## 4 Wave-Current Force on Pile
### Wave-Current Force Analysis  
#### 4.3.1 Effects of Current Velocity  
To investigate how the current velocity affects the wave-current forces on the pile, Figures 9 and 10 compare the peak values of the inline and transverse wave-current forces on the three pile columns (i.e., \(D\) = 4.5 cm, 7.5 cm, and 10.5 cm) at a water depth of \(d\) = 1.0 m and the wave-current parameters of \(H_{0}\) = 8.9 cm and 13.0 cm, \(T\) = 1.35 s, and \(U\) = 0-0.4 m/s (\(U/c_{0}\) = 0-0.194). As shown in Figure 9, when the current is relatively slow (\(U\) = 0.1 m/s), the current-affected wave height is reduced compared to that of pure waves, which means that the energy of the flow field is reduced. However, the increase in the velocity of water particles in the \(x\)-direction due to the current effect is not enough to offset the decrease in inline force caused by the decrease in wave height. Therefore, the peak inline force \(F_{x}\) on the pile under the combination of waves and a slow current is slightly less than that under wave-only conditions. As the current becomes faster (\(U\) = 0.3-0.4 m/s), although the current-affected wave height decreases further, the faster water particles in the \(x\)-direction make a more significant contribution to the forward wave flow force in the whole water depth. The  
Figure 10: Transverse wave-current forces on single-pile models versus relative velocity: (**a**) \(H_{0}\) = 8.9 cm, \(U/c_{0}\) = 0–0.194; (**b**) \(H_{0}\) = 13.0 cm, \(U/c_{0}\) = 0–0.194. Figure 9: inline wave–current forces on single-pile models versus relative velocity: (**a**) \(H_{0}\) = 8.9 cm, \(U/c_{0}\) = 0–0.194; (**b**) \(H_{0}\) = 13.0 cm, \(U/c_{0}\) = 0–0.194. peak value of the inline wave-current force \(F_{x}\) is significantly higher than that induced by pure waves (\(U\) = 0 m/s), and the increase rate of the peak value accelerates obviously as the current velocity increases. For the pile with \(D\) = 10.5 cm, the wave-current-induced inline forces for the cases of \(H_{0}\) = 8.9 cm, \(U\) = 0.4 m/s and \(H_{0}\) = 13.0 cm, \(U\) = 0.4 m/s, increase by ca. 51% and 31%, respectively, compared to the wave-only forces (for the cases of \(H_{0}\) = 8.9 cm and 13.0 cm). The variation trends of inline wave-current force \(F_{x}\) versus current velocity for the three piles are different. In the current velocity range of \(U\) = 0.2-0.4 m/s, the larger the pile diameter, the more obvious the upward trend of the forward wave-current force, as shown in Figure 9a,b. In addition, because of the current effect, when the current velocity reaches \(U\) = 0.1 m/s, the peak value of the inline wave-current force \(F_{x}\) increases rapidly for the pile with \(D\) = 4.5 cm, while for the pile with \(D\) = 10.5 cm, the growth rate of the peak value of \(F_{x}\) accelerates obviously when the flow velocity \(U\) exceeds 0.2 m/s, lagging behind that for the pile with \(D\) = 4.5 cm. Figure 10 shows that, similar to the inline wave-current force, the peak values of the transverse wave-current force \(F_{y}\) on the three piles increase as the current becomes faster, and the growth trend becomes more significant as the pile diameter increases. Compared with that under wave-only conditions or the combination of waves and a slow current, \(F_{y}\) increases greatly under the combination of waves and a fast current, and the increase can be by a factor of several tens. For \(H_{0}\) = 13.0 cm and \(U\) = 0.4 m/s, the peak transverse forces on the piles with \(D\) = 4.5 cm, 7.5 cm, and 10.5 cm are ca. seven, 31, and 20 times those under wave-only conditions, respectively. This indicates that the current velocity is an important factor affecting the increase of the transverse wave-current force. Moreover, comparing Figures 9b and 10b shows that the peak transverse force on the pile can reach the same value as the peak inline force under the combination of a strong wave and a strong current (i.e., \(H_{0}\) = 13.0 cm and \(U\) = 0.4 m/s).