its vicinity can also be described by means of a horseshoe vortex, whose bound vortex lies on the wing half-chord. The effect of the two free vortices is restricted, however, to the range within the Mach cones originating at the wing tips. Only the downwash on the x axis will be computed for this arrangement. This can be done by means of the results for the horseshoe vortex at incompressible flow according to Eq. (7-23), which may be applied to supersonic flow by referring to the corresponding discussion of Sec. 4-5. Thus, the distribution of the downwash angle on the x axis behind the wing becomes

o) = rh 4 te – («<4 – i) (74i)

where с^/ЪгА = сс{(0). The downwash distribution according to this equation is shown in Fig. 7-30 for several Mach numbers. These curves demonstrate that, as has
already been discussed in connection with Fig. 7-29, no downwash at all exists on the middle section over a certain stretch closely behind the wing (down to £o = ІМаІо — 1). For large distances, £ > £0, first the downwash increases strongly and then reaches the asymptotic value aw = —2а{ =сьІтгА for §-*°°, which is the value for incompressible flow (see Fig. 7-14).

To show more accurately an induced velocity field of a free vortex at supersonic flow, the velocity distribution will now be considered in a Mach cone originating, as shown in Fig. 7-31, at the tip of a semi-infinite wing. This flow was first studied by Schlichting [33]. In Fig. 7-3lc the streamline pattern is shown in a lateral plane x = const, normal to the Mach cone axis. Here the cone shell is a singular surface because it is formed completely by Mach lines. The streamline pattern within the Mach cone consists partially of closed streamlines encircling the vortex filament and partially of streamlines entering the cone on one side and leaving it on the other. Near the cone axis, the flow is comparable to that in the vicinity of a vortex filament in incompressible flow. The distribution of the downwash velocity over the Mach cone diameter for the plane z = 0 is obtained according to [33] as

(7-42)

This distribution is shown in Fig. 7-3lc?, where x tan ii—R is the radius of the Mach cone at the distance x. Because w = Г0/27гр in the potential vortex, it can be concluded from Eq. (7-42) that, at supersonic flow, the distribution of the induced velocity near the axis у = 0 deviates only a little from that at incompressible flow. Both distributions are given in Fig. 7-3 Id.

Lagerstrom and Graham [17] gave an exact solution for the downwash field of the inclined plate of semi-infinite span. They obtained it by means of the cone-symmetric flow (Sec. 4-5-2) by first establishing the solution for the laterally cut-off plate of infinite chord, which is

(743a)

Figure 7-30 Downwash at the longi­tudinal axis of a wing of constant circulation distribution (horseshoe vor­tex) at supersonic velocities of several Mach numbers Маса, from Eq. (7-41).

Figure 7-31 Velocity distribution within the Mach cone of a free vortex at supersonic flow. Semi – infinitely long wing of constant circulation distribu­tion, from Schlichting. (a) Circulation distribution. (b) Wing planform and Mach cone, (c) Streamline pattern of the section x = const, (d) Downwash and upwash velocities in the plane z = 0. Solid curve, from Eq. (7-42). Dashed curve, plane potential vortex.

Here, as in Fig. 7-32, t = уfx tani= уjR.

This solution leads to that for the downwash field of the laterally cut-off flat plate of finite chord by superposition. In Fig. 7-32, the distribution of the downwash factor bawfda in the plane of the plate is shown for several distances xfc behind the plate. There are downwash velocities within the inner half of the Mach cone, upwash velocities within the outer half. The curve for xjc = 1 applies on the inner half to points immediately behind the trading edge, whereas, from Eq. (7-43a), daw/da — — 1 for points on the surface. At a very large distance (*-*«>), the following expressions are obtained:

Here R0 = c tan ji is the radius of the Mach cone at the wing trailing edge.

The downwash field of the rectangular wing of finite chord and finite span is
obtained from the above solution by superposition. In Fig. 7-33, the downwash factor daw/da for the middle section according to Laschka [18] is plotted against the distance x/c and with Л s/Mal, — 1 as the parameter. Here the downwash factor shows the same trend as seen in Fig. 7-30. For Л^/Ма/L — 1 < 2, Mach lines originating at the 2 forward comers intersect each other on the wing. Thus there is

Figure 7-33 Distribution of the downwash factor on the longitudinal axis behind rectangular wings at supersonic incident flow for various values of the parameter л-jMalo — 1, from [18]. Asymptotic values for x -*• from Eq. (7-45).

no zone behind the wing in which the downwash is zero. At a very large distance behind the wing there is, for у — 0,

s*w _ __£ Л ____ і/t_________ 2

71 I Л УМа% – 1

For /1 ZMalo — 1 <2 the result is daw/3a = —■4/tt = const. For a rectangular wing of aspect ratio /1 = 2, the downwash factor daw/da is given in Fig. 7-34 at several distances x/c as a function of the Mach number Max. The very strong influence of the Mach number on the efficiency factor of the horizontal tail is obvious.

Experimental studies about the downwash behind the rectangular wing at supersonic velocities have been conducted by Davis [1] and by Adamson and Boatright [1].

The above theoretical results have been obtained with the lifting-surface theory. Mirels and Haefeli [24] developed a lifting-line theory that has been applied to both rectangular and delta wings. The results of this lifting-line theory agree with the lifting-surface theory at some distance behind the wing, as would be expected. Another computational method for the downwash, applying dipole distributions, has been given by Lomax et al. [20]. By using this method, comprehensive computations of examples have been conducted on delta wings with subsonic leading edges. Likewise, delta wings with subsonic leading edges have been treated by Robinson and Hunter-Tod [29] and by Ward [41 ]. Some results for delta wings with a supersonic leading edge are found in Lagerstrom and Graham [17].

