4- 5-1 Fundamentals of Wing Theory at Supersonic Flow

Mach cone (influence range) There is an essential physical difference between flows of subsonic and supersonic velocities, namely, that the disturbances of a sound point source in the former flow propagate in all directions, but in the latter flow only within a cone that lies downstream of the sound source (Fig. 1-9b and d). This so-called Mach cone has the apex semiangle fi, which, by Eq. (1-33), is given by

sin« = ~— and tan ц = ■■■ ■ ———- (4-80)

I/Ma*, – 1

with Маж = Uoojaoo. The state of affairs just discussed may also be interpreted (see Fig. 4-57) that a given point in a supersonic flow, U„ >#«, can influence only the space within the downstream cone, whereas it can itself be influenced only from the space within the upstream cone. Application of this basic fact of supersonic flow on a wing of finite span is demonstrated in Fig. 4-58. The flow conditions at a point x,

Figure 4-56 Profile drag coefficients vs. Mach number for an unswept and a swept-back wing Op = 45°), t/c = 0.12, a = 4.

y, z = 0 on the wing can be influenced only from the crosshatched area A’ of the wing that is cut out of the wing by the upstream cone. When the Mach line M. L. lies before the wing leading edge, as in Fig. 4-58, the area between this Mach line and the leading edge also contributes to the influence on point x, y, z — 0. Downstream, the influence range is bounded by the two Mach lines through the point x, y, z = 0.

Subsonic and supersonic edge The conditions of Fig. 4-57 find an important application in oblique incident flow on a wing edge. If, as in Fig. 4-59a, a Mach line lies before the wing edge, the component vn of the incident flow velocity £/«, normal to the edge is smaller than the speed of sound аю. Such an edge is termed subsonic edge. Conversely, if, as in Fig. 4-596, the Mach line lies behind the wing edge, then vn is larger than а*,. In this case, the edge is termed supersonic edge. With /г as the Mach angle and 7 as the angle of the edge with the incident flow direction (Fig. 4-59), the expression

m = – = tan у Ma^ — 1 (4-81)

Figure 4-57 Upstream cone and downstream cone of a point in supersonic flow. fj. = Mach angle.

Figure 4-58 Wing in supersonic incident flow. A’ = influence range.

allows one to determine whether the edge is subsonic or supersonic. Thus the edges are characterized as follows.

Subsonic edge: vn < a« /<« > 7 m < 1 ■ (4-82a)

Supersonic edge: ип>ак, іл<.ут>1 (4-82 b)

The special case 7 = 0° (m = 1) is a subsonic edge for all supersonic Mach numbers, and the case 7 = 90° (m — °°) is a supersonic edge. The concept of subsonic and supersonic edges is of significance not only for the leading edge, but also for the trailing and side edges. This fact is explained in Fig. 4-60. Here, the subsonic edges are drawn as dashed fines, the supersonic edges as solid fines. For the same wing planform, the Mach fines for three different Mach numbers are drawn. At the lowest


Figure 4-60 Example for the explanation of sub­sonic and supersonic edges of swept-back wings. Dashed lines: subsonic edges; solid lines: super­sonic edges, (a) Subsonic leading edge and sub­sonic trailing edge. (6) Subsonic leading edge and supersonic trailing edge, (c) Supersonic leading edge and supersonic trading edge.

Mach number (Fig. 4-60a), all edges are subsonic, at the highest Mach number (Fig.

4- 60c), the leading and trailing edges are supersonic, but the side edges are still subsonic. Distinction between subsonic and supersonic edges is conditioned by the difference in flow patterns in the vicinity of the edges. In Fig. 4-61, the various types of flow patterns are sketched, which are the sections normal to the leading and trailing edges, respectively. In close vicinity to the section plane, the flow may be considered to be approximately two-dimensional. The basically different character of subsonic and supersonic flows over an inclined flat plate was demonstrated in Fig. 4-22. Based on this figure, Fig. 4-61 shows the subsonic leading edge, at which flow around the leading edge is incompressible accord­ing to Fig. 2-9a. An essential characteristic of this flow is the formation of an upstream-directed suction force on the nose (see Fig. 4-22a). Figure 4-61 b shows the subsonic trailing edge with smooth flow-off according to the Kutta condition (see Sec. 2-2-2). At such a trailing edge, the pressure difference between the lower and upper surfaces is equal to zero (Fig. 4-22г). Complete pressure equalization between the lower and upper surfaces is achieved. In Fig. 4-6lc and d, the supersonic leading edge and the supersonic trailing edge, respectively, are shown. In both cases, neither flow around the edge nor smooth flow-off is achieved, but Mach lines originate at the edges along which the flow quantities change unsteadily. Between the lower and upper surfaces, a finite pressure difference exists (see Fig.

4- 22b).

Finally, the pressure distributions over a wing section are shown schematically for the three different cases of Fig. 4-60. For the section with subsonic leading and

Figure 4-61 Typical flow patterns at subsonic and supersonic edges (see Fig. 4-59). (a) Subsonic leading edge, vn < ctoo, flow around edge. (6) Subsonic trailing edge, ип<аю, smooth flow-off (Kutta condition), (c) Supersonic leading edge, vn > with Mach lines, (d) Supersonic trailing edge, vn > a«, with Mach lines.

trailing edges, Fig. 4-62^, the pressure distribution is similar to that of incompressible flow, as would be expected. The rear Mach line, however, causes a break in the pressure distribution. In the case of the section with supersonic leading and trailing edges (Fig. 4-6 2c), the pressures at the leading and trailing edges have finite values. The front Mach line again produces a break in the pressure distribution.

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