A supersonic flow in a constant area duct at Mach number $M_1$ encounters a compression ramp of angle $\theta_1$ generating an oblique shock wave. Which of the following statements is TRUE across the shock wave?
Options:
(A) Total pressure increases and static temperature decreases
(B) Static pressure increases and total pressure decreases
(C) Mach number increases and density decreases
(D) Static enthalpy decreases and entropy decreases
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2019 Q.37 / AE Q.27)**
**1. Intuitive Physical Setup & Given Parameters**:
- Supersonic flow past a compression ramp generates an oblique shock wave.
**2. Step-by-Step Physical Derivation**:
1. **Shock Wave Conservation Principles**:
- An oblique shock wave is an irreversible, dissipative compression process.
2. **Static Property Changes across Shock**:
- Static pressure increases ($P_2 > P_1$).
- Static temperature increases ($T_2 > T_1$).
- Static density increases ($\rho_2 > \rho_1$).
- Downstream Mach number decreases ($M_2 < M_1$).
3. **Stagnation Property Changes**:
- Total temperature is constant in adiabatic flow without work addition: $T_{02} = T_{01}$.
- Entropy strictly increases due to viscous and thermal dissipation within the shock thickness: $\Delta s > 0$.
- By the thermodynamic relation $\Delta s = -R \ln(P_{02}/P_{01})$, total pressure strictly decreases:
P_{02} < P_{01}
4. Hence, static pressure increases while total pressure decreases.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** (total pressure increases): ❌ Incorrect. Total pressure can never increase across a shock (violates Second Law).
- **(B)** (static pressure increases and total pressure decreases): ✅ **CORRECT**. Fundamental signature of all shock waves.
- **(C)** (Mach number increases): ❌ Incorrect. Mach number always drops across a compression shock.
- **(D)** (entropy decreases): ❌ Incorrect. Violates Second Law of Thermodynamics.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **B**
- **Exam Rule**: Across any shock wave: $P \uparrow, T \uparrow, \rho \uparrow, s \uparrow$, while $M \downarrow$ and $P_0 \downarrow$.