# Effect of above-waterline hull shape on broaching-induced roll in irregular stern-quartering waves

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## Abstract

In this paper, the effect of the above-waterline hull shape on broaching danger in irregular stern-quartering waves was numerically investigated using the US Office of Naval Research (ONR)-tumblehome and ONR-flare hulls. To indicate the danger of broaching, the probability of a broaching-induced large roll angle of the two vessels was examined along with the probability of broaching. The numerically estimated broaching-induced roll angles were compared with the time histories of the free-running model experiments. Then, the effects of above-waterline hull shape on broaching danger in the North Atlantic were simulated for various speeds, autopilot courses and rudder gains.

## Keywords

Effect of above-water hull form ONR-tumblehome ONR-flare Broaching Broaching-induced roll Irregular stern-quartering waves## Abbreviations

*a*_{H}Rudder force increase factor

*A*_{R}Rudder area

*F*_{N}Rudder normal force

*f*_{α}Rudder lifting slope coefficient

*J*_{P}Advance coefficient of propeller

*l*_{R}Correction factor for flow-straightening effect due to yaw rate

*N*_{R}Rudder-induced yaw moment

*U*Ship speed

*U*_{R}Resultant rudder inflow velocity

*u*_{R}Longitudinal inflow velocity component to rudder

*v*_{R}Lateral inflow velocity component to rudder

*w*_{P}Wake coefficient at propeller position

*w*_{R}Wake coefficient at rudder position

*X*Calm-water ship resistance

*x*_{HR}Longitudinal position of additional lateral force due to rudder

*x*_{R}Longitudinal position of rudder

*Y*_{R}Rudder-induced sway force

*α*_{R}Rudder inflow angle

*β*Drift angle

- \(\gamma\)
Flow-straightening coefficient for drift angle

- \(\delta\)
Rudder angle

*ε*Ratio of wake fraction at propeller and rudder position

- \(\rho\)
Water density

*θ*Pitch angle

- \(\kappa\)
Propeller inflow speed increase constant

*η*Ratio of propeller diameter to rudder span

*ξ*_{G}Longitudinal position of the centre of the ship’s gravity from a wave trough

- \(\varphi\)
Roll angle

*χ*Heading angle from wave direction

## Notes

### Acknowledgements

This work was supported by the US Office of Naval Research Global Grant no. N62909-13-1-N257 under the administration of Dr. Woei-Min Lin and a Grant-in Aid for Scientific Research from the Japan Society for Promotion of Science (JSPS KAKENHI Grant no. 15H02327). The authors would like to thank Enago (http://www.enago.jp) for the English language review.

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