Topic: Fluid Mechanics
A sphere is moving in water with a velocity of $1.6~m/s$. Another sphere of twice the diameter is placed in a wind tunnel and tested with air which is 750 times less dense and 60 times less viscous than water. The velocity of air that will give dynamically similar condition is
- A. $5~m/s$
- B. $10~m/s$
- C. $20~m/s$
- D. $40~m/s$
Answer: B. $10~m/s$
Explanation: For dynamic similarity in this scenario, Reynolds numbers must be equated: $Re_1 = Re_2$.
$\frac{\rho_1 V_1 D_1}{\mu_1} = \frac{\rho_2 V_2 D_2}{\mu_2}$.
$V_2 = V_1 \left(\frac{\rho_1}{\rho_2}\right) \left(\frac{D_1}{D_2}\right) \left(\frac{\mu_2}{\mu_1}\right)$.
Given: $V_1 = 1.6~m/s$, $D_2 = 2D_1$, $\rho_2 = \frac{\rho_1}{750}$, $\mu_2 = \frac{\mu_1}{60}$.
$V_2 = 1.6 \times 750 \times \frac{1}{2} \times \frac{1}{60} = 1.6 \times \frac{750}{120} = 1.6 \times 6.25 = 10~m/s$.