GATE AE 2017 • Q37 NAT • 2 Marks

Aerodynamics & Fluid Mechanics

Wind Tunnel Testing & Instrumentation

Subtopic: Reynolds Number Matching in Fluid Testing

Question Statement:

A football, meant to be thrown at 100 km/h in sea level air ($\rho = 1.22$ kg/m$^3$, $\mu = 1.8 \times 10^{-5}$ Pa-s), is tested in a water tunnel with a 1:2 scale model. If the kinematic viscosity of water is $1.0 \times 10^{-6}$ m$^2$/s, the water flow speed (in m/s) required for dynamic similarity is _________ (round off to 1 decimal place).

Official Answer Key:

3.8

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2017 Q.37)** **1. Intuitive Physical Setup & Given Parameters**: - Prototype speed: $V_p = 100\text{ km/h} = \frac{100 \times 1000}{3600} = 27.78\text{ m/s}$. - Air kinematic viscosity: $\nu_{air} = \frac{\mu}{\rho} = \frac{1.8 \times 10^{-5}}{1.22} \approx 1.4754 \times 10^{-5}\text{ m}^2/\text{s}$. - Scale ratio: $\frac{L_m}{L_p} = \frac{1}{2} = 0.5$. - Water kinematic viscosity: $\nu_w = 1.0 \times 10^{-6}\text{ m}^2/\text{s}$. **2. Step-by-Step Mathematical Derivation**: 1. **Reynolds Number Dynamic Similarity**: $Re_m = Re_p \implies \frac{V_m L_m}{\nu_w} = \frac{V_p L_p}{\nu_{air}}$ 2. **Solving for Model Water Velocity ($V_m$)**: $V_m = V_p \left( \frac{L_p}{L_m} \right) \left( \frac{\nu_w}{\nu_{air}} \right)$ $V_m = 27.78 \times 2 \times \left( \frac{1.0 \times 10^{-6}}{1.4754 \times 10^{-5}} \right) = 55.56 \times 0.06778 \approx 3.766\text{ m/s}$ Rounding to one decimal place: $V_m \approx 3.8\text{ m/s}$ This falls inside the official acceptable interval **[3.6, 4.0]**. **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Official Range**: **3.6 to 4.0 m/s**. **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **3.8** (Acceptable Range: **3.6 to 4.0**) - **Key Formula**: $V_m = V_p \left(\frac{L_p}{L_m}\right) \left(\frac{\nu_m}{\nu_p}\right)$.

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