## 3 Application of the TGF_N_V for Solving Roll Damping of Hull Sections with Bilge Keels
### Hull Section Under Large Amplitude Roll  
Figure 5 depicts an S60 midship section with bilge keels. The beam and mean draught of the hull section are \(b=0.237\) m and \(d=0.057\) m, respectively. The bilge radius and bilge keel depth are \(R_{\text{b}}=0.035\) m and \(K_{\text{d}}=0.01\) m, respectively. The single degree of freedom model experiment was carried out by [PERSON] et al. [15], in which the S60 midship section model is fixed by a forced rolling device, and the roll moment is measured through the torque transducer. The S60 midship section model harmonically rolls on the water surface with large amplitudes \(\theta=0.15\) rad \((8.59^{\circ}),\ 0.20\) rad \((11.6^{\circ}),\ 0.25\) rad \((14.4^{\circ})\). The axis of rotation is located at the center of waterplane of the hull section. More detailed description on the model experiments can be found in [PERSON] et al. [15].  
As in the previous case, the S60 midship section is discretized into 160 segments (30 for each wall, 20 for each bilge keel, 60 for the flat bottom). It is worth mention that the bilge keels in this case are simplified to lines without thickness, and each side of the bilge keel is discretized into 10 segments. The numerical error may occur when the source point and field point located on the two sides of the bilge keel are overlapped. To avoid this, the value of the Green's function is set as zero once the source point and field point are overlapped. Note that for the body-nonlinear methods, the segments on walls could be stretched or compressed with the roll motion of the S60 midship section. The second viscosity distribution approach, i.e., a non-zero viscosity for all segments on the wetted surface, is employed for TGF_N_V in this case. It is found that for TGF_N_V the non-dimensional viscosity coefficient \(e=0.2,\ 0.25,\ 0.3\) for the roll motion with amplitude \(\theta=0.15,\ 0.20,\ 0.25\) (rad), respectively, could lead to desirable results. It can be noted that for larger roll amplitudes, it requires larger non-dimensional viscosity coefficient for TGF_N_V. The roll frequency \(\omega\) is taken as the reference frequency \(\omega_{0}\) in TGF_N_V.  
Figure 6 portrays the equivalent roll damping using TGF, TGF_N and TGF_N_V as compared with CFD [15] and EFD [15]. One can find that, under all roll amplitudes, the linear inviscid method TGF and body-nonlinear inviscid method TGF_N dramatically underestimate the roll damping as compared with EFD data. Nonetheless, as compared to TGF, the TGF_N results are closer to EFD, which suggests that there exist nonlinear effects in the roll damping. On the other hand, the TGF_N_V results agree well with EFD data, though at high roll frequency TGF_N_V slightly underestimates the equivalent roll damping. In contrast, the CFD results also agree well with EFD data, though it overestimates the damping at high roll frequency.  
Figure 5: A semi-submerged S60 midship section with bilge keels harmonically rolls in the presence of a free surface; beam \(b=0.237\) m, mean draught \(d=0.057\) m, bilge radius \(R_{\text{b}}=0.035\) m, bilge keel depth \(K_{\text{d}}=0.01\) m, roll amplitudes \(\theta=0.15,\ 0.20,\ 0.25\) (rad). The rotation axis is located at the center of the waterplane.  
The numerical results in this case suggest that for the hull section with bilge keels under large amplitude roll, the viscous effects in the roll damping are much more important than the nonlinear effects, and one could obtain desirable numerical results only when both of nonlinear and viscous effects are considered. The study also confirms the effectiveness of TGF_N_V in evaluating the equivalent roll damping of a hull section with bilge keels rolling on the free surface with large amplitude.