Froude Scaling & Model Testing

The scaling conflict that makes towing-tank testing possible only by treating friction and waves separately.

Marine EngineeringResistance & Powering
⏱️ About 16 min

You can't tow a 180-metre ship through a test basin — so towing tanks test a small model instead. But a model can't match both the Reynolds number and the Froude number of the real ship at once, in the same water. Froude's method of comparison is how naval architects work around that.

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The big idea: Froude's law of comparison matches the Froude number (not the Reynolds number) between model and ship, because that preserves wave-pattern similarity — residuary resistance coefficients transfer directly, while frictional resistance is calculated separately at each object's own Reynolds number.
🎯 By the end, you'll be able to
  • State Froude's law of comparison and why Froude number, not Reynolds number, is matched between model and ship
  • Compute the corresponding speed for a model given the ship's speed and geometric scale ratio
  • Describe the model-to-ship resistance extrapolation procedure
  • Explain what the correlation allowance CA accounts for
📎 Helpful to know first

A Scaling Conflict With No Perfect Solution

Dynamic similarity between a model and a full-scale ship would ideally require matching both the Froude number Fn = V/√(gL) (which governs wave-making) and the Reynolds number Re = VL/ν (which governs viscous friction). But for a model built at scale ratio λ = L_ship/L_model in the same fluid, matching Fn requires the model to run much slower than the ship (V_model = V_ship/√λ), while matching Re would require the model to run much faster (V_model = V_ship × λ) — the two requirements point in opposite directions and can't both be satisfied. There is no way around this with a single-fluid tank test.

Froude's law of comparison resolves the conflict by choice: match Fn (preserving the wave pattern, which is what a geometrically scaled model can actually reproduce), and treat frictional resistance analytically instead, using the ITTC line at each object's own Reynolds number.

\[ Fn = \frac{V}{\sqrt{gL}}, \qquad V_{model} = \frac{V_{ship}}{\sqrt{\lambda}}, \qquad \lambda = \frac{L_{ship}}{L_{model}} \]
V_model is the 'corresponding speed' — the model speed at which Fn_model = Fn_ship.
📝 Worked example: A ship is 180 m long and cruises at 7.5 m/s. A 1:25 scale model is tested in a towing tank. Find the model's length and its corresponding speed.
  1. L_model = L_ship/λ = 180/25 = 7.2 m
  2. V_model = V_ship/√λ = 7.5 ÷ √25 = 7.5 ÷ 5 = 1.5 m/s
✓ L_model = 7.2 m; V_model = 1.5 m/s
✏️ Practice: A ship is 140 m long and cruises at 6.2 m/s. Its 1:20 scale model is tested in a towing tank. Find the corresponding model speed (in m/s).
m/s
Solution
  1. V_model = V_ship/√λ = 6.2/√20 = 6.2/4.4721 ≈ 1.387 m/s

The Extrapolation Procedure

At the corresponding speed, model and ship share the same Froude number, so the Froude assumption holds: their residuary resistance coefficients CR are equal. That single fact is what makes a five-step extrapolation from model test to ship prediction possible:

  1. Tow the model at the corresponding speed and measure its total resistance; compute its total coefficient C_Tm = R_Tm/(½ρ_mS_mV_model²).
  2. Compute the model's own frictional coefficient C_Fm from the ITTC line, at the model's Reynolds number.
  3. Subtract to isolate the residuary coefficient: CR = C_Tm − C_Fm.
  4. Compute the ship's frictional coefficient C_Fs from the ITTC line, at the ship's (much larger) Reynolds number.
  5. Add CR (assumed equal to the model's, since Fn matched) plus a small correlation allowance CA to get the ship's total coefficient: C_Ts = C_Fs + CR + CA.
🔑 What the correlation allowance CA is for

The ITTC line is itself an empirical correlation, and a real ship's hull is never as smooth or as perfectly geometrically similar to the model as the theory assumes — plating roughness, appendages, and other full-scale effects the model can't reproduce all add a small extra drag. CA is a modest empirical correction (commonly a few ten-thousandths) added at the ship-prediction step to account for this model-ship correlation gap. It is not a substitute for accurate model testing — just a standard, small correction on top of it.

📝 Worked example: For the 180 m ship above (V = 7.5 m/s, seawater ν = 1.19 × 10⁻⁶ m²/s, S = 4500 m², ρ = 1025 kg/m³), a model test at the corresponding speed (1.5 m/s, model length 7.2 m, fresh water ν = 1.14 × 10⁻⁶ m²/s) gives a measured total coefficient C_Tm = 0.004500. Using a correlation allowance CA = 0.0004, find the ship's predicted total resistance RT.
  1. Model Re: Re_m = (1.5 × 7.2)/(1.14 × 10⁻⁶) ≈ 9.474 × 10⁶; log₁₀Re_m ≈ 6.9765, so C_Fm = 0.075/(4.9765)² ≈ 0.003029
  2. Residuary coefficient: CR = C_Tm − C_Fm = 0.004500 − 0.003029 = 0.001471
  3. Ship Re: Re_s = (7.5 × 180)/(1.19 × 10⁻⁶) ≈ 1.1345 × 10⁹, giving C_Fs ≈ 0.001507 (from the previous lesson)
  4. Ship total coefficient: C_Ts = C_Fs + CR + CA = 0.001507 + 0.001471 + 0.0004 = 0.003378
  5. RT = ½ρ_sS_sV_s²C_Ts = 0.5 × 1025 × 4500 × 7.5² × 0.003378 = 129,726,562.5 × 0.003378 ≈ 438,200 N
✓ CR ≈ 0.001471; ship C_Ts ≈ 0.003378; predicted RT ≈ 438.2 kN

Check your understanding

1. A towing-tank model can't match both the Froude number and the Reynolds number of the ship at once because:
Fn-matching gives V_model = V_ship/√λ (slower); Re-matching would require V_model = V_ship×λ (much faster) — for λ > 1 these can't both hold.
2. In Froude's method of comparison, the model's residuary coefficient CR is assumed equal to the ship's CR because:
Fn governs wave-making; matching it between model and ship makes their wave patterns dynamically similar, so CR transfers directly.
3. The correlation allowance CA is added to account for:
CA is a small empirical correction covering the model-ship correlation gap not captured by the ITTC line or the model test alone.
🔎 Related lessons
Effective, Delivered, Shaft & Brake Power · soon
✅ Key takeaways
  • A model can't match both Froude and Reynolds number against the ship at once — Froude's law of comparison chooses to match Fn, preserving wave-pattern similarity
  • Corresponding speed: V_model = V_ship/√λ, where λ is the geometric scale ratio
  • Extrapolation: model C_Tm − model C_Fm gives CR (assumed equal for the ship); ship C_Ts = ship C_Fs + CR + a small correlation allowance CA
➡️ With the ship's total resistance predicted, the next step is converting that resistance into the power figures that actually size an engine.
Want to test yourself on this? Try the Marine Engineering Aptitude test →