The GATE Grind

GATE 2024 AE – Question 44

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

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 $\delta$ (the angle between the Forward and Rearward Mach lines) in degrees?

The values of the Prandtl-Meyer functions are $\nu(3) = 49.76^o$ and $\nu(2) = 26.38^o$.

An expansion fan at a corner. The flow comes from the left at $M = 2$ along a flat wall that turns down at the corner. $\delta$ is the angle between the forward Mach line (the first line of the fan) and the rearward Mach line (the last line, next to the downstream wall).
  1. 23.38
  2. 19.47
  3. 53.38
  4. 33.91

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) 33.91

Explanation

The flow turns by $\theta = \nu(3) - \nu(2) = 23.38^\circ$. The forward Mach line makes $\mu_1 = \sin^{-1}\frac{1}{2} = 30^\circ$ with the incoming flow. The rearward line makes $\mu_2 = \sin^{-1}\frac{1}{3} = 19.47^\circ$ with the downstream flow, which has turned by $\theta$ away from the incoming direction. So the rearward line is at $\mu_2 - \theta = -3.91^\circ$ from the original direction, and $\delta = 30 + 3.91 = 33.91^\circ$.