## 4 Results and Discussion
### Lift Coefficients for Semi-Submerged Cylinder  
Through the above identified method described in Section 4.2, the lift coefficients and phase angle are determined and summarized in Figures 14-16. Figure 14 shows the lift coefficients and phase angle changing with KC number under the same Stokes number. The maximum lift coefficient can reach 1.5, which is even larger than the typical value of drag coefficient for fully submerged cylinder (\(C_{D}=1.2\)) as shown in Figure 14a. Therefore, such large lift force for semi-submerged cylinder cannot be ignored in the related structure design and engineering projects.  
Similar to the drag coefficient, the discrete distribution can be also observed in lift coefficients when KC \(<15\). The lift coefficient varies from 1.19 to 1.51. At this region, the lift coefficients are significantly affected by the Stokes number and seem to increase with the value of the parameter \(\beta\). When KC \(>15\) and \(\beta<11\),364, the lift coefficients are stable at approximately 1.40 with KC number. The effects of Stokes number and KC number seem minor. Under higher Stokes number (\(\beta=11\),364), an unexpected drop of C\({}_{\rm L}\) occurs. C\({}_{\rm L}\) decreases from 1.45 to 1.25. This phenomenon will be discussed  
Figure 16: Lift coefficient and phase difference angle changing with Froude number under different KC number ((**a**) lift coefficients; (**b**) phase difference angle).  
Figure 14: Lift coefficient and phase difference angle changing with KC number under different Stokes number ((**a**) lift coefficients; (**b**) phase difference angle).  
Figure 15: Lift coefficient and phase difference angle changing with Froude number under different KC number ((**a**) lift coefficients; (**b**) phase difference angle).  
in next section. As for phase difference angle, the angles are stable with KC number under the same Stokes number. \(\varphi\) stabilizes at approximately 16\({}^{\circ}\), 19\({}^{\circ}\) and 26\({}^{\circ}\) for \(\beta\) = 11,364, 7353 and 5435, respectively. Different from lift coefficient, the angle decreases with Stokes number. Slight fluctuation can be found in the angle distribution under smaller Stokes number (\(\beta\) = 5435).  
To further investigate the effects of Froude number, Figure 15 illustrates the above two values versus Froude number under the same Stokes number. The stable trend of \(C_{\rm L}\) can be seen in the region of Fr ranging from 0.3 to 1.3 as presented in Figure 15a. Relatively dispersed distribution is also found under lower Froude number. Sharp downward trend happens when Fr is around 1.5. In addition, there is slightly obvious phenomenon that the lift coefficient rises with Stokes number when the Froude number is greater than 0.5. Figure 15b shows the phase angle with Froude number under the same Stokes number. More discrete features of the phase distribution can be observed under Fr \(<\) 0.5, while stable and slight downtrend appears under Fr \(>\) 0.5.  
Under the same KC number, the distribution of \(C_{L}\) versus Froude number is presented in Figure 16. \(C_{L}\) increases with Froude number when KC \(\leq\) 31.4, while decreases with \(Fr\) for KC \(>\) 31.4. Similar evolutionary phenomenon with drag and added mass coefficient can also be found in lift coefficient. However, there is no linear relationship between \(C_{L}\) and \(Fr\). The trend of the \(C_{L}\) changes from rapid rise to flat increase and finally shows a downward trend. As for the phase angle, \(\varphi\) decreases with Froude number, especially under smaller KC number (KC \(\leq\) 18.8) shown in Figure 16b. Rapid descend of the phase angle changes to gradual fall with KC number increasing. According to above comparison and analysis, it is more reasonable to use the Froude number and KC number to predict the lift coefficient and phase angle when the Stokes number and Froude number are in the conflict for the semi-submerged cylinder.