Question Statement:
A rectangular wing of 1.2 m chord length and aspect ratio 5 is tested in a wind
tunnel at an air speed of 60 m/s. The density and the dynamic viscosity of air are
1.3 kg/m3 and 1.8×10-5 kg/m-s, respectively. A second rectangular wing of the
same span, but with an aspect ratio of 6, is to be tested in the same tunnel at the
same Reynolds number. The air speed at which the second test should be
performed is _______ m/s (answer in integer).
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.47)**
**1. Intuitive Physical Setup & Given Parameters**:
- Wing 1: Rectangular wing, chord $c_1 = 1.2\text{ m}$, aspect ratio $AR_1 = 5$, tested at $V_1 = 60\text{ m/s}$.
- Tunnel air: $\rho = 1.3\text{ kg/m}^3$, $\mu = 1.8 \times 10^{-5}\text{ kg/m}\cdot\text{s}$ (identical for both tests).
- Wing 2: Rectangular wing, **same span** ($b_2 = b_1$), aspect ratio $AR_2 = 6$.
- Tested in the same tunnel at the **same Reynolds number** ($Re_2 = Re_1$).
We need to find the air speed $V_2$ for the second test in $\text{m/s}$ (integer).
**2. Step-by-Step Aerodynamic Similarity Derivation**:
1. **Geometric Relations for Rectangular Wings**:
For a rectangular wing, span $b = AR \times c$:
b_1 = AR_1 \times c_1 = 5 \times 1.2 = 6.0\text{ m}
Since Wing 2 has the same span ($b_2 = b_1 = 6.0\text{ m}$) and $AR_2 = 6$:
c_2 = \frac{b_2}{AR_2} = \frac{6.0}{6} = 1.0\text{ m}
2. **Reynolds Number Equality**:
Re_1 = \frac{\rho V_1 c_1}{\mu}, \quad Re_2 = \frac{\rho V_2 c_2}{\mu}
Since fluid density $\rho$ and viscosity $\mu$ are identical:
V_1 c_1 = V_2 c_2
3. **Solve for $V_2$**:
V_2 = V_1 \left(\frac{c_1}{c_2}\right) = 60 \times \left(\frac{1.2}{1.0}\right) = 60 \times 1.2 = 72\text{ m/s}
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $V_2 = 72\text{ m/s}$.
- **Rounding Requirement**: Nearest integer $\implies \mathbf{72}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **72.0** (Acceptable Range: **72 to 72**)
- **Exam Takeaway**: For equal Reynolds number in the same fluid medium, velocity is inversely proportional to chord: $V_2 = V_1 (c_1 / c_2) = 60 \times 1.2 = 72\text{ m/s}$.