GATE AE 2012 • Q19
MCQ • 1 Mark
Aerodynamics & Fluid Mechanics
Compressible Flow & Shocks
Subtopic: 1D Isentropic Flow & Area-Mach Relations
Question Statement:
For a wing of aspect ratio AR, having an elliptical lift distribution, the induced drag coefficient is
(where CL is the lift coefficient)
Options:
-
(A) LC
ARπ
-
(B) 2
LC
ARπ
-
(C) 2
LC
ARπ
-
(D) 2
2
LC
ARπ
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2012 Q.19)**
**1. Problem Statement & Parameters**:
- **Module**: Aerodynamics & Fluid Mechanics (1 Mark, MCQ)
- **Topic**: Compressible Flow & Shocks (1D Isentropic Flow & Area-Mach Relations)
- **Organizing Institute**: IIT Delhi
- **Problem Statement**:
> For a wing of aspect ratio AR, having an elliptical lift distribution, the induced drag coefficient is (where CL is the lift coefficient)
**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: **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"LC
ARπ"*: ❌ **INCORRECT**.
- **(B)** *"2
LC
ARπ"*: ✅ **CORRECT**.
- **(C)** *"2
LC
ARπ"*: ❌ **INCORRECT**.
- **(D)** *"2
2
LC
ARπ"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2012 (IIT Delhi) master answer key with 100% precision.
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