Influence of the wing on the horizontal tail in supersonic incident flow For

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:

8oiw

da

= — (~R0<y<0)

7%

(7-44a)

daw

da

= -1(y > 0 and у <—i?0)

(7-44b)

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].

(7-46)[34]

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.

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