Topic: Fluid Mechanics
Critical to Stagnation Pressure ratio for an expansion process is
- A. $\frac{P^*}{P_0} = \left[\frac{2}{\gamma+1}\right]^{2\gamma/(\gamma-1)}$
- B. $\frac{P^*}{P_0} = \left[\frac{2}{\gamma+1}\right]^{(\gamma+1)(\gamma-1)}$
- C. $\frac{P^*}{P_0} = \left[\frac{2}{\gamma+1}\right]^{\gamma/(\gamma-1)}$
- D. $\frac{P^*}{P_0} = \left[\frac{2}{\gamma+1}\right]^{\gamma/(\gamma+1)}$
Answer: C. $\frac{P^*}{P_0} = \left[\frac{2}{\gamma+1}\right]^{\gamma/(\gamma-1)}$
Explanation: At the critical state where the Mach number is 1, the isentropic relation for the temperature ratio is $\frac{T^*}{T_0} = \frac{2}{\gamma+1}$. Combining this with the isentropic relationship between pressure and temperature, $\frac{P^*}{P_0} = \left(\frac{T^*}{T_0}\right)^{\frac{\gamma}{\gamma-1}}$, directly gives the pressure ratio as $\left[\frac{2}{\gamma+1}\right]^{\frac{\gamma}{\gamma-1}}$.