# The Horizontal Tail in Subsonic Incident Flow

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„.

dccu = U*(l – Mai) + 4-2 8oi І Л2 (1 — Ma%) + 4 4- 2 |

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.