In a converging-diverging nozzle, Nozzle A displays oblique shock waves at the exit plane, whereas Nozzle B displays an expansion fan at the exit plane. Nozzles A and B, respectively, are operating in:
Options:
(A) over-expanded mode and under-expanded mode
(B) under-expanded mode and perfectly expanded mode
(C) perfectly expanded mode and under-expanded mode
(D) under-expanded mode and over-expanded mode
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2021 Q.14)**
**1. Intuitive Physical Setup & Given Parameters**:
We analyze supersonic nozzle flow exiting into an ambient atmosphere with back pressure $p_a$.
- Nozzle A exhibits oblique shock waves attached to the exit lip.
- Nozzle B exhibits a Prandtl-Meyer expansion fan centered at the exit lip.
**2. Step-by-Step Mathematical Derivation**:
1. **Exit Pressure Matching Mechanics**:
Supersonic flow inside a converging-diverging nozzle expands according to the area-ratio $A_e/A^*$ independently of the back pressure $p_a$, fixing the exit static pressure $p_e$.
2. **Nozzle A Analysis (Oblique Shocks)**:
If $p_e < p_a$, the gas has expanded 'too much' inside the nozzle, reaching a pressure below atmospheric. To match the higher ambient pressure $p_a$, the flow must undergo compressive turning via oblique shock waves at the nozzle lip. This defines the **over-expanded mode**.
3. **Nozzle B Analysis (Expansion Fan)**:
If $p_e > p_a$, the gas has not expanded enough inside the nozzle, leaving the exit plane with pressure higher than atmospheric. The flow expands outward through an isentropic Prandtl-Meyer expansion fan to drop its pressure to $p_a$. This defines the **under-expanded mode**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) over-expanded mode and under-expanded mode**: ✅ **CORRECT**. Shocks compress over-expanded flow ($p_e < p_a$); expansion fans depressurize under-expanded flow ($p_e > p_a$).
- **(B) under-expanded mode and perfectly expanded mode**: ❌ **INCORRECT**. Perfectly expanded flow produces no exit wave pattern ($p_e = p_a$).
- **(C) perfectly expanded mode and under-expanded mode**: ❌ **INCORRECT**. Nozzle A has clear oblique shocks.
- **(D) under-expanded mode and over-expanded mode**: ❌ **INCORRECT**. Inverts the identification between Nozzle A and B.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Rule of thumb: Shocks compress (used when exit pressure is too low: over-expanded). Expansion fans expand (used when exit pressure is too high: under-expanded).
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