GATE 2026 AE – Question 64

Aerodynamics · Compressible flows: isentropic flow, normal and oblique shocks, Prandtl-Meyer flow, nozzles and diffusers · 2 marks · Numerical answer

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]	M	$\nu$ [deg]	M	$\nu$ [deg]
1.72	18.40	1.82	21.30	1.92	24.15
1.74	18.98	1.84	21.88	1.94	24.71
1.76	19.56	1.86	22.45	1.96	25.27
1.78	20.15	1.88	23.02	1.98	25.83
1.80	20.73	1.90	23.59	2.00	26.38

A flow turning away at a corner. The expansion fan spreads from the corner. $\theta$ is the wall deflection below the incoming flow direction and $\psi$ is the angle between the incoming stream and the last wave of the fan.

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Show answer and explanation

Correct answer: 27.26 to 27.54

Explanation

At $M_1 = 1.78$, $\nu_1 = 20.15^\circ$. After turning by $\theta = 4^\circ$, $\nu_2 = 24.15^\circ$, which corresponds to $M_2 = 1.92$. The last wave makes the Mach angle $\mu_2 = \sin^{-1}\frac{1}{1.92} = 31.4^\circ$ with the downstream flow, which is itself turned $4^\circ$ away from the incoming stream. So $\psi = \mu_2 - \theta = 31.4 - 4 = 27.4^\circ$.