The results presented so far in Figs. 7-32-7-34 apply to the conditions on the vortex sheet (z = 0). In conclusion, a few data will now be given for the downwash factor outside the vortex sheet. In Fig. 7-35, daw/da is plotted against the vertical coordinate £ for several values of the parameter As/MoL — 1. As for incompressible

Figure 7-34 Distribution of the downwash factor on the longitudinal axis behind a rectangular wing of aspect ratio л = 2 at supersonic incident flow for several Mach numbers Маж, from [18].

Figure 7-35 Downwash factor in the root section 0 = 0) behind rectangular wings vs. the high position at supersonic velocities, from [18].

flow (Fig. 7-21), the downwash factor decreases strongly with increasing distance from the vortex sheet. Corresponding results for delta wings are found in [20].

Now a computational method that is analogous to that for incompressible flow will be briefly described. The transformation from incompressible to supersonic flow has been explained in Sec. 4-5. Accordingly, Eq. (7-31) for the downwash velocity is also valid for supersonic flow if the function Gx is replaced by the function G of Eq. (4-95) and the function G2, corresponding to Eq. (7-32), by

Here xQ(y’) is the location of the Mach line according to Eq. (4-96). Laschka [18] suggests that one compute G in Eq. (4-95) and G2 in Eq. (7-46) by taking the vortex density к as a constant over the chord x and as a variable over the span у, that is, k(x, y) = k(y). Thus Gx and G2 can be integrated in closed form. For the determination of the downwash velocity w in Eq. (7-31), only an integration over the span coordinate remains to be done. Ferrari [7] gives a summary survey of the downwash in compressible flow.

Fundamentals The influence on the horizontal tail of the forward airplane components (wing and fuselage) is, at supersonic incident flow, generally greatly different from that at subsonic incident flow. This difference is a result of the limited influence zones at supersonic incident flow as shown in Fig. 7-28. The flow at a point of the horizontal tail can be affected only by the parts of the airplane lying within the upstream cone of this point. This cone is, from Fig. 4-58, the Mach cone of the generating semiangle q, located upstream of the control point under consideration and with axis parallel to the incident flow direction. The relation between incident flow Mach number and Mach angle is given by Eq. (4-80). The upstream cone cuts out of the airplane the influence zone that affects the horizontal tail (see also Fig. 4-58). This influence zone is marked in Fig. 7-28 for two Mach numbers (Mach lines and m2, respectively). The influence zone shrinks with increasing Mach number; that is, it would be expected that the effect on the horizontal tail, particularly of the upstream-lying wing, decreases with increasing Mach number. Furthermore, Fig. 7-28 demonstrates that the distance between the horizontal tail and the wing is of paramount importance for the magnitude of the interference. At constant Mach number, p = const, the horizontal

Figure 7-27 Effect of Mach number on the efficiency factor of the horizontal tail behind a delta wing of aspect ratio л = 2.31.

Figure 7-28 Effect of wing and fuselage on the horizontal tail at supersonic velocity.

tail is less affected when it is close to the wing than when it is farther away. To establish computational methods for the determination of the downwash at the location of the horizontal tail, those for incompressible flow must be modified to take into account whether, as in Fig. 7-28, the influence zone of the horizontal tail encloses, at the respective Mach number, only a part of the wing (jrii) or the whole wing (m2).

As a first step, the physical character of the downwash field generated by a wing in supersonic incident flow will be discussed qualitatively by means of Fig.

7- 29. Here a rectangular wing is sketched with its circulation distribution as in Fig. 4-79a. It generates downwash and upwash velocities only within the two Mach cones originating at the two forward corners. In the middle part of the wing of width b* the flow is purely two-dimensional, and according to Fig. 4-21 does not generate a downwash behind the wing. Thus the triangular zone I of Fig. 7-29 remains without downwash (aw = 0). From the triangular surface zones at the wing tips in which the circulation drops off, free vortices are shed downstream as in incompressible flow. Thus downwash velocities (aw < 0) are in zone II behind the wing. Conversely, upwash velocities (aw > 0) prevail in the two zones III that contain the outer halves of the two Mach cones. In the entire range IV before and beside the wing, outside of the Mach cones aw = 0.

The horizontal tail without interference in supersonic Cow According to Sec. 7-2-1, the contribution of the horizontal tail to the pitching moment and to the lift of the whole airplane depends on the lift slope of the tail surface dcuf/dottf and on the efficiency factor Ъан! да= 1 +Эotwjda. First, a few data will be given on the lift slope dciHjd<xH of the horizontal tail without interference. They may be taken from Sec. 4-5-4, in which the theory of wings of finite span at supersonic incident flow

Figure 7-29 Induced downwash and upwash fields in the vicinity of a rectangular wing in supersonic inci­dent flow (schematic).

was discussed. For a horizontal tail of rectangular planform as in Eq. (4-112), the lift slope becomes

Лещ _ 4 L_____________ 1_____

ІМа’Ь – 1 2 Ан ІMd^ – 1

if /іН/Магж — 1 > 1. The first factor represents the lift slope in plane flow, the second the correction for the finite aspect ratio of the horizontal tail. This relationship is illustrated in Fig. 4-78c.

