## 4 Results and Discussion
### Empirical Prediction of the Lift Force for Semi-Submerged Cylinder in Oscillatory Flow  
As previously mentioned in Section 4.1, the lift force fluctuates at a frequency twice the forced motion frequency and an obvious phase difference between the force and velocity motion can be observed. Moreover, the mean value is found to be the main feature of the lift force in the CF direction. However, there is no empirical formula for lift force on semi-submerged cylinder in an oscillatory flow. In the real engineering project design, it is very difficult to efficiently determine the hydrodynamic force on these relevant marine structures. To reasonably describe the relationship between the lift force and flow velocity, above main features should be satisfied and thus we further propose the application of this empirical prediction of the lift force to partially submerged cylinder under an oscillatory flow:  
\[F_{YP}=\frac{1}{2}\rho C_{L}DL[U(t+\tau)]^{2}=\frac{1}{2}\rho C_{L}DLU_{m}^{2} \Big{[}\cos\Big{(}\frac{2\pi}{T}t+\frac{2\pi}{T}\tau\Big{)}\Big{]}^{2} \tag{16}\]  
where \(F_{YP}\) is the predicted lift force in the CF direction. \(C_{L}\) is lift force coefficient and \(\tau\) is a time delay representing a phase difference. For convenience of description, the phase difference \(\varphi\) is expressed as,  
\[\varphi=\frac{2\pi}{T}\tau\times\frac{180}{\pi}(^{\circ}) \tag{17}\]  
The Equation (16) can be simplified as,  
\[F_{YP}=\frac{1}{2}\rho C_{L}DLU_{m}^{2}[\cos(\omega t+\varphi)]^{2},\omega= \frac{2\pi}{T} \tag{18}\]  
Expanding Equation (18) out, we obtain  
\[F_{YP}=\frac{1}{4}\rho C_{L}DLU_{m}^{2}+\frac{1}{4}\rho C_{L}DLU_{m}^{2}\cos (2\omega t+2\varphi) \tag{19}\]  
From Equation (19), the predicted force can satisfy the main features observed in Section 4.1 and can be divided into two parts--the mean lift force \(F_{YPM}\) and twice the forced motion frequency fluctuating lift force \(F_{YPo}\). \[F_{YP}=F_{YPM}+F_{YP^{\circ}} \tag{20}\]Through Equation (20), the lift coefficient can be calculated by,  
\[C_{L}=F_{YM}/(1/4\rho DLU_{m}^{2}) \tag{21}\]  
where \(F_{YM}\) is the measured mean lift force in the CF direction. Here, Equation (18) only has two parameters that remain to be determined, the lift coefficient \(C_{L}\) and the phase difference \(\varphi\). However, if we attempt to directly describe the lift force as the mean and fluctuating component, we need to determine four parameters including the mean value, fluctuating frequency, fluctuating amplitude and phase difference. This novel empirical formula in Equation (18) directly assigns the fluctuating frequency as twice of the frequency of oscillatory flow and the fluctuating amplitude the same as the mean value, which reduced undetermined parameters from four to two. This reduction is directly based on our observation of the main feature of the measured lift force; for example, red line in Figure 6b. In the following, we demonstrate that such a reduction of parameters still well predicts the fluctuating lift force in a range of KC number, Froude number and Stokes number. To determine the value of \(\varphi\), the sum of the squared difference \(J_{1}(\varphi)\) between the predicted and measured fluctuating force, which can be expressed as,  
\[J_{1}(\varphi)=\sum\left[(F_{Y}-F_{YM})-(F_{YP}-F_{YPM})\right]^{2},\quad 0 \leq\varphi\leq 180^{\circ} \tag{22}\]  
Figure 8 presents the value of \(J_{1}(\varphi)\) versus phase difference angle \(\varphi\) under KC = 18.8 at \(\beta\) = 11,364. It can be seen that \(J_{1}(\varphi)\) minimizes at \(\varphi\) = 21\({}^{\circ}\). Thus, the phase difference angle at this case is adopted as 21\({}^{\circ}\). Using the same method, the value of \(\varphi\) for each test case can be obtained. To validate the predicted method, the identified lift coefficient and phase angle are substituted into the predicted formula of Equation (18).