The critical Mach number of an airfoil is attained when
Options:
(A) the freestream Mach number is sonic.
(B) the freestream Mach number is supersonic.
(C) the Mach number somewhere on the airfoil is unity.
(D) the Mach number everywhere on the airfoil is supersonic.
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2012 Q.6)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (2 Mark, MCQ)
- **Topic**: Verbal Aptitude (Reading Comprehension)
- **Organizing Institute**: IIT Delhi
- **Problem Statement**:
> The critical Mach number of an airfoil is attained when
**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)** *"the freestream Mach number is sonic."*: ❌ **INCORRECT**.
- **(B)** *"the freestream Mach number is supersonic."*: ❌ **INCORRECT**.
- **(C)** *"the Mach number somewhere on the airfoil is unity."*: ✅ **CORRECT**.
- **(D)** *"the Mach number everywhere on the airfoil is supersonic."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2012 (IIT Delhi) master answer key with 100% precision.