The effect of compressibility on the aerodynamic coefficients had been determined by means of the Prandtl-Glauert-Gothert rule for the wing in Sec. 4-4 and for the wing-fuselage system in Sec. 6-3-1. In the same way, this effect can be determined for the horizontal tail. Through a transformation, the subsonic similarity rule allows one to reduce the compressible subsonic flow about the whole airplane to incompressible flow. Here the incompressible flow is computed for a transformed airplane as shown by an example in Fig. 7-24 for Маж = 0.8. The transformation of the geometric data is given in Eqs. (6-29)-{6-31). For the geometric data on the horizontal tail, Eqs. (б-ЗОд)-(б-ЗОе) apply accordingly. For the transformation of the distance of the tail surface from the wing, the relationship rH-mc = rH has to be added, observing Eq. (6-29). The same relationship as for the wing alone applies to the dependence of the lift slope of the horizontal tail without interference on the Mach number Ma„. Hence, with Eq. (4-74), the relationship

dcjH ‘■In/.lH

doiH У(1- M04 + 4 – b 2

is obtained, which is shown in Fig. 4-45. By computing the incompressible flow for the transformed airplane at the angle of attack of the subsonic flow, that is, for ainc — on, the induced downwash angle in the vortex sheet becomes

7) aw inclines ‘bine)
Oiw — 2Q! jinc (I -*■ °°)

This relationship allows one to determine in a very simple manner the downwash field of compressible flow from that of incompressible flow. A simple approxima­tion formula for the downwash of incompressible flow at some distance behind the wing has been given by Eq. (7-25b). With the above transformation and with Eq.

Figure 7-24 The Prandtl-Glauert rule at subsonic incident flow velocities. (a) Given airplane, (b) Transformed airplane.

Figure 7-25 Effect of Mach number on the downwash angle at the longitudinal axis behind a wing of elliptic circulation distribu­tion, from Eq. (7-38).

(4-72я), this formula can be reduced to subsonic flow. For elliptic lift distribution there results

In Fig. 7-25 the downwash angles so computed for £ = 1, 1.5, and 2 have been plotted against the Mach number Ma„.

As a further result, in Fig. 7-26 the efficiency factors of the horizontal tail from Eq. (7-28a) are plotted against the Mach number for several aspect ratios. The analytical expression is

Figure 7-26 Efficiency factor of the hori­zontal tail vs. Mach number for elliptic wings of various aspect ratios A, from Eq. (7-39) for

—V CO t

This figure indicates the remarkable result that the efficiency factor decreases strongly with increasing Mach number at all aspect ratios A. For Маж = 1, the efficiency factor of the horizontal tail becomes zero at all aspect ratios, a result in agreement with slender-body theory (see also Sacks [32]). Finally, in Fig. 7-27, the efficiency factor of the horizontal tail dajj/da for a delta wing of aspect ratio A = 2.31 is given for several Mach numbers as a function of the tail surface distance. Accordingly, the efficiency factor changes only a little with Mach number in the range 0<Ma«, <0.8.

The horizontal tail without interference The further discussions on the aero­dynamics of the horizontal tail of this section’will deal first with incompressible flow and then with compressible flow at subsonic and supersonic velocities. The horizontal tail without interference from fuselage and wing will be treated first, followed by an account of the effect of the wing on the horizontal tail.

For the horizontal tail in incompressible flow without interference, the three-dimensional wing theory of Chap. 3 can largely be applied. Of the aerodynamic coefficients, first the lift slope dciHjdaH for small and moderately large aspect ratios AH is required. In Fig. 7-8 a few theoretical curves are given for the lift slope of the horizontal tail as a function of the aspect ratio Ah- A

Figure 7-8 Lift slope of a horizontal tail without interference for incompressible flow vs. aspect ratio of the tail surface л ц (lifting-surface theory).

rectangular, a swept-back, and an elliptic wing are described. The elliptic wing follows, from Eq. (3-98), the simple formula

dciu ттЛя… , nAu 1 .

—— = ■ ■ – H—— with TcB = – r – ^ —As

“b ^ 1 ^

Further information on the lift slope and comparisons with measurements have been given in Sec. 3-3. There, the neutral-point position can also be found, which is required for the determination of the tail-surface lever arm.

The above data for the lift slope can be applied to a horizontal tail without a vertical tail surface and also to a horizontal tail with a single vertical tail.

For a horizontal tail with two fins, as shown in Fig. 7-9, the lift slope is considerably larger because of the end-plate effect. Theoretical investigations on wings with end plates have been conducted by Mangier [22]. The effect of end plates on the lift slope can be taken into account approximately by introducing, besides the geometric aspect ratio AH, a so-called effective aspect ratio Afj – For a horizontal tail with end plates, these two values Ajj and A% are related by the empirical formula

Measurements on the effect of end plates were first published by Prandtl and Betz [27]. In Fig. 7-9, the lift slopes dclH{d<xH, based on those measurements, are given as a function of the effective aspect ratio A%. The solid curve applies to the rectangular wing of Fig. 7-8.

Effect of the fuselage on the horizontal tail The interference of the wing and fuselage with the horizontal tail consists of a reduction of the dynamic pressure at the location of the tail surface and also in an altered incident flow direction of the tail surface. The reduction in dynamic pressure is caused mainly by the boundary layer at the wing-fuselage interface, and the change in incident flow direction of the
tail surface by the induced velocity field of the wing-fuselage system. Whereas the induced velocity field can be reasonably well determined theoretically, the dynamic pressure reduction must be found experimentally.

It is desirable that the value of the ratio be as close to unity as possible

and that it be essentially independent of the angle of attack of the airplane. Both requirements can be satisfied through suitable selection of the horizontal tail relative to the wing and the fuselage; compare Hafer [13].

Now, the influence of the fuselage on the horizontal tail will be discussed first. The arrangement of a horizontal tail on the fuselage corresponds basically to a wing-fuselage system as treated in Sec. 6-2. There is the difference, however, that the fuselage usually does not extend behind the tail surface. It is very difficult to establish a general procedure for the computation of the influence of the fuselage on the tail plane because of the many different arrangements of the horizontal tail (high, mid, low surface) and the various shapes of the tail of the fuselage. Therefore, a review of some test results on this influence must suffice.

