Consider a centered Prandtl-Meyer expansion fan at a \(\theta = 4^{\circ}\) corner in a Mach \(1.78\) air flow, as shown in the figure below. The angle \(\psi\) (see figure) made by the ending wave of the fan with respect to the incoming stream is _______ degrees (rounded off to 1 decimal place).
An excerpt from the table of Prandtl-Meyer function for air is provided below.
M
\(\nu\) [deg]
1.72
18.40
1.74
18.98
1.76
19.56
1.78
20.15
1.80
20.73
1.82
21.30
1.84
21.88
1.86
22.45
1.88
23.02
1.90
23.59
1.92
24.15
1.94
24.71
1.96
25.27
1.98
25.83
2.00
26.38
Show Hint
Use \(\nu(M_2) = \nu(M_1) + \theta\) to find the downstream Mach number, then note the ending wave sits at the Mach angle \(\mu_2\) measured from the downstream (turned) flow direction, so \(\psi = \mu_2 - \theta\) measured from the incoming stream.
Step 1: Set up a common reference direction.
Measure all angles from the original upstream flow direction, taking the sense in which the corner turns the flow (away from the extended upstream line, following the wall) as negative. So the downstream flow direction sits at $-\theta$ from the reference.
Step 2: Get the downstream Mach number from the table.
$\nu_1$ at $M_1=1.78$ is $20.15^{\circ}$ from the table. Adding the turn angle, $\nu_2 = 20.15+4 = 24.15^{\circ}$, which the table lists exactly at $M_2 = 1.92$.
Step 3: Place each Mach wave using this signed-angle convention.
Each Mach wave leans at the local Mach angle $\mu$ from its local flow direction, on the side away from the wall (the positive side). The very first wave of the fan forms right where the flow is still at the upstream direction and Mach number, so it sits at $0 + \mu_1 = \mu_1$ from the reference. The very last wave forms once the flow has fully reached the downstream direction and Mach number, so it sits at $(-\theta) + \mu_2$ from the reference:
\[ \mu_2 = \arcsin(1/1.92) = 31.4^{\circ} \]
\[ \psi = \mu_2 - \theta = 31.4 - 4 \]
Final Answer:
\[ \boxed{\psi \approx 27.4^{\circ}} \]
