GATE 2025 AE – Question 46
An aircraft is flying at an altitude of 4500 m above sea level, where the ambient pressure, temperature and density are 57 kPa, 259 K and 0.777 kg/m$^3$, respectively. The speed of the aircraft ($V$) is 230 m/s. Gas constant $R = 287$ J/kg/K, and specific heat ratio $\gamma = 1.4$. If the stagnation pressure is $p_o$, and static pressure is $p$, the value of $\left[\frac{p_o - p}{\frac{1}{2}\rho V^2}\right]$ is ________ (rounded off to two decimal places).
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Correct answer: 1.11 to 1.13
Explanation
The speed of sound is $a = \sqrt{1.4 \times 287 \times 259} = 322.6$ m/s, so $M = \frac{230}{322.6} = 0.713$. The isentropic stagnation pressure ratio is $\frac{p_o}{p} = (1 + 0.2M^2)^{3.5} = 1.4034$. Then $p_o - p = 0.4034 \times 57000 = 22994$ Pa and $\frac{1}{2}\rho V^2 = 0.5 \times 0.777 \times 230^2 = 20552$ Pa. The ratio is 1.12 (it is 1 for incompressible flow).