Koloska [13] reports three-component measurements on fuselage-tail surface systems. The tail surfaces were rectangular of aspect ratio Л#= 2 and 1.2, attached to a partial fuselage. The lift slopes as affected by the fuselage, dciHjdaH, are considerably smaller than those for the horizontal tail without interference as shown in Fig. 7-8. In Fig. 7-10, values of dciH/daH under the influence of the fuselage are given as a function of the aspect ratio of the horizontal tail AH and the relative fuselage width bp/bn – Accordingly, to give an example, at an aspect ratio AH — 2 and a relative fuselage width bp/bH = 0.3, the fuselage effect reduces the lift slope by about 20%.

Effect of the wing on the horizontal tail The effect of the wing on the tail surface consists essentially of a change of the angle of incidence of the horizontal tail

Figure 7-9 Measured lift slope of hori­zontal tail with end plates, from [27], vs. effective aspect ratio of the tail surface Лд-, from Eq. (7-19). Theoreti­cal curves from Fig. 7-8 for Ajj.

Figure 7-10 Lift slope of the horizontal tail as affected by the fuselage vs. aspect ratio of the horizontal tail for several relative fuselage widths bp/bh, from Koloska.

because of the induced downwash velocity behind the wing. The relationship between the angle of incidence of the horizontal tail aH and that of the wing a is given by Eq. (7-4); the change of the angle of incidence of the horizontal tail with the angle of attack of the airplane is given by Eq. (7-7). In general, the coefficient daw/da is negative (daw/da< 0) and, for a given wing, depends only on the position of the tail surface. The coefficient дан/да acts as an efficiency factor of the horizontal tail [Eq. (7-6); see also Eq. (7-13)]. Its value is usually between 0 and 1 and signifies that the downwash reduces the stabilizing effect of the horizontal tail.

The aim of the remainder of this section is the determination of this efficiency factor as a function of geometric and aerodynamic, data of the wing and of the position of the horizontal tail relative to the wing.

The induced downwash velocity is generated by the vortex system of the wing (bound and free vortices). Figure 7-11 illustrates schematically the vortex system of a given circulation distribution. Figure 7-1 la shows the free, not yet rolled-up vortex sheet, whereas in Fig. 7-1 lb the free vortex sheet is rolled up into two single vortices at a certain distance behind the wing.

A plane vortex sheet as in Fig. 7-1 Ід is unstable and tends to roll up into two single vortices (see also Figs. 3-8 and 3-22). From the known vortex system of a wing, the field of the induced downwash velocities is obtained with the Biot-Savart law. The vortex system of a given wing is obtained from the lift distribution as described in Sec. 3-3. In Fig. 7-12, the induced downwash and upwash angles on the longitudinal axis (x axis) are shown for an elliptic wing without twist. The induced downwash angle aw is referred to the induced angle of attack at = cLjnA of the wing by Eq. (3-3ід). The ratio aw/q is dependent on the angle of attack of the

wing. The relative downwash angle OLwlat is given as a function of the dimensionless longitudinal coordinate £=jc/s. The solid curve represents the induced downwash angle of the total vortex system (bound and free vortices) from Eq. (3-96), and the dashed curve represents the contribution of the free vortices. The difference between the solid and the dashed curves is the contribution of the bound vortices. This latter contribution becomes meaningless for £> 1. For such distances of the tail surface, the induced downwash angle is determined predominantly by the contribution of the free vortices, so that

aw = —2on (£ -> со) (7-20)

To obtain the induced downwash angle at the location of the horizontal tail, the position of the horizontal tail relative to the vortex sheet must also be known. Here it must be realized that, in general, the vortex sheet behind the wing lies neither in the wing plane nor in a plane parallel to the incident flow direction. Its shape is curved as shown in Fig. 7-13. Its distance from the wing chord z — 0 is given by zx{x, y). Because of the kinematic flow condition, the vortex sheet at the wing trading edge xr is tangent to the wing plane; farther downstream it is deflected more and more upward from the wing plane. Its position may be easily determined from the equation

(7-21)*

whose validity is obvious from Fig. 7-13. The location of the wing trailing edge is given by xr(y). Once the position of the vortex sheet is found, the distance of the horizontal tail from the vortex sheet, needed to determine the induced downwash angle at the location of the horizontal tail, is given by (z — zx).

Now, by means of theoretical results and measurements, we shall discuss the influence of the wing shape and of the lift distribution on the distribution of the downwash angle behind the wing.

For the not-rolled-up vortex sheet with a given circulation distribution Г(у) = bUooj(y), the downwash angle at z — zx is obtained from lifting-line theory by the Biot-Savart law from Eqs. (341), (3-50д), and (3-50h) as

(7-22)

constant.

Figure 7-13 Position of the vortex sheet behind the wing (schematic).

Here £=x/s, ri=y/s, and $=z/s are the dimensionless coordinates, and I/ = |/(У) gives the location of the lifting line in the wing from Fig. 3-29. For unswept wings the coordinate origin lies on the lifting line and = 0. For the numerical evaluation of this equation, a quadrature procedure has been developed by Multhopp [25]. Other computational methods and results have been published by Glauert [11], Lotz and Fabricius [21], and Helmbold [21].

