Question 1 of 9: Froude vs. Reynolds Scaling in Model Resistance Tests

Question 1: Froude vs. Reynolds Scaling in Model Resistance Tests (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(i) In practice the model is towed at corresponding (Froude-identical) speed: the model Froude number $F_n=V/\sqrt{gL}$ is matched to the ship's, because for a displacement hull the dominant scale-sensitive component is the free-surface wave-making (residuary) resistance, and wave patterns are governed by gravity and therefore by $F_n$. Matching $F_n$ while also matching Reynolds number $R_n=VL/\nu$ would require a model kinematic viscosity roughly $\lambda^{3/2}$ times smaller than the ship's (where $\lambda=L_S/L_M$ is the geometric scale) — no practical test fluid exists with that property (even mercury or superheated water fall far short for typical scale ratios of 20–50) — so Reynolds similarity is abandoned.

(ii) Because $F_n$ is matched, the model's Reynolds number is always far below the ship's, so the model's frictional boundary layer is disproportionately thick and part of it would naturally remain laminar at model scale even though the full-scale hull is fully turbulent — a laminar model boundary layer would give a frictional resistance coefficient that does not extrapolate correctly. Two standard procedures correct for this:

1. Forced (artificial) turbulence stimulation. A trip wire, a row of studs, or a sand strip is fitted near the bow of the model (typically at 5% of length from the forward perpendicular) to force the boundary layer to become fully turbulent from very near the bow, so that the model's flow regime matches the ship's over its whole length.
2. Separate scaling of the frictional and residuary components (ITTC 1957/1978 extrapolation). Rather than scaling the total resistance coefficient $C_T$ directly (which would carry the model's non-representative friction law with it), the total is split as $C_T=(1+k)C_F+C_R$ (form-factor method). The residuary/wave component $C_R$ is assumed to be a function of $F_n$ only and is carried unchanged from model to ship (Froude similarity), while the frictional component $C_F$ is computed separately, at the model's and the ship's own (very different) Reynolds numbers, from the ITTC 1957 model-ship correlation line $C_F=0.075/(\log_{10}R_n-2)^2$. This is exactly the extrapolation carried out numerically in Question 4 below.