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

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.

Appendix

The rudder forces and moments were calculated by the following equations [12]:

$${Y_{\text{R}}}= - (1+{a_{\text{H}}}){F_{\text{N}}}\cos \delta$$

(1)

$${N_{\text{R}}}= - ({x_{\text{R}}}+{a_{\text{H}}}{x_{{\text{HR}}}}){F_{\text{N}}}\cos \delta$$

(2)

$${F_{\text{N}}}=(1/2)\rho {A_{\text{R}}}U_{{\text{R}}}^{2}{f_\alpha }\sin {\alpha _{\text{R}}}$$

(3)

$${u_{\text{R}}}=\varepsilon u(1 - {w_{\text{p}}}) \cdot \sqrt {\eta {{\left\{ {1+\kappa \left( {\sqrt {1+\frac{{8{K_{\text{T}}}}}{{\pi J_{{\text{P}}}^{2}}}} - 1} \right)} \right\}}^2}+(1 - \eta )}$$

(4)

$${v_{\text{R}}}=U\gamma (\beta - l_{{\text{R}}}^{\prime }{r^\prime })$$

(5)

$$\varepsilon =(1 - {w_{\text{R}}})/(1 - {w_{\text{P}}})$$

(6)