Question

A 100 m long ship has a cruising speed of 25 knots. A geometrically similar model of 4 m length is used for resistance prediction in a towing tank. The corresponding speed of the model is ………….. knots.

Hint:

1. Use Froude number similarity to determine the speed relationship between a ship and its model.
2. Ensure that the length scales and speeds are consistent in units when applying the formula.
3. Froude number ensures dynamic similarity in resistance prediction.

Answer 1

Step 1: Use of Froude Number Similarity. In ship model testing, the Froude number similarity is used to predict the corresponding speed of a model. The Froude number is defined as: \[ F_r = \frac{V}{\sqrt{gL}} \] where: - \( V \) is the speed, - \( g \) is the acceleration due to gravity, and - \( L \) is the length of the ship or model. For geometric similarity, the Froude number for the ship and the model must be equal: \[ F_r^{\text{ship}} = F_r^{\text{model}} \] This gives: \[ \frac{V_{\text{ship}}}{\sqrt{gL_{\text{ship}}}} = \frac{V_{\text{model}}}{\sqrt{gL_{\text{model}}}} \] Step 2: Rearrange for \( V_{\text{model}} \). \[ V_{\text{model}} = V_{\text{ship}} \cdot \sqrt{\frac{L_{\text{model}}}{L_{\text{ship}}}} \] Step 3: Substitute the given values. - \( V_{\text{ship}} = 25 \, \text{knots} \) - \( L_{\text{ship}} = 100 \, \text{m} \) - \( L_{\text{model}} = 4 \, \text{m} \) \[ V_{\text{model}} = 25 \cdot \sqrt{\frac{4}{100}} \] \[ V_{\text{model}} = 25 \cdot \sqrt{0.04} \] \[ V_{\text{model}} = 25 \cdot 0.2 = 5 \, \text{knots} \] Conclusion: The corresponding speed of the model is \( \mathbf{5 \, \text{knots}} \).