In the figure shown below, the flow at the nozzle exit is ________.
Options:
(A) overexpanded
(B) underexpanded
(C) ideally expanded
(D) subsonic
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.21)**
**1. Intuitive Physical Setup & Given Parameters**:
The figure depicts flow exiting a convergent-divergent supersonic nozzle into ambient backpressure $p_a$.
At the nozzle exit plane, the static pressure of the jet is $p_e$.
In the illustration, oblique shock waves originate from the nozzle exit lip and cross into the plume, compressing the flow inward.
We need to classify the expansion regime of the nozzle.
**2. Step-by-Step Fluid Dynamic Derivation**:
1. **Classification of Nozzle Operating Regimes**:
- **Ideally Expanded ($p_e = p_a$)**: The jet static pressure matches ambient backpressure exactly. Flow leaves smoothly in parallel streamlines with no shock waves or expansion fans at the exit lip.
- **Underexpanded ($p_e > p_a$)**: Jet exit pressure is higher than ambient pressure. The flow expands further into the atmosphere through Prandtl-Meyer expansion fans emanating from the nozzle lip.
- **Overexpanded ($p_e < p_a$)**: Jet exit pressure is lower than ambient pressure. The surrounding higher ambient pressure must compress the jet, forcing oblique shock waves to emanate from the nozzle exit lip towards the jet axis.
2. **Identification from the Wave Structure**:
- Since the flow experiences shock compression (oblique shocks forming at the lip) to match the higher atmospheric backpressure, $p_e < p_a$.
- By definition, this flow regime is **overexpanded**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) overexpanded**: ✅ **CORRECT**. $p_e < p_a$, leading to compressive oblique shock waves originating at the nozzle exit corners.
- **(B) underexpanded**: ❌ **INCORRECT**. Underexpansion occurs when $p_e > p_a$, which produces expansion fans (Mach waves), not compressive shock waves.
- **(C) ideally expanded**: ❌ **INCORRECT**. Ideally expanded nozzles have $p_e = p_a$ with neither shock waves nor expansion waves at the exit.
- **(D) subsonic**: ❌ **INCORRECT**. A convergent-divergent nozzle operating with supersonic outflow produces wave patterns (shocks/expansion fans); subsonic exit flows cannot support oblique shock structures.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: "Over" means the nozzle expanded the gas *too much* (dropping exit pressure below ambient), so nature recompresses it with shocks. "Under" means it didn't expand enough ($p_e > p_a$), so it expands further with Prandtl-Meyer fans.
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