The prediction of roll motion of a ship with bilge keels is particularly difficult because of the nonlinear characteristics of viscous damping. Flow separation and vortex shedding caused by bilge keels significantly affect the roll damping and the magnitude of the roll response. To predict roll damping and motion of a ship, the Slender-Ship Free-Surface Random Vortex Method (SSFSRVM) was employed. It is a free-surface viscous-flow solver with low computational cost so that it can run on a standard desktop computer. It features a quasi-three dimensional formulation that allows the decomposition of the three-dimensional hull problem into a sequence of two-dimensional computational planes, in which the two-dimensional free-surface Navier-Stokes solver FSRVM can be applied. In this work, the SSFSRVM methodology has been further developed to model multi-degrees of freedom of free-body motion in the time domain. This version of SSFSRVM model does not require the assumption of small amplitude motion, and is capable of having viscosity turned on or off in the solution procedure. Because FSRVM uses a grid-free formulation, there is no issue with numerical viscosity.

We validated the SSFSRVM in simulating the free roll decay motion of a naval vessel without forward speed. The numerically predicted vorticity distributions at different time instants near a bilge keel closely matched experimental PIV images. We found that the SSFSRVM model is capable of predicting the roll motion of a hull, and capturing the behavior of the vortical structures in the fluid. Further, we examined how the roll decay coefficients and the flow field were altered by the span of the bilge keels, based on the time-domain simulation of the coupled hull and fluid motion. Plots of vorticity contours and iso-surfaces along the three-dimensional hull were presented to reveal the motion of fluid particles and vortex filaments near the keels. In addition, the generation of the quadratic roll damping was investigated by showing the bilge-keel hydrodynamic moment and the pressure distribution on the hull surface and bilge keels.

Finally, the predicted roll time histories of a naval hull with three different forward speeds were compared with those obtained from experimental measurements. The numerical predictions were in good agreement with the experimental measurements for all three speeds. In addition, the numerical model also successfully produced the divergent waves with the same angles as those measured in the experiment, and accurately predicted the locations of the peaks and troughs of the divergent waves. The motion of the sonar-dome and bilge-keel vortex filaments, as well as their interactions, were presented to investigate the effect of forward speed. Significant influences of forward speed on the roll motion and roll damping were noted and explained.