Question 9 of 9: Planning an Open-Water Propeller Test

Question 9: Planning an Open-Water Propeller Test (20 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.

Given. Model diameter $D_M=0.200$ m; full-scale diameter $D_S=4.0$ m; full-scale chord at $0.75R$, $c_{0.75R,S}=1.30$ m; minimum local Reynolds number $3\times10^5$; test range $J=0$ to $1.1$; dynamometer limits: torque $\le10$ N·m, thrust $\le250$ N; a similar (not identical) propeller's bollard-pull coefficients $K_{T,0}=0.35$, $K_{Q,0}=0.04$. Fresh water, 15 °C ($\nu=1.139\times10^{-6}\ \text{m}^2/\text{s}$, $\rho=999\ \text{kg/m}^3$).

Find. The minimum and maximum shaft speeds for the test, and the values of the other test parameters.

Approach. Scale the full-scale $0.75R$ chord down to the model to get a local Reynolds-number constraint; since advance speed only adds to the rotational velocity component, the Reynolds number is smallest — and the dynamometer loads are largest — at bollard pull ($J=0$), so both the minimum and maximum shaft-speed limits are set at that single worst-case condition.

1. Model chord at 0.75R. Geometric similarity at scale ratio $\lambda=D_S/D_M=4.0/0.200=20$: $$c_{0.75R,M}=\frac{c_{0.75R,S}}{\lambda}=\frac{1.30}{20}=\boxed{0.065\ \text{m} = 65\ \text{mm}}.$$
2. Minimum shaft speed (Reynolds-number criterion, worst case at bollard). At bollard pull ($J=0$, $V_A=0$) the section's resultant velocity is purely rotational, $V_{R,0.75}=0.75\pi n D_M$ — the smallest resultant velocity the section ever sees at a given $n$, since any advance speed only adds to it. Requiring the local Reynolds number $R_n=c_{0.75R,M}V_{R,0.75}/\nu\ge3\times10^5$ there (generalising the data sheet's $0.7R$ formula to the $0.75R$ station specified in this question): $$n_{min}=\frac{(3\times10^5)(1.139\times10^{-6})}{c_{0.75R,M}\,(0.75\pi D_M)}=\frac{0.3417}{(0.065)(0.4712)}=\boxed{11.16\ \text{rps}\ (669\ \text{rpm})}.$$
3. Maximum shaft speed (dynamometer-capacity criterion, worst case at bollard). $K_T$ and $K_Q$ are highest at bollard pull for a normal propeller curve, so the reference propeller's bollard values bound the expected loads there. From $T=K_{T,0}\rho n^2D_M^4\le250$ N and $Q=K_{Q,0}\rho n^2D_M^5\le10$ N·m: $$n_{max,T}=\sqrt{\frac{250}{K_{T,0}\,\rho\,D_M^4}}=\sqrt{\frac{250}{(0.35)(999)(0.200)^4}}=21.14\ \text{rps},$$ $$n_{max,Q}=\sqrt{\frac{10}{K_{Q,0}\,\rho\,D_M^5}}=\sqrt{\frac{10}{(0.04)(999)(0.200)^5}}=27.96\ \text{rps},$$ so the thrust cell governs: $n_{max}=\boxed{21.1\ \text{rps}\ (1268\ \text{rpm})}$.
4. Choose a single test shaft speed within range. Select $\boxed{n_{test}=18.0\ \text{rps}\ (1080\ \text{rpm})}$, comfortably inside $[11.16,\,21.14]$ rps, leaving margin on both the Reynolds-number floor and the dynamometer ceiling.
5. Carriage (advance) speed range to sweep $J=0$ to 1.1 at fixed $n$. With $n$ held constant, $J$ is swept by varying the towing-carriage speed $V_A=JnD_M$: $$V_{A,max}=(1.1)(18.0)(0.200)=\boxed{3.96\ \text{m/s at } J=1.1},$$ starting from $V_A=0$ (bollard pull, $J=0$).

Assumptions. (1) The open-water test is run in fresh water at the same 15 °C used elsewhere on the data sheet. (2) Both the Reynolds-number floor and the dynamometer ceiling are governed by the bollard-pull ($J=0$) condition, since advance speed only increases the section's resultant velocity (helping Reynolds number) while $K_T$, $K_Q$ — and hence dynamometer load — are highest at $J=0$ and fall as $J$ increases. (3) The un-named "similar" propeller's bollard $K_T$, $K_Q$ are representative enough of the test propeller to size the dynamometer ceiling conservatively. (4) Shaft (propeller) submergence is set to at least $1.5D_M\approx0.30$ m below the free surface to avoid ventilation/free-surface effects on the open-water measurement.