# The Horizontal Tail in Supersonic Incident 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.