Can a scale model match both Re and Fr?

A matching shape is only the starting point

Imagine testing a geometrically similar, 1 m model of a 10 m surface body moving at 5 m/s. Should the model speed be one tenth of the full-scale speed, or should it stay the same? We will compare Reynolds and Froude numbers to answer a narrower question: what speed matches each ratio? This is an illustrative scaling exercise, not a performance prediction for a particular vessel.

Use ρ = 1000 kg/m³, μ = 0.001 Pa·s and g = 9.81 m/s² for both cases. The kinematic viscosity is ν = 10⁻⁶ m²/s. L denotes a representative body length throughout, not an open-channel depth.

1. Establish the full-scale reference

The first ratio concerns viscosity; the second concerns gravity. A body-length Fr below one does not label the surrounding flow as subcritical open-channel flow. Check the chosen length and the conditions attached to a calculator's regime description.

2. Choose a speed that matches Fr

With equal gravity, matching Fr requires v_model/√L_model = v_full/√L_full. Define λ = L_model/L_full = 0.1. The required speed ratio is √λ.

At this speed, model Fr is approximately 0.5048, but model Re is approximately 1,581,139. Check the ratio directly: Re_model/Re_full = λ^3/2 ≈ 0.03162. The viscosity-related scale is only about 3.16% of the full-scale value despite the matching Fr.

3. Choose a speed that matches Re

With the same fluid, matching Re requires v_modelL_model = v_fullL_full. Therefore the model speed must be 5 × 10/1 = 50 m/s. Model Fr then becomes 50/√9.81 ≈ 15.9638, about 31.62 times full-scale Fr.

The Fr-matched row uses the unrounded speed 5√0.1. Entering the displayed 1.5811 produces a small last-digit difference.

4. What would matching both require?

Keeping the Fr-matched speed while matching Re would require ν_model = λ^3/2ν_full = 3.1623×10⁻⁸ m²/s. This is an algebraic requirement, not a recommendation for an available working fluid. If density also changes, use μ = ρν consistently.

Adjusting speed alone cannot match every important dimensionless ratio. Surface tension may require a Weber-number comparison; compressibility may require Mach number. Choose the phenomenon to reproduce before deciding which additional ratios matter.

Keep a comparison record

• State whether λ means model/full scale or its reciprocal.
• Record the definition of L, geometric ratios and boundary conditions.
• List both matched ratios and unmatched ratios.
• Avoid applying circular-pipe Reynolds thresholds or open-channel Froude classifications to body-length calculations.

A matching Fr alone does not validate total drag, skin friction or full-scale performance. The table quantifies the scaling mismatch under equal-fluid and equal-gravity assumptions; it is a starting point for deciding what further evidence is needed.