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).
Show Hint
Correct Answer: 50
Solution and Explanation
Vortex shedding behind a bluff body (like a landing gear strut) is characterised by the Strouhal number, a dimensionless group defined as
\[ St = \frac{f D}{V} \]
where \(f\) is the shedding frequency, \(D\) is a characteristic size of the body, and \(V\) is the freestream velocity. For dynamically similar flows (same flow regime, which is the standard assumption made when scaling wind tunnel model results to full scale for this kind of problem), the Strouhal number is the same for the model and the full scale (prototype) body.
Step 2: Write the similarity condition.
Equating the Strouhal numbers of the model (subscript 1) and the actual landing gear (subscript 2):
\[ \frac{f_1 D_1}{V_1} = \frac{f_2 D_2}{V_2} \]
Step 3: Substitute the given values.
The model has \(f_1=50\) Hz, \(V_1=5\) m/s, and some size \(D_1\). The actual landing gear has size \(D_2=10D_1\) and operates at \(V_2=50\) m/s. Substituting:
\[ \frac{50\times D_1}{5} = \frac{f_2 \times 10D_1}{50} \]
\[ 10D_1 = \frac{10 f_2 D_1}{50} \]
Step 4: Solve for \(f_2\).
Cancel \(D_1\) (non-zero) from both sides:
\[ 10 = \frac{10 f_2}{50} \]
\[ 10 = \frac{f_2}{5} \]
\[ f_2 = 50 \text{ Hz} \]
Final Answer:
The expected vortex shedding frequency behind the actual landing gear is 50 Hz. \[ \boxed{f_2 = 50 \text{ Hz}} \]
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