## Supersonic Flow Over Wedges and Cones

For the supersonic flow over wedges, as shown in Figures 9.10 and 9.11, the oblique shock theory developed in Section 9.2 is an exact solution of the flow field; no simplifying assumptions have been made. Supersonic flow over a wedge is characterized by an attached, straight oblique shock wave from the nose, a uniform flow downstream of the shock with streamlines parallel to the wedge surface, and a surface pressure equal to the static pressure behind the oblique shock p2. These properties are summarized in Figure 9.14a. Note that the wedge is a two-dimensional profile; in Figure 9.14a, it is a section of a body that stretches to plus or minus infinity in the direction perpendicular to the page. Hence, wedge flow is, by definition, two-dimensional flow, and our two-dimensional oblique shock theory fits this case nicely.

In contrast, consider the supersonic flow over a cone, as sketched in Figure 9.14b. There is a straight oblique shock which emanates from the tip, just as in the case of a wedge, but the similarity stops there. Recall from Chapter 6 that flow over a three-dimensional body experiences a “three-dimensional relieving effect.” That is, in comparing the wedge and cone in Figure 9.14, both with the same 20° angle, the flow over the cone has an extra dimension in which to move, and hence it more easily adjusts to the presence of the conical body in comparison to the two-dimensional wedge. One consequence of this three-dimensional relieving effect is that the shock wave on the cone is weaker than on the wedge; that is, it has a smaller wave angle, as compared in Figure 9.14. Specifically, the wave angles for the wedge and cone are 53.3 and 37°, respectively, for the same body angle of 20° and the same upstream Mach number of 2.0. In the case of the wedge (Figure 9.14a), the streamlines are deflected by exactly 20° through the shock wave, and hence downstream of the shock the flow is exactly parallel to the wedge surface. In contrast, because of the weaker shock on the cone, the streamlines are deflected by only 8° through the shock, as shown in Figure 9.14b. Therefore, between the shock wave and the cone surface, the streamlines must gradually curve upward in order to accommodate the 20° cone. Also, as a consequence of the three-dimensional relieving effect, the pressure on the surface of the cone, pc, is less than the wedge surface pressure P2, and the cone surface Mach number Mc is greater than that on the wedge surface М2. In short, the main differences between the supersonic flow over a cone and wedge, both with the same body angle, are that (1) the shock wave on the cone is weaker, (2) the cone surface pressure is less, and (3) the streamlines above the cone surface are curved rather than straight.

The analysis of the supersonic flow over a cone is more sophisticated than the oblique shock theory given in this chapter and is beyond the scope of this book. For details concerning supersonic conical flow analysis, see Chapter 10 of Reference 21. However, it is important for you to recognize that conical flows are inherently different from wedge flows and to recognize in what manner they differ. This has been the purpose of the present section.

Consider a wedge with a 15° half angle in a Mach 5 flow, as sketched in Figure 9.15. Calculate | Example 9.5 the drag coefficient for this wedge. (Assume that the pressure over the base is equal to freestream static pressure, as shown in Figure 9.15.)

Solution

Consider the drag on a unit span of the wedge D’. Hence,

D’ D’

From Figure 9.15,

D’ = 2p2l sin6? — 2pl sinfl = (21 sin$)(p2 — P)

(Note: The drag is finite for this case. In a supersonic or hypersonic inviscid flow over a two-dimensional body, the drag is always finite. D’Alembert’s paradox does not hold for freestream Mach numbers such that shock waves appear in the flow. The fundamental reason for the generation of drag here is the presence of shock waves. Shocks are always a dissipative, drag-producing mechanism. For this reason, the drag in this case is called wave drag, and cd is the wave-drag coefficient, more properly denoted as cdtW.)