The vortex shedding frequency behind a landing gear model is found to be 50 Hz
when tested in a wind tunnel operating at 5 m/s. If the actual landing gear size is
10 times that of the model, and it is designed to operate at 50 m/s, then the
expected vortex shedding frequency behind it is ________ Hz (rounded off to the
nearest integer).
Official Answer Key:
50.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.34)**
**1. Intuitive Physical Setup & Given Parameters**:
- Wind tunnel model test:
- Velocity: $V_m = 5 ext{ m/s}$
- Vortex shedding frequency: $f_m = 50 ext{ Hz}$
- Characteristic dimension: $L_m$
- Prototype landing gear:
- Scale factor: $L_p = 10 L_m$
- Operating velocity: $V_p = 50 ext{ m/s}$
- We need to find the expected vortex shedding frequency $f_p$ behind the prototype in Hz.
**2. Step-by-Step Fluid Dynamic Derivation**:
1. **Dynamic Similarity & Strouhal Number**:
- Vortex shedding in bluff body cross-flow (e.g., von Kármán vortex streets behind landing gear cylinders) is governed by the non-dimensional **Strouhal number**:
$St = rac{f L}{V}$
- Under dynamic similarity, the Strouhal number is invariant between model and prototype:
$St_m = St_p \implies rac{f_m L_m}{V_m} = rac{f_p L_p}{V_p}$
2. **Solve for Prototype Frequency $f_p$**:
$f_p = f_m \left( rac{V_p}{V_m}
ight) \left( rac{L_m}{L_p}
ight)$
3. **Substitute Numerical Values**:
- Given $ rac{V_p}{V_m} = rac{50}{5} = 10$
- Given $ rac{L_m}{L_p} = rac{1}{10}$
$f_p = 50 imes 10 imes rac{1}{10} = 50 ext{ Hz}$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $f_p = 50 ext{ Hz}$.
- **Rounding Requirement**: Nearest integer $\implies \mathbf{50}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **50.0** (Acceptable Range: **50 to 50**)
- **Exam Takeaway**: When both velocity and geometric scale increase by the identical factor ($10 imes$ speed, $10 imes$ length), their effects on Strouhal frequency precisely cancel out, leaving vortex shedding frequency identical.
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