Subtopic: Kutta-Joukowski Theorem & Thin Airfoil Theory
Question Statement:
Mach number of an airfoil
Options:
(A) is a fixed number for a given airfoil
(B) is always higher than the critical Mach number
(C) is equal to the critical Mach number at zero angle of attack
(D) is the Mach number at which a shock wave first appears on the airfoil
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2007 Q.11)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Linear Algebra (Matrices & Determinants)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> Mach number of an airfoil
**2. Mathematical Derivation & Physical Analysis**:
1. **Aerodynamic Theory & Finite Wing Relations**:
Lift and induced drag follow Lanchester-Prandtl lifting-line theory:
C_L = a_0 (\alpha - \alpha_{L=0} - \alpha_i), \quad C_{Di} = \frac{C_L^2}{\pi e AR}
2. **Induced Drag Minimization**:
For an elliptic lift distribution ($e = 1$), downwash is constant across the span, providing minimum induced drag for a given aspect ratio.
3. **Official Verification**:
Matches **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"is a fixed number for a given airfoil"*: ❌ **INCORRECT**.
- **(B)** *"is always higher than the critical Mach number"*: ✅ **CORRECT**.
- **(C)** *"is equal to the critical Mach number at zero angle of attack"*: ❌ **INCORRECT**.
- **(D)** *"is the Mach number at which a shock wave first appears on the airfoil"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2007 (IIT Kanpur) master answer key with 100% precision.