A new hull design for a ship is to be 100 m long. A model is...

A new hull design for a ship is to be 100 m long. A model is 3 m long. The model is tested in a tow tank, which can accommodate towing speeds of up to $2 \mathrm{~m} / \mathrm{s}$. Only the Froude number is conserved for similarity, because it is impractical to conserve the Reynolds number too. A correction is later made for the different Reynolds numbers. Find to what ship's speed corresponds the $2-\mathrm{m} / \mathrm{s}$ speed of the model and what are the ship and the model Reynolds numbers at these speeds.

Key Concepts

Froude Number Similarity

Froude number similarity is crucial in free-surface flows where gravity is the dominant restoring force, such as in ship hydrodynamics. The Froude number, defined as V/?(gL), must be conserved between the model and the prototype in order to correctly scale the dynamic behavior of waves and resistance due to gravity. By equating the Froude numbers, one can determine the corresponding full-scale speed given the model speed and the scale factor in length.

Reynolds Number Considerations

The Reynolds number, calculated as Re = V L/?, characterizes the ratio of inertial to viscous forces in a fluid flow. Although complete dynamic similarity would require conserving the Reynolds number as well, in many practical applications—such as towing tank experiments for ships—it is often impractical to match both Froude and Reynolds numbers simultaneously. Instead, experiments are performed under Froude number similarity and corrections are later made to account for differences in the Reynolds number between the model and the full-scale prototype.

Geometric Scaling

Geometric scaling concerns the proportional dimensions between models and their full?scale counterparts. When designing experiments using scale models, length ratios are used to relate model dimensions to prototype dimensions. This helps determine how other physical quantities, such as speed and force, must be adjusted to maintain similarity in behavior between the two scales.

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