## 4 Wave-Current Force on Pile
### Analysis of Hydrodynamic Coefficients
#### 4.4.1 Data Dispersion Analysis  
Similar to the experimental results of [PERSON] et al. [11; 12] and [PERSON] et al. [27], in this study the drag coefficient \(C_{D}\) (see Figure 13), inertia coefficient \(C_{M}\) (see Figure 14), and lift coefficient \(C_{L}\) (see Figure 15) are also somewhat dispersed. For the drag coefficient \(C_{D}\) and inertia coefficient \(C_{M}\), the wave surface fluctuations in adjacent cycles are different under wave-current conditions (see Figure 5), so naturally there is also a difference in the fluctuations of the inline force in adjacent cycles (see Figure 7), and it is clear from Figure 7 that this difference becomes more obvious with increasing current velocity. Therefore, the values of \(C_{D}\) and \(C_{M}\) estimated by using the time-domain least-squares method have some discreteness. Figure 13: Drag coefficient \(C_{D}\) versus \(KC^{*}\) under wave–current action: (**a**) \(D=4.5\) cm; (**b**) \(D=7.5\) cm; (**c**) \(D=10.5\) cm. Figure 14: Inertia coefficient \(C_{M}\) versus \(K\)C\({}^{*}\) under wave–current action: (**a**) \(D\) = 4.5 cm; (**b**) \(D\) = 7.5 cm; (**c**) \(D\) = 10.5 cm. Figure 15: Lift coefficient \(C_{L}\) versus \(K\)C\({}^{*}\) under wave–current action: (**a**) \(D\) = 4.5 cm; (**b**) \(D\) = 7.5 cm; (**c**) \(D\) = 10.5 cm. In addition, the distribution of \(C_{D}\) is similar to those of [PERSON] et al. (2012) and [PERSON] et al. (2018), who investigated the characteristics of inline forces on circular cylinders with small diameters under wave-only and wave-current conditions. For low \(K\)C*, the inertial forces play a dominant role in the Morison formula, so the error function in the least-squares method is insensitive to any error in \(C_{D}\), which is the main reason for the great fluctuation of \(C_{D}\) for low \(K\)C*; e.g., for \(K\)C* \(<\) 10, the fluctuation of \(C_{D}\) for the pile with \(D\) = 10.5 cm is significantly greater than that for \(K\)C* \(>\) 10. However, these discrete \(C_{D}\) values for low \(K\)C* have little impact on the total inline force predicted by the Morison formula. It is clear from Figures 13-15 that compared with the values of \(C_{D}\) and \(C_{M}\), those of \(C_{L}\) are more scattered. Using flow-visualization experiments, [PERSON] et al. (2017) examined the vortices around a circular cylinder in a flow field with co-existing waves and current. For either waves only or waves and a low-speed current, pairs of symmetric vortices formed at the sides of the cylinder and caused symmetric pressure on the cylinder surface. As the current velocity increased, large asymmetric vortices continuously formed and detached at the sides of the cylinder, and these increased the complexity of the flow field and significantly increased the asymmetry of the pressure on the cylinder surface. This is reflected in the increased difference in the fluctuations of the transverse force time series between adjacent cycles (see Figure 8), resulting in a significant increase in the dispersion of \(C_{L}\) under the combination of waves and high current velocity, e.g., for \(K\)C* \(>\) 10, the fluctuation of \(C_{L}\) for the pile with \(D\) = 10.5 cm is significantly greater than that for \(K\)C* \(<\) 10. Experimentally, [PERSON] et al. (2019) showed that the pressure difference caused by asymmetric vortex shedding has a significantly greater impact on the transverse force on a large-diameter cylinder than that on a small-diameter cylinder. In the present study, this is shown by the dispersion of \(C_{L}\) for the super-large-diameter pile (\(D\) = 10.5 cm) being greater than that for the smaller-diameter pile (\(D\) = 4.5 cm). Here, the data dispersion of \(C_{D}\), \(C_{M}\), and \(C_{L}\) as calculated from experimental data under wave-current conditions were explained in detail.