Topic: Fluid Mechanics
In an isentropic flow with a stagnation pressure of 6 MPa, find the static pressure, at the point where flow is sonic. (Assume $\gamma=1.4$; consider $1.2^{7}\simeq1.89^{2})$
- A. 2.17 MPa
- B. 3.17 MPa
- C. 1.58 MPa
- D. 2.58 MPa
Answer: B. 3.17 MPa
Explanation: At sonic condition ($M = 1$), the critical pressure ratio is given by $\frac{P^*}{P_0} = \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}}$.
$\frac{P_0}{P^*} = \left(\frac{\gamma+1}{2}\right)^{\frac{\gamma}{\gamma-1}} = \left(\frac{1.4+1}{2}\right)^{\frac{1.4}{0.4}} = (1.2)^{3.5}$.
Given $1.2^7 \simeq 1.89^2$, taking square root on both sides yields $1.2^{3.5} \simeq 1.89$.
Therefore, $\frac{P_0}{P^*} = 1.89 \Rightarrow P^* = \frac{P_0}{1.89} = \frac{6~MPa}{1.89} = 3.174~MPa$.