Discussion
Air flowing at Mach number M = 2 from left to right accelerates to M = 3 across an expansion corner as shown in the figure. What is the value of δ (the angle between the Forward and Rearward Mach lines) in degrees?
The values of the Prandtl-Meyer functions are ν(3) = 49.76° and ν(2) = 26.38°.
Answer: D
The angle between the Mach lines is 33.91°.
The wall turns the flow through
\[ \theta = \nu(3) - \nu(2) = 49.76 - 26.38 = 23.38^{\circ} \]
The forward Mach line makes the Mach angle \(\mu_{1} = \sin^{-1}(1/2) = 30^{\circ}\) with the incoming flow. The rearward Mach line makes \(\mu_{2} = \sin^{-1}(1/3) = 19.47^{\circ}\) with the outgoing flow, which is itself turned 23.38° toward the wall. So
\[ \begin{aligned} \delta &= \theta + \mu_{1} - \mu_{2} \\ &= 23.38 + 30 - 19.47 = 33.91^{\circ} \end{aligned} \]
23.38° is the flow turning angle alone, 19.47° is the downstream Mach angle, and 53.38° adds the turning angle to the upstream Mach angle without subtracting the downstream one.
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