The effect of the lift distribution on the downwash distribution in the plane of symmetry of the wing (77 = 0) and in the vortex sheet f = fi is shown in Fig. 7-14 for the rectangular, elliptic, and parabolic lift distributions. The downwash angle aw is referred to the induced angles of attack a( in the middle of the wing (7? = 0), whose values are also given in the figure. Hence, for ah three lift distributions the ratio aw/az = 2 far downstream of the wing. This result, which has been given in Eq. (7-20), is obtained by setting i–*00 in Eq. (7-22) and comparing with Eq. (3-71c). The curves of Fig. 7-14 demonstrate that the kind of lift distribution over the wing span has a considerable influence on the values of the downwash angle at small distances from the wing. For a constant circulation distribution the downwash is expressed by the simple formula

-(f, 0) = (1 + tli+ij IL. (Г = const) (7-23)

where cLl2nA= az-(0). This formula is obtained from Eq. (7-22), but also directly from the horseshoe vortex by means of the Biot-Savart law. For the elliptic circulation distribution the downwash angle becomes, according to Glauert [11],

– <*„(!, 0) = (l + -| – в) ^ (Г = r0V 1 – n2) (7-24)

where E is the complete elliptic integral of the second kind with the module 1/Vl2 + 1. In the present case, аг-(0) = cLln/i. The downwash angle at some distance behind the wing is given by an approximation formula of Truckenbrodt [25] as

This last expression applies to elliptic circulation distributions with a* = cLliu. The result of this formula is added in Fig. 7-14 as an approximation.

The effect of the wing planform on the distribution of the downwash angle over the span at a distance £ = 1 behind the wing is shown in Fig. 7-15. The three wings have an aspect ratio A — 6 and taper ratios X = 1.0, 0.6, and 0.2. This figure shows that the shape of the wing planform decisively affects the distribution of the downwash angle over the span. Hence the effectiveness of the horizontal tail is much smaller for a highly tapered trapezoidal wing than for a rectangular wing.

The solid curves were determined by a computational procedure of Multhopp [25], whereas the dashed curves were computed using the approximation formula Eq. (7-25a),

Figure 7-16 shows the effect of the sweepback angle on the distribution of the downwash angle behind the wing. For simplicity, constant circulation distribution over the span has been assumed for all those sweepback angles. The distribution of the downwash angle over the longitudinal axis shows that the downwash is much

greater at a backward-swept wing than at a forward-swept wing. The Biot-Savart law leads to the following simple formula for the downwash distribution:

where clI2ttA = a*-(0). Systematic measurements on the downwash of swept-back wings have been conducted by Trienes [40] by the probe surface method. Note also the investigations of Silverstein and Katzoff [38] and of Alford [2].

The results obtained so far were based on the flow with a not-rolled-up vortex sheet. A few data will now be given of the influence of the vortex sheet roll-up on the downwash at the location of the horizontal tad. As has been described by Fig.

7- lib and in more detail in Sec. 3-2-1, the vortex sheet rolls up into two single vortices at some distance behind the wing. They have the circulation Г0 of the root section of the wing, and, from Eq. (3-58), are apart by b0 far behind the wing. In Fig. 7-17, the ratio bQjb is plotted against the aspect ratio for a rectangular wing according to Glauert [11]. For an elliptic circulation distribution the ratio is constant:

For rectangular wings, b0/b increases from this value when the aspect ratio A becomes larger. For very large A, it approaches unity asymptotically, which is the value of the constant circulation distribution. A simpler computation of the downwash at a rolled-up vortex sheet is possible by considering a horseshoe vortex as in Fig. 7-17 of strength Г0 whose free vortices have the distance bQ. This quite idealized picture of the roll-up process has not been fully confirmed by measurements of Rohne [16], as seen from Fig. 7-18. Here, the ratio b0/b and the distance a0 behind the wing at which the rolling-up process has been completed

Figure 7-17 Aerodynamics of the rolled – up vortex sheet behind a wing (sche­matic). Ratio b0lb vs. aspect ratio of the wing л. Rectangular wing from [11].

have been plotted against the lift coefficient. The measured ratio b0jb is noticeably larger than the theoretical value of Fig. 7-17. A summary report on early downwash measurements is given by Fliigge-Lotz and Kuchemann [8].

Studies of the physical explanation of the roll-up process were first made by Kaden [16] and Betz [16], somewhat later by Kaufmann [16] and Spreiter and Sacks [39]. More recently, additional insight has been gained, to some extent, through the use of efficient computers [3, 4, 12, 30, 42]. To convey a feeling for

the magnitude of the effect of the wing on the horizontal tail, the efficiency factor of the horizontal tail from Eq. (7-7) is plotted in Fig. 7-19 against the aspect ratio.

for wings with elliptic circulation distributions. With the value for the lift slope of Eq. (3-98), the efficiency factor of the horizontal tail for not-rolled-up vortex sheets becomes

For the rolled-up vortex sheet (horseshoe vortex) it is

(7-286)

with 60/6 = 7r/4 from Eq. (7-27). At small wing aspect ratios, the efficiency factor of the horizontal tail is relatively small; it increases strongly with A.

All the results on downwash obtained so far apply to control points in the vortex sheet. The horizontal tail lies, depending on the angle of attack of the airplane, in, above, or below the vortex sheet. Outside the vortex sheet the downwash is always smaller than in the sheet. This will be shown by the following examples. Before pursuing this matter, however, the position of the vortex sheet (Fig. 7-13) will be discussed. With the help of Eq. (7-21), the position of the vortex sheet is obtained from the distribution of the downwash behind the wing. In Fig.

7- 20 the position of the vortex sheet in the root section t? = 0 behind the wing is shown for an elliptic wing. The distance between vortex sheet and the wing plane is proportional to the angle of attack of the wing. For the downwash angle outside of the vortex sheet, the following equation is obtained for a given circulation distribution by generalization of Eq. (7-22) according to lifting-line theory:

where r = V(| — %i)2 + (v — ri)2 + (f — )2

Figure 7-19 Efficiency factor of the hori­zontal tail bdfjjba in incompressible flow vs. aspect ratio of the wing for rolled-up and not-rolled-up vortex sheets. Computation from lifting-line theory for elliptic circula­tion distribution at a very large distance behind the wing (£ ->• °°).

Figure 7-20 Position of the vortex sheet behind elliptic wings of several aspect ratios л (see Fig. 7-13).

