Consider supersonic flow above a rigid-body surface that remains flat everywhere upstream of point O but then starts to bend up as shown in Figure 4.31. According to the Prandtl–Meyer theory,...

Consider supersonic flow above a rigid-body surface that remains flat everywhere upstream of point O but then starts to bend up as shown in Figure 4.31. According to the Prandtl–Meyer theory, deceleration of the flow over a curved part of the wall should be observed, causing the characteristics of the first family to converge. Consider two of them, namely the characteristic emerging from point O and the neighbouring characteristic emerging from point O′ situated a small distance ∆x downstream of O. They intersect at point C; see Figure 4.31. Using  s (4.4.48), (4.4.49), and (4.4.34), show that the distance d between points O and C may be calculated as d = 2(M2 − 1) (γ + 1)M3θ ′ w(0). Here θw(x) is the angle made by the tangent to the body contour with the x-axis and M is the Mach number in the uniform flow upstream of OC. Assume that the derivative θ ′ (x) of the wall slope angle θx(x) is finite at point O.