# The Horizontal Tail in Incompressible Flow

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 . 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 . 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 .

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  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 , 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

 Figure 7-11 The vortex system behind a wing (schematic), (a) Not-iolled-up vortex sheet, (b) Rolled-up vortex sheet.

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)

 Figure 7-12 Induced downwash angle aw on the x axis of a wing of elliptic planfoim, from . Contribution of free and bound vortices.

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).

 Figure 7-14 Downwash angle in the vortex sheet (f = ft) for 17 = 0 (plane of symmetry of the airplane) behind unswept wings, from Truckenbrodt, computed by lifting-line theory. Curve 1, constant circulation distribution. Curve 2, elliptic circulation distribution. Curve 3, parabolic circulation distribution.

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 . Other computational methods and results have been published by Glauert , Lotz and Fabricius , and Helmbold .

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 ,

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

 (7-25д) (7-25*)

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  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 , 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  by the probe surface method. Note also the investigations of Silverstein and Katzoff  and of Alford .

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 . 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 , 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 .

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 .

Studies of the physical explanation of the roll-up process were first made by Kaden  and Betz , somewhat later by Kaufmann  and Spreiter and Sacks . 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.

 Э*н _ 1 , Sccw _ V/T – – f 4 – 2

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 , 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 . They have been obtained by the probe surface method, which is

described in  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  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 . An extension of this quadrature method for the computation of the downwash outside the vortex sheet has been developed by Gersten  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  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  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.  have studied this problem in detail.