The quantities used in this equation are defined in connection with Eq. (7-22). Equation (7-29) is converted into Eq. (7-22) by f ^. According to Multhopp [25], the change in downwash with distance from the vortex sheet is given by

g«w _ _ C — Ci d2y

wli-b 1C — Cl! dv2

Thus the curves of the downwash angle aw against the distance from the vortex sheet have, in general, a break at the station of the vortex sheet. Experimental results of this kind for unswept and swept-back wings are plotted in Fig. 7-21, from Trienes [40]. They have been obtained by the probe surface method, which is

described in [40] and, therefore, are mean values of the downwash angle aw over the span of the horizontal tail surface. These experimental results confirm that the downwash angle has a peak value in the vortex sheet. Finally, in Fig. 7-22, theoretical downwash distributions from Glauert [11] are included for the transverse plane far behind the elliptic wing. They show that, for any high posi­tion, downwash prevails within the wing span range and upwash outside this range.

To compute the downwash in the vortex sheet, as pointed out above, a quadrature method based on lifting-line theory has been given by Multhopp [25]. An extension of this quadrature method for the computation of the downwash outside the vortex sheet has been developed by Gersten [10] for both the theories of the lifting line and of the lifting surface.

The induced downwash velocity according to lifting-surface theory is obtained from the velocity potential of Eq. (3-46), where w = дФ/dz, as

Here, Gi is the expression of Eq. (3-47), and

In analogy to the lifting-surface method of Sec. 3-3-5, Gersten [10] based the evaluation of Eq. (7-31) on two fundamental functions for the vortex density k. In this way he succeeded in developing a relatively simple computational procedure to determine the downwash.

Stabilization by the horizontal tail (neutral-point displacement) This discussion of the downwash will now be concluded with a simple reflection on the displacement of the neutral point of the airplane caused by the horizontal tail xNH (see Fig. 7-6). The analytical expression for this quantity has been given by Eq. (7-13). Let the wing and horizontal tail be of elliptic planform and the distance between the two neutral, points be rHN. The aerodynamic coefficients in Eq. (7-13) have already been discussed in detail. The lift slope of the airplane without horizontal tail (dcLlda)ofj is taken to be equal to that of the wing according to Eq. (3-98). The lift slope of the horizontal tail without interference has been given in Eq. (7-18) and the efficiency factor of the horizontal tail (1 + daw/da) in Eq. (7-28д). Under the assumption that = 1, introduction of these expressions into Eq. (7-13)

yields, after some intermediate steps,

Here

Equation (7-33) expresses a remarkably simple relationship between the neutral – point shift caused by the horizontal tail and the four geometric parameters: aspect ratio of the wing A and of the tail surface Ah, respectively; ratio of the areas of horizontal tail and wing AHfA and distance between the neutral points of the tail surface and the wing гНн – This relationship is shown in Fig. 7-23. In this diagram is also shown the neutral-point displacement that would be obtained without interference. It is computed, for simplicity, by the stripe method, in which the lift slopes of wing and horizontal tail are set equal to 2rr. This case is obtained from Eq. (7-33) witha^ = aH – 1 as

Figure 7-23 Neutral-point displacement caused by the horizontal tail of wing-horizontal tail systems vs. the area ratio Apj/A, from Eq. (7-33). Stripe method from Eq. (7-35).

The difference between this curve and the others indicates the interference effect of the wing on the horizontal tail with respect to the neutral-point displace­ment, including the influences of the finite aspect ratios of wing and tail surface.

Stability at nose-high flight attitude (stall) When an airplane gets into the nose-high flight attitude, safety requires that the pitching-moment curves in this range still be stable (dcM/dcL <0). For many wing shapes, for example, swept-back wings of large aspect ratio, this condition is not fulfilled. There are a number of measures, such as, for example, boundary-layer fences and slat wings, that lead to a wing stall behavior ensuring that no nose-up (tail-heavy) pitching moment (pitch up) can occur. Particular attention must be paid to the effect of the downwash as changed by the partial flow separation from the wing on the horizontal tail. Besides the wing planform, the position of the horizontal tail relative to the wing plays an important role, and particularly the high position of the tail surface. Furlong and McHugh [9] give a detailed report on this problem.

Severe stability problems can arise, particularly for swept-back-wing airplanes with a tail surface in extreme high position (T fin) at very large angles of attack. Here the horizontal tail lies in the separated flow of the wing, and its incident flow has a very low velocity. This leads to an unstable action and an almost complete loss of maneuverability. Then the angle of attack increases more and more until, eventually, at a very large angle of attack, a stable – flight attitude is again established. Because of the lack of control effectiveness, it is impos­sible to change this extreme flight attitude, and the airplane is in danger of crashing. This flight attitude is termed “super-stall” or “deep stall.” Byrnes et ai. [6] have studied this problem in detail.

7- 2-1 Contribution of the Horizontal Tail to the
Aerodynamics of the Whole Airplane

Airplane in straight flight The lift acting on the horizontal tail adds considerably to the pitching moment of the whole airplane because of its large lever arm compared

to the wing chord (see Fig. 7-1). Let Lя be the lift of the horizontal tail and rH the distance of this lift force from the moment reference axis (usually the lateral axis through the wing center of gravity). Then, from Fig. 7-5, the contribution of the horizontal tail to the pitching moment of the whole airplane is

MH = – r’„LH (7-1)

where the nose-up pitching moment is taken as positive. Here the contribution of the tangential force of the horizontal tail to the pitching moment has been disregarded because of the small high position of the tail surface relative to the fuselage axis. For the contributions of the horizontal tail to the lift LH and to the pitching moment MH, dimensionless coefficients are introduced through

Lh = сшАнЧн (7-2я)*

Мн = cMHAc^qm (7-2Z?)

