GATE AE 2017 • Q38 NAT • 2 Marks Aerodynamics & Fluid Mechanics

Aerodynamics & Fluid Mechanics

Airfoil & Wing Theory

Subtopic: Finite Wing Theory & Induced Drag

Question Statement:

An aircraft model was tested in a low speed wind-tunnel (Reynolds number based on mean chord $Re = 10^6$). The lift curve slope of the wing is $C_{L\alpha} = 4.5$ rad$^{-1}$. If the aspect ratio of the wing is 6 and Oswald efficiency factor is 0.9, the profile drag coefficient $C_{D0}$ is 0.015. At $C_L = 0.6$, the total drag coefficient $C_D$ of the wing is _________ (round off to 3 decimal places).

Official Answer Key:

0.036

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2017 Q.38)** **1. Intuitive Physical Setup & Given Parameters**: - Zero-lift parasite drag coefficient: $C_{D0} = 0.015$ - Aspect ratio: $AR = 6$ - Oswald efficiency factor: $e = 0.9$ - Operating lift coefficient: $C_L = 0.6$ **2. Step-by-Step Mathematical Derivation**: 1. **Parabolic Drag Polar Formulation**: C_D = C_{D0} + C_{Di} = C_{D0} + \frac{C_L^2}{\pi e AR} 2. **Induced Drag Calculation**: C_{Di} = \frac{(0.6)^2}{\pi \times 0.9 \times 6} = \frac{0.36}{5.4 \pi} = \frac{0.36}{16.9646} \approx 0.02122 3. **Total Drag Coefficient ($C_D$)**: C_D = 0.015 + 0.02122 = 0.03622 Rounding to three decimal places: C_D \approx 0.036 Official acceptable range: **[0.035, 0.038]**. **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Official Range**: **0.035 to 0.038**. **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **0.036** - **Key Formula**: $C_D = C_{D0} + \frac{C_L^2}{\pi e AR}$.
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