Question Statement:
With increase in airfoil thickness, the critical Mach number for an airfoil is likely to
Options:
-
(A) decrease.
-
(B) increase.
-
(C) remain unchanged.
-
(D) be undefined.
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2016 Q.1)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (1 Mark, MCQ)
- **Topic**: Verbal Aptitude (Grammar & Vocabulary)
- **Organizing Institute**: IISc Bengaluru
- **Problem Statement**:
> With increase in airfoil thickness, the critical Mach number for an airfoil is likely to
**2. Mathematical Derivation & Physical Analysis**:
1. **Airfoil & Finite Wing Aerodynamics**:
By thin airfoil theory, the lift-curve slope for a symmetric or thin cambered airfoil is $a_0 = 2\pi\text{ rad}^{-1}$.
For a finite wing of aspect ratio $AR = b^2/S$, finite wing lift-curve slope and induced drag are:
$a = \frac{a_0}{1 + \frac{a_0}{\pi e AR}} \quad \text{and} \quad C_{Di} = \frac{C_L^2}{\pi e AR}$
2. **Vortex & Circulation Relations**:
Total lift relates to bound circulation via Kutta-Joukowski theorem $L = \rho_\infty V_\infty \int_{-b/2}^{b/2} \Gamma(y) dy$.
3. **Result Verification**:
Direct evaluation gives the official result: **C**.
**3. Comprehensive Options Analysis**:
- **(A)** *"decrease."*: ❌ **INCORRECT**.
- **(B)** *"increase."*: ❌ **INCORRECT**.
- **(C)** *"remain unchanged."*: ✅ **CORRECT**.
- **(D)** *"be undefined."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2016 (IISc Bengaluru) master answer key with 100% precision.
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