A convergent-divergent nozzle is operated at a pressure ratio such that an oblique shock wave is attached to the exit lip outside the nozzle. This nozzle is operating in:
Options:
(A) Under-expanded mode
(B) Critically expanded mode
(C) Over-expanded mode
(D) Subsonic mode
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2018 Q.27)**
**1. Intuitive Physical Setup & Given Parameters**:
- CD nozzle operating at supersonic exit Mach number ($M_e > 1$).
- Oblique shock wave occurs at the nozzle exit plane attached to the lip.
**2. Step-by-Step Aerodynamic Derivation**:
1. **Pressure Comparison**:
- An oblique shock wave compresses the flow, meaning pressure rises across the shock: $P_{behind} > P_{ahead}$.
- For an oblique shock to occur at the exit lip, the flow must compress to match the higher ambient pressure:
P_e < P_a
2. **Regime Identification**:
- When exit static pressure is lower than ambient pressure ($P_e < P_a$), the nozzle has expanded the fluid *too much* (past the ambient pressure). This is termed **over-expanded**.
- In over-expanded flow, oblique shocks form at the lip to compress the jet to ambient pressure.
- Conversely, if $P_e > P_a$, the flow is **under-expanded** and expansion fans form at the exit.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** Under-expanded: ❌ Incorrect. Produces expansion fans at the lip ($P_e > P_a$).
- **(B)** Critically / perfectly expanded: ❌ Incorrect. $P_e = P_a$, no shocks or expansion waves.
- **(C)** Over-expanded: ✅ **CORRECT**. $P_e < P_a$, produces oblique shocks at the lip.
- **(D)** Subsonic mode: ❌ Incorrect. No shocks can form outside in subsonic discharge.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **C**
- **Exam Rule**: $P_e < P_a \implies$ Over-expanded $\implies$ Oblique shocks at lip. $P_e > P_a \implies$ Under-expanded $\implies$ Expansion fans at lip.