Ship Resistance Prediction
Theory: two similarity laws that can't both hold
Overview
Ship resistance is measured by towing models in a tank. Can a small model's result be applied straight to the big ship?
Not directly. Ship resistance includes wave-making resistance from creating surface waves and frictional resistance from water's viscosity. Reproducing wave resistance requires matching Froude number, and friction requires matching Reynolds number, but with the same water both can't match at once. So the test matches Froude number and friction is replaced by calculation at full scale. Let's work through this for a 150 m ship and a 1:30 model.
Froude number, Reynolds number and the friction line
Extrapolation to full scale
The 2D method assumes the residual coefficient is equal at both scales; the 3D method uses the form factor (1+k) to scale the whole viscous resistance with Reynolds number. $P_E$ is effective power (resistance × speed).
Froude's towing tank
Britain's William Froude built the world's first proper towing tank near his home in the 1870s and measured model resistance. He split resistance into friction and the rest, reasoning that the rest is similar between model and ship when the speed-to-length ratio (today's Froude number) matches. He confirmed it by actually towing the Royal Navy ship HMS Greyhound. A century and a half later, his approach still underpins ship design.
Worked example: a 150 m ship at 20 knots
Ship: 150 m, wetted area 5,000 m², 20 knots (10.29 m/s), sea water. Model: 1:30, fresh water. The model's total resistance coefficient is assumed as 4.0×10⁻³ (in place of a measurement).
| Item | Model | Ship |
|---|---|---|
| Length | 5 m | 150 m |
| Speed | 1.88 m/s | 10.29 m/s |
| Froude number | 0.268 | 0.268 |
| Reynolds number | 8.2×10⁶ | 1.3×10⁹ |
| Friction coefficient (ITTC-1957) | 0.00310 | 0.00148 |
Effective power estimates
| Method | Ship total coefficient | Resistance | Effective power |
|---|---|---|---|
| Model coefficient as-is | 0.00400 | 1,085 kN | 11.2 MW |
| 2D (Froude) | 0.00238 | 645 kN | 6.6 MW |
| 3D (k = 0.2) | 0.00205 | 557 kN | 5.7 MW |
Adding a correlation allowance C_A = 0.0002 for hull roughness gives 7.2 MW (2D) and 6.3 MW (3D).
Using the model coefficient directly nearly doubles the power estimate.
At its low Reynolds number the model has a relatively thick boundary layer and a high friction coefficient. Here nearly 80% of model resistance is friction, and that coefficient more than halves at full scale. Without recalculating friction at the ship's Reynolds number, resistance and power are badly overestimated. The 2D/3D difference lies in how hull-shape viscous effects are treated: in 3D the whole viscous resistance shrinks with Reynolds number, lowering the full-scale value further. Today CFD at full-scale Reynolds number is also used to check these extrapolations.
Treatment in CFD
- Validate model-scale CFD against tank tests, then apply the same extrapolation.
- Full-scale CFD has very thin boundary layers; watch wall-cell y⁺.
- Solve the free surface (e.g. VOF) to capture wave resistance.
- Including sinkage and trim improves high-speed predictions.
- Hull roughness and fouling raise full-scale friction; include a correlation allowance.
“Hull fouling hurt fuel economy”
One cargo ship needed over 10% more fuel for the same speed about a year after entering service. In dry dock its hull was found thickly covered with barnacles and algae. Most full-scale resistance is friction, so a rougher surface raises it considerably. Cleaning the hull and recoating with antifouling paint restored fuel economy. Correlation allowances in resistance prediction partly account for such full-scale surface conditions.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Using the coefficient directly | Power about doubled | Correct friction |
| Matching Reynolds number | Wave resistance lost | Match Froude number |
| Ignoring sea/fresh water | Few-percent error | Distinguish ρ and ν |
| Ignoring form factor | Ship overestimated | Compare 3D method |
| Ignoring roughness | Ship underestimated | Add correlation allowance |
I'd like to learn related topics.
Related pages include propeller CFD, wave modelling, VOF waves, sloshing and mooring analysis.
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