Here qH is the dynamic pressure at the location of the tail surface. It is, in general, smaller than the dynamic pressure of the undisturbed flow qx because of the effect of the wing on the tail surface. The moment coefficient of the tail surface referred to the wing quantities is obtained from Eqs. (7-l)-(7-2h) as

The lift coefficient of the horizontal tail сщ depends on, in addition to the geometric data, its angle of attack ccH and the elevator deflection 1?Я (see Fig. 7-6a). The term dclH/daH represents the lift slope of the horizontal tail without interference, and (0ая/Эря)т? я the change in the direction of the horizontal tail for zero lift caused by the elevator deflection. For the plane problem of the airfoil with control surface (flap), this coefficient has been given as a function of the control-surface chord ratio; for additional information see Chap. 8.

Generally, the incident flow direction of the horizontal tail is considerably different from that of the wing because the tail surface is strongly influenced by the wing and fuselage and lies in the wing downwash (interference). The incident flow directions of the wing and horizontal tail differ, as shown in Fig. 7-6я, by the downwash angle aw = w/U^, induced by the wing and fuselage at the location of

*Note that the index l has been chosen for the lift coefficient with reference to the tail-surface quantities.

Figure 7-5 Contribution of the horizontal tail to the pitching moment (schematic). C. G. = center of gravity of the airplane. W = weight of the airplane.

Figure 7-6 Aerodynamics of a horizontal tail in straight flight. WJ5 = geometric neutral point, N = aerodynamic neutral point, (N2S )д = geometric neutral point of horizontal tail, (a) Incident flow direction of the horizontal tail, ад = а + Єд + а у/, (b) Aerodynamic forces on the wing and horizontal tail.

the tail surface. Here, w< 0 means downwash and w> 0 means upwash. The angle of attack of the horizontal tail thus becomes

&h — + єн + №w

where Єд is the setting angle of the horizontal tail relative to the wing chord and a is the angle of attack of the wing. Hence the contribution of the horizontal tail to the pitching moment at zero elevator deflection becomes

where r’H is the distance of the neutral point of the horizontal tail from the

moment reference axis of the airplane. The change of the moment with the angle of attack at fixed setting angle of the horizontal tail (stability coefficient) is then obtained as

The quantity

is termed the efficiency factor of the horizontal tail. For the moment change with setting angle of the horizontal tail at constant angle of attack, Eq. (7-5) yields

(7-8)

Comparison of Eqs. (7-6) and (7-8) shows that the moment change with angle of attack (stability contribution of the horizontal tail) depends on the interference between the wing and the horizontal tail. It is proportional to the efficiency factor of the horizontal tail, дан/да = (1 + 9aw/3a). The efficiency factor of the horizontal tail is generally considerably less than unity, as will be shown more accurately later. The moment change with setting angle of the horizontal tail (control), however, is not affected by the interference if the ratio ЯнІЯ<*> is disregarded.

To establish the contribution of the horizontal tail to the lift of the whole airplane, it is advantageous to define the lift coefficient of the horizontal tail, in analogy to Eq. (7-2b), as

Lh – clhM

In analogy to Eq. (7-5),

(7-10)

Here the comments made in connection with Eq. (7-5) apply also to the derivatives of сїн with respect to a and єн.

In the investigations made so far of the contribution of the horizontal tail to the pitching moment and the lift of the whole airplane, the respective coefficients have been established as functions of the angle of attack of the airplane and the setting angle of the horizontal tail. For some problems it is more favorable, however, to establish the contribution of the tail surface to the angle of attack and to the pitching moment as a function of the lift coefficient of the whole airplane and of the setting angle of the horizontal tail.

The lift coefficient of the whole airplane is composed of that of the airplane without the horizontal tail Ci0H, and the contribution of the horizontal tail cLH, that is, Ci = cLoH + cLff. Hence the lift slope of the whole airplane, without consideration of the effect of the tail surface on the wing at fixed setting angle of the horizontal tail, is obtained from Eq. (7-10) as

The sought change of the angle of attack with the lift coefficient of the whole

The change of the angle of attack with the setting angle of the horizontal tail zH at constant lift coefficient of the whole airplane becomes

(7-12)*

Here the second factor is given by Eq. (7-10).

Like the wing alone, the whole airplane has a neutral point, that is, a point on which that portion of the lift force of the whole airplane acts that is proportional to the angle of attack (compare Sec. 1-3-3). As shown in Fig. 7-6b, let the distance of the neutral point of the whole airplane from the neutral point of the airplane without the tail unit be designated as xNH. This distance is identical to the neutral-point displacement caused by the horizontal tail. This neutral-point displacement xNH, as shown in Fig. 7-6b, can be determined from the moment equilibrium about the neutral point N0H of the airplane without the tail unit xjyHL — rHN^H – Here rHN is the distance of the neutral point of the horizontal tail from the neutral point of the airplane without a horizontal tail. The result is

Introducing Eqs. (7-10) and (7-11) into this equation finally leads to the neutral-point displacement caused by the horizontal tail,

(7-13)

In this equation, the first fraction on the right-hand side determines the percentage of the tail-surface lever arm by which the airplane neutral point is shifted rearward relative to the neutral point of the airplane without a horizontal tail. In many cases, the second term of the sum of the denominator can be disregarded in comparison with the first term. The neutral-point position of the whole airplane is obtained from the neutral-point position of the wing alone (Chaps. 3 and 4), from the neutral-point displacement caused by the fuselage (including the wing-fuselage interference, Chap. 6), and from the neutral-point displacement caused by the tail surface as given above.

