## 4 Results
### Validation of the Numerical Model
Figure 6: Comparison of breaking wave forces between the present result and the published data for the vertical cylinder. The two conducted validation studies show that the present numerical model has good performance in predicting free surface elevations and breaking wave forces on the vertical and inclined cylinders. Therefore, the present model is also considered to give reasonable accuracy for simulating breaking wave past a cylinder with different transverse inclined angles. Characteristics of the Free Surface Elevations and the Breaking Wave Forces on the Standing Cylinder with Different Transverse Inclined Angles  
In this section, the free surface elevations and the wave forces on the cylinder with different transverse inclined angle \(\theta\) are studied. Note that the transverse inclined angle \(\theta\) is different from the angle \(\varphi\) mentioned in Section 4.1.2. \(\theta\) is the inclined angle normal to the wave propagating direction and \(\varphi\) is the inclined angle parallel to the wave propagating direction. Figure 9 depicts the variations of the free surface elevations at WG1 and WG3 locations over one wave period. The peak value of the free surface elevation for the vertical cylinder (\(\theta=0^{\circ}\)) is significantly higher than that for cylinders with other transverse inclined angles at WG1 location. The relative difference of the wave height is reduced by 48.8% when the angle \(\theta\) varies from \(0^{\circ}\) to \(45^{\circ}\). A small part  
Figure 8: Comparison of breaking wave forces between the present result and the published data for the inclined cylinder (\(\varphi=-22.5^{\circ}\)). Figure 7: Comparison of free surface elevations at WG2 between the present result and the published data for the inclined cylinder (\(\varphi=-22.5^{\circ}\)). of the water mass will be reflected due to the existence of the cylinder. Therefore, the second peak of the free surface elevation is also observed in Figure 9a. The maximum value of the wave height at WG3 occurs when the breaking wave passes the vertical cylinder (\(\theta\) = 0\({}^{\circ}\)). Figure 10 shows the peak values of the free surface elevations at WG1 and WG3 for the cases of the cylinder with different \(\theta\). The free surface elevation at WG1 decreases first and then increases slightly. The minimum surface elevation is obtained at \(\theta\) = 45\({}^{\circ}\). The free surface elevation at WG3 gradually decreases with \(\theta\) varying from 0\({}^{\circ}\) to 30\({}^{\circ}\) and increases when \(\theta\) increases from 30\({}^{\circ}\) to 75\({}^{\circ}\). This can be explained as given below. When the breaking wave passes a standing cylinder, it will produce a chute-like jet at the back of the cylinder. The height of the chute-like jet is related to transverse inclined angle of the cylinder and decreases with increasing \(\theta\). This is the reason that the free surface elevation at WG3 reduces first. With the further increase of \(\theta\), the cylinder's resistance to waves in the vertical direction is weaker, and waves can propagate more easily through the cylinder, which increases the surface elevation. Figure 9: Variations of the free surface elevations around the cylinder with normalized time: (**a**) free surface elevations before the cylinder (WG1); (**b**) free surface elevations behind of cylinder (WG3). The forces on the cylinder are normalized by \(F^{\prime}=F_{c}/\big{(}0.5\rho_{water}(L/T)^{2}S_{h}\big{)}\), where \(F_{c}\) represents different compositions of wave forces, \(L\) is the wavelength, and \(S_{h}\) is the submerged projected area in the direction of wave propagation. Figure 11 shows the normalized \(F_{T}\) on the cylinder versus \(\theta\) over one wave period. The peak of the normalized \(F_{T}\) is the largest for the case of the cylinder with \(\theta=0^{\circ}\).