GATE AE 2013 • Q13 MCQ • 1 Mark Aerodynamics & Fluid Mechanics

Aerodynamics & Fluid Mechanics

Airfoil & Wing Theory

Subtopic: Kutta-Joukowski Theorem & Thin Airfoil Theory

Question Statement:

Consider a thin flat plate airfoil at a small angle to an oncoming supersonic stream of air. Assuming the flow to be inviscid, 2 d l C C is

Options:

  • (A) zero
  • (B) independent of 
  • (C) proportional to 
  • (D) proportional to 2

Official Answer Key:

B

Step-by-Step Solution & Explanation

### **Dedicated Step-by-Step Solution (GATE AE 2013 Q.13)** **1. Problem Statement & Parameters**: - **Module**: Engineering Mathematics (1 Mark, MCQ) - **Topic**: Calculus (Limits, Continuity & Differentiability) - **Organizing Institute**: IIT Bombay - **Problem Statement**: > Consider a thin flat plate airfoil at a small angle to an oncoming supersonic stream of air. Assuming the flow to be inviscid, 2 d l C C is **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)** *"zero"*: ❌ **INCORRECT**. - **(B)** *"independent of "*: ✅ **CORRECT**. - **(C)** *"proportional to "*: ❌ **INCORRECT**. - **(D)** *"proportional to 2"*: ❌ **INCORRECT**. **4. Official Answer Verification**: - **Official Master Answer**: **B** - **Evaluation Status**: Matches official GATE 2013 (IIT Bombay) master answer key with 100% precision.
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