The change of the moment coefficient of the horizontal tail as a function of the lift coefficient from Eq. (1-29) is obtained immediately from the neutral-point displacement caused by the horizontal tail in the form

with bcJbEfj – ЭсЕЯ/Э£Я-

Finally, the change of the moment with the setting angle of the horizontal tail may be determined for a fixed lift coefficient of the whole airplane. This is a free moment because the total lift remains constant. As shown in Fig. 7-66, MH = —t’hnLh, where LH is the lift of the horizontal tail caused by the change of the setting angle of the horizontal tail, and r’HN is the distance between the neutral points of the horizontal tail and of the whole airplane. Thus, observing Eq. (7-10) and with rHN = rHN — xNH, the following relationship is obtained:

(7-15)

This relationship is also valid for (dcMldeH)CL-const. The two coefficients of Eqs. (7-14) and (7-15) can be taken from Fig. 7-26, the first as the difference of the slopes of the curves cM(c£) with and without tail surface and the second from the curves with different setting angles zH of the horizontal tail.

To evaluate the above equations for the contribution of the horizontal tail to the lift and the moment, attention must be paid to the ratio qHlq„. Special attention must be paid, however, to the angle of incidence aH of the horizontal tail, because it depends strongly on the interference between the wing and the tail surface. The angle of incidence of the horizontal tail aH is decisively affected by the induced downwash angle aw < 0 caused by the wing at the location of the horizontal tail [see Eq. (74)]. In Fig. 1-la, the change of aH with the rearward position of the tail surface is shown under the assumption of a horizontal tail chord parallel to the wing chord (eh = 0). At the wing trailing edge, aH = 0 because here the kinematic flow condition requires that a + aw = 0. With increasing distance

from the wing trailing edge, aH increases and assumes a constant value at a large distance that is considerably smaller than a. The distribution of aw and thus of aH behind the wing can be computed with wing theory. This matter will be discussed below.

The lift slope of the horizontal tail dc[HjdaH for a horizontal tail free of interference may be determined with wing theory.

Airplane in pitching motion So far only the influence of the airplane angle of attack on the aerodynamics of the horizontal tail has been considered. In addition, however, the rotational motion of the airplane about the lateral axis is particularly important for the aerodynamics of the horizontal tail. During rotation of the airplane about the lateral axis with angular velocity ojy, an angle-of-incidence distribution aH of the horizontal tail is created as shown in Fig. 1-lb that increases linearly with distance from the axis of rotation. This angle of incidence at the location of the horizontal tail (three-quarter point) becomes, with the distance from the axis of rotation rH,

(7-16a)

into Eq. (7-36) and the resulting formula into Eq. (7-3a), the change of the moment coefficient with the pitching angular velocity is obtained, with as

(7-17)

This coefficient is termed the contribution of the horizontal tail to the pitch damping. Comparison of this formula with Eq. (7-6) shows that the contribution of the horizontal tail to the stability is proportional to (Ая/А)(^/см), and that to the damping is proportional to {AHjA){r’HICfj)2.

The geometry of the horizontal and the vertical tails may be described basically like that of a wing (see Sec. 3-1). In general, the horizontal tail has a symmetric planform and the vertical has an asymmetric side elevation (Fig. 7-1).

The planform of the horizontal tail is defined, in analogy to Sec. 3-1, by the following main quantities:

Span of the horizontal tail bH

Area of the horizontal tail AH

Aspect ratio of the horizontal tail AH = bjj/Ajj

Setting angle of the tail plane (Fig. 7-6a) £H

Deflection of the elevator (Fig. 7-6a) rH

The position of the horizontal tail relative to the airplane is given by the lever arm Tfj of the horizontal tail, defined as the distance between the geometric neutral points of the horizontal tail and the wing. The geometric neutral point is defined in Sec. 3-1.

For some airplanes, the high position of the horizontal tail relative to the wing plays some role.

For the aerodynamic effects of the horizontal tail, the following two dimensionless quantities, which express size and location of the horizontal tail relative to the wing quantities, are particularly important: area ratio AHjA and relative tail-surface distance гн/сц. Here, A is the wing area and сц is the reference wing chord according to Eq. (3-5Й).

For a large number of airplanes, the area ratio lies between AHfA «0.15 and 0.25 and the relative tail distance between rH/c^ « 2 and 3.

The side elevation of the vertical tail is described by the following quantities (Fig. 7-1):

Height of the vertical tail h v Area of the vertical tail A v Deflection of the rudder 7]v

The location of the vertical tail relative to the airplane is given by the lever arm rv of the vertical tail, defined as the distance between the geometric neutral points of the vertical tail and the wing. A general definition of the aspect ratio of the vertical tail is not feasible because of the great variety of tail-surface shapes and the various positions of the vertical tail relative to the fuselage and to the horizontal tail (see Sec. 7-3-2).

For the aerodynamic effect of the vertical tail the following two dimensionless quantities are important, as for the quantities for the horizontal tail: area ratio Ay IA and relative tail-surface distance r-yjs, where s=b/ 2 is the wing semispan. Approximately, AvjA = 0.1-0.2 and rv]s = 0.5-1.0.

On many newer airplanes, the horizontal tail has been eliminated so that the airplane has only a vertical tail as shown in Fig. 7-3. Such an airplane is termed an all-wing airplane (flying wing). Here the function of the elevator (control about the lateral axis) has been assigned to a control surface (elevator) of width bH.

Besides the most commonly used central arrangement of the vertical tail as shown in Figs. 7-1 and 7-3, various other arrangements are also found. For instance, Fig. 7-4a shows two fins (vertical tail surfaces) at the tips of the horizontal tail. Figure 7-4b illustrates a tail surface with large dihedral (V tail surface), combining the functions of both the horizontal and the vertical tails.

Figure 7-3 The geometry of the empennage of an all-wing airplane.

For the aileron, to be discussed in more detail in Chap. 8, the aileron span sд as shown in Figs. 7-1 and 7-3 is important in addition to the aileron chord ratio.