## 3 Application of the TGF_N_V for Solving Roll Damping of Hull Sections with Bilge Keels
### Hull Section Under Small Amplitude Roll  
A rectangular hull section with bilge keels is shown in Figure 2. The beam and mean draught of the hull section are \(b=0.4\) m and \(d=0.2\) m, respectively. Two bilge keels are set at the corners that connect the wall and bottom of the hull section, and the bilge radius and bilge keel depth are \(R_{\mathrm{b}}=0.02b\) and \(K_{\mathrm{d}}=0.04b\), respectively. The hull section harmonically rolls on the water surface with small amplitude \(\theta=0.05\) rad \(\approx 2.87^{\circ}\). The axis of rotation is located at the center of waterplane of the hull section. The detailed description of the experiments can be found in [PERSON] et al. [13]. As abovementioned, the novel [PERSON]'s function method TGF_N_V and two contrast methods (TGF and TGF_N [25]) are employed for the roll damping calculation. In each [PERSON]'s function method, the hull section is discretized into 160 segments (30 for each wall, 20 for each bilge keel, 60 for the flat bottom). In the linear method TGF, all segments are located on the mean wetted surface of the hull section and kept constant. On the other hand, in the nonlinear methods TGF_N and TGF_N_V, all segments move with roll motion, and the segments on two sidewalls need to be remeshed due to the change of draught. Two viscosity distribution approaches are adopted for TGF_N_V in this case. As shown in Figure 3a, in the first approach a non-zero viscosity is set only on the segments in the vicinity of bilge keels (\(16\leq i\leq 60,\ 101\leq i\leq 145\)), while on the rest segments the viscosity is set to zero. In the second approach (see Figure 3b) a non-zero viscosity is set on all segments (\(1\leq i\leq 160\)). The non-dimensional viscosity coefficient \(\epsilon\) should be empirically selected or obtained through the CFD or EFD methods. It is found that desirable results can be achieved if taking the non-dimensional viscosity coefficient as \(\epsilon=0.25\) for the first approach and \(\epsilon=0.17\) for the second approach. The roll frequency \(\omega\) is taken as the reference frequency \(\omega_{0}\) in TGF_N_V (see Equation (27)). Figure 2: A rectangular hull section with bilge keels harmonically rolls on the water surface, with beam \(b=0.4\) m, mean draught \(d=0.2\) m, bilge radius \(R_{\mathrm{b}}=0.02b\), bilge keel depth \(K_{\mathrm{d}}=0.04b\), and roll amplitude \(\theta=0.05\) rad. The rotation axis is located at the center of waterplane. Figure 4 portrays the equivalent roll damping using TGF, TGF_N, and TGF_N_V as compared with EFD. The lines labeled with \"TGF_N_V\({}_{1}\)\" and \"TGF_N_V\({}_{2}\)\" denote results from TGF_N_V by using the first and second viscosity distribution approach, respectively. Comparing the TGF_N_V\({}_{1}\) result with the TGF_N_V\({}_{2}\) one, one can find that both of two viscosity distribution approaches are effective for evaluating the roll damping of a hull section with bilge keels. Nonetheless, it can be observed that the second approach can yield smoother results. Thereby, in the next case, only the second viscosity distribution approach will be employed for study. From Figure 4, one can observe that, as compared with EFD data, the linear inviscid method TGF and body-nonlinear inviscid method TGF_N significantly underestimate the roll damping, though the discrepancy between TGF_N result and EFD data is smaller than that between the TGF result and EFD data, which suggests that the numerical results could be improved when the nonlinear effects are taken into account. Moreover, the TGF and TGF_N results decrease when \(\widetilde{\omega}>0.8\), which are in the opposite trend as compared to EFD data. On the other hand, the TGF_N_V results agree well with EFD data under the given non-dimensional viscosity coefficients, even when \(\widetilde{\omega}>0.8\). The numerical results in this case suggest that, for the hull section with bilge keels under roll motion of small amplitude, the viscous effects in the roll damping are more important than the nonlinear effects.