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25-Nav-A2 Hydrodynamics of Ships (I)_ Resistance and Propulsion · Undated paper

Question 2 of 10: Froude and Reynolds Similarity in Model Resistance Tests

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Notes on this paper

National Exams — May 2019 — 16-Nav-A2 Hydrodynamics of Ships I: Resistance and Propulsion. Three-hour, closed-book exam; a data sheet, a Wageningen B4-55 propeller chart and a Burrill cavitation chart are supplied. Format: Questions 1–8 are compulsory (attempt all eight), then one of Questions 9 or 10. All ten are solved below for completeness.

Reference texts: Larsson & Raven, Ship Resistance and Flow (SNAME) — model-scale resistance testing, Froude/Reynolds scaling, boundary-layer estimates and the ITTC 1978 performance-prediction method; Lewis (ed.), Principles of Naval Architecture, Vol. II — Resistance, Propulsion and Vibration (SNAME) — propeller geometry, wave-pattern interference, open-water B-series design and wake-induced blade loading; Carlton, Marine Propellers and Propulsion (Butterworth-Heinemann) — Wageningen B-series charts, the Burrill back-cavitation criterion and open-water model testing.

Question 2: Froude and Reynolds Similarity 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.

Approach. Write the model-speed requirement implied by each similarity law separately, then show the two requirements are algebraically incompatible unless the scale ratio is unity; the practical resolution (obeying Froude, correcting friction separately) follows directly from which physical mechanism each number governs.

  1. Froude-similarity requirement on model speed. With $\lambda=L_S/L_M$ the linear scale ratio, $F_{n,M}=F_{n,S}$ gives $$\frac{V_M}{\sqrt{gL_M}}=\frac{V_S}{\sqrt{gL_S}}\ \Rightarrow\ V_M=\frac{V_S}{\sqrt\lambda}.$$
  2. Reynolds-similarity requirement on model speed. Testing in the same fluid ($\nu_M=\nu_S$), $R_{n,M}=R_{n,S}$ gives $$\frac{V_ML_M}{\nu_M}=\frac{V_SL_S}{\nu_S}\ \Rightarrow\ V_M=V_S\lambda.$$
  3. Show the two are incompatible. Both can hold only if $V_S/\sqrt\lambda=V_S\lambda$, i.e. $\lambda^{3/2}=1\ \Rightarrow\ \boxed{\lambda=1}$. For any real model ($\lambda>1$), Froude similarity demands a slower model speed while Reynolds similarity demands a much faster one — the two cannot be satisfied together with the same fluid. (Even allowing a different model fluid, matching both would require $\nu_M=\nu_S/\lambda^{3/2}$ — for a typical $\lambda\approx30$, a kinematic viscosity over 160 times smaller than water's, which no practical fluid provides.)
  4. (b) Which is obeyed, and why. In practice, model basins obey Froude similarity. Wave-making (residuary) resistance is governed by gravity/free-surface effects, so matching Froude number reproduces a geometrically similar wave pattern between model and ship — this is essential, because there is no other practical way to correct wave-making resistance after the fact. Frictional resistance, by contrast, can be estimated separately and added back analytically (next part).
  5. (c) Scalable vs. non-scalable components. The residuary (wave-making) resistance coefficient $C_R$ is assumed to depend on Froude number only, so it transfers directly from the model test to the ship at equal $F_n$: $C_{R,S}=C_{R,M}$. The frictional resistance cannot be scaled this way because it is Reynolds-number-dependent, and the model's Reynolds number (run at Froude-corresponding, i.e. much lower, speed) is far below the ship's. To account for it, the ITTC 1978 method computes model and ship friction separately from a flat-plate correlation line (ITTC-57) evaluated at each one's own actual Reynolds number, then recombines: $$C_{TS}=(1+k)C_{FS}+C_{TM}-(1+k)C_{FM}+C_A+C_{AA}=(1+k)C_{FS}+C_R+C_A+C_{AA}.$$