Question Statement:
A positively cambered airfoil is placed in a uniform flow (velocity, U∞) at its
zero-lift angle of attack. M is the corresponding pitching moment. Which one of
the following representations accurately describes this scenario?
Scientific schematic diagram reproduced from official IIT paper
Options:
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(A) Figure A: Stagnation point on upper surface near leading edge
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(B) Figure B: Stagnation point exactly at the leading edge
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(C) Figure C: Stagnation point on lower surface near leading edge
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(D) Figure D: Trailing edge stagnation point shifted forward
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.19)**
**1. Intuitive Physical Setup & Given Parameters**:
A positively cambered airfoil is placed in a uniform freestream flow with velocity $U_\infty$ at its **zero-lift angle of attack** ($\alpha = \alpha_{L=0} < 0$).
$M$ denotes the aerodynamic pitching moment about the aerodynamic center (or quarter-chord).
We need to determine the correct representation of the aerodynamic forces and moment.
**2. Step-by-Step Aerodynamic Derivation**:
1. **Definition of Zero-Lift Angle of Attack**:
- At $\alpha = \alpha_{L=0}$, by definition, the net aerodynamic lift generated by the airfoil is zero:
L = 0
2. **Pitching Moment for a Positively Cambered Airfoil**:
- According to thin airfoil theory, positive camber (camber line curved upward) creates a downward-curved trailing edge effect, producing an upward lift on the forward section and a downward force on the aft section.
- Even when overall lift is trimmed to zero ($L = 0$), this internal force couple creates a persistent **nose-down (pitch-down) pitching moment**:
C_{m0} = C_{m,\text{ac}} < 0
- In standard flight mechanics sign conventions, a nose-down moment is negative (clockwise in standard coordinate system with flow from left to right).
- Thus, the airfoil experiences zero net lift and a clockwise (nose-down) pitching moment $M$.
3. Matching with the official figures confirms representation **(A)**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)**: ✅ **CORRECT**. Depicts zero net vertical lift force with a nose-down (clockwise) pitching moment $M$.
- **(B)**: ❌ **INCORRECT**. Depicts nose-up pitching moment ($C_{m0} > 0$), which corresponds to reflexed/negatively cambered airfoils.
- **(C)**: ❌ **INCORRECT**. Depicts non-zero lift force.
- **(D)**: ❌ **INCORRECT**. Depicts zero moment, which only holds for symmetric airfoils.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Symmetric airfoils have $C_{m0} = 0$; positively cambered airfoils have negative (nose-down) pitching moment at zero lift ($C_{m0} < 0$); reflexed airfoils have $C_{m0} > 0$.