__Oblique shocks and corresponding equations:__

A shock wave will make an oblique angle with respect to the upstream flow is called oblique shock wave. A normal shock wave is simply a special case of the general family of oblique shocks, namely, the case where the wave angle is 90°.

In addition to oblique shock waves, where the pressure increases discontinuously across the wave, supersonic flows are also characterized by oblique expansion waves, where the pressure decreases continuously across the wave. Let us examine these two types of waves further.

Consider a supersonic flow over a wall with a corner at point A, as sketched in Fig. 9.1. In Fig. 9.1a, the wall is turned upward at the corner through the deflection angle θ; i.e., the corner is concave. The flow at the wall must be tangent to the wall; hence, the streamline at the wall is also deflected upward through the angle θ. The bulk of the gas is above the wall, and in Fig. 9.1a, the streamlines are turned upward, into the main bulk of the flow. Whenever a supersonic flow is “turned into itself” as shown in Fig. 9.1a, an oblique shock wave will occur. The originally horizontal streamlines ahead of the wave are uniformly deflected in crossing the wave, such that the streamlines behind the wave are parallel to each other and inclined upward at the deflection angle θ. Across the wave, the Mach number discontinuously decreases, and the pressure, density, and temperature discontinuously increase. In contrast, Fig. 9.1b shows the case where the wall is turned downward at the corner through the deflection angle θ; i.e., the corner is convex. Again, the flow at the wall must be tangent to the wall; hence, the streamline at the wall is deflected downward through the angle θ. The bulk of the gas is above the wall, and in Fig. 9.1b, the streamlines are turned downward, away from the main bulk of the flow. Whenever a supersonic flow is “turned away from itself” as shown in Fig. 9.1 b, an expansion wave will occur. This expansion wave is in the shape of a fan centered at the corner. The fan continuously opens in the direction away

from the corner, as shown in Fig. 9.1b. The originally horizontal streamlines ahead of the expansion wave are deflected smoothly and continuously through the expansion fan such that the streamlines behind the wave are parallel to each other and inclined downward at the deflection angle θ. Across the expansion wave, the Mach number increases, and the pressure, temperature, and density decrease. Hence, an expansion wave is the direct antithesis of a shock wave.

Examining Fig. 9.3b, the Mach wave, i.e., the envelope of disturbances in the