## 3 Results and Discussions
### Combined Non-Sinusoidal Pitching and Sinusoidal Plunging Motion  
The effects of the five different non-sinusoidal pitching motions on the extraction performance of parallel foils are considered. The motion of the foil is combined non-sinusoidal pitching and sinusoidal plunging motion. The non-sinusoidal pitching motion is controlled by the parameter \(K_{\theta}\). The energy extraction performance values at different \(K_{\theta}\) and \(k\) are simulated, as shown in Figure 9. It can be found that the \(C_{Pm}\) at four kinds of motions ascends firstly and then decreases with increasing \(k\), except the case at \(K_{\theta}=-1\). The peak values of \(C_{Pm}\) at these four motions all appear at \(k=0.8\). This means \(k=0.8\) is the optimal frequency which could allow for the maximal power output in general. When \(K_{\theta}=2\), the maximal \(C_{Pm}\) is obtained, which is \(0.623\). When the frequency is low, the effects of different oscillation motions on the energy extraction performance are less. With the increase in \(k\), the influence from the non-sinusoidal pitching motion  
Figure 8: The \(C_{l}\) curves of the upper foil under different \(K_{h}\) at \(k=0.8\): (**a**) \(K_{h}=0\); (**b**) \(K_{h}=2\). on \(C_{Pm}\) increases. Foils go into the energy consumption situation earlier and faster in the case with \(K_{\theta}=-1\) and \(K_{\theta}=2\) at high frequencies. The extraction efficiency \(\eta\) has the same trend with \(C_{Pm}\). The optimal extraction efficiency \(\eta\) achieves 0.312 at \(k=0.8\) with \(K_{\theta}=2\). To investigate the influence of plunging motion and pitching motion on energy extraction, we separate \(C_{Pm}\) into the plunging part \(C_{Plim}\) and pitching part \(C_{Plim}\), as shown in Figure 10. In the plunging oscillation process, the energy extracted by the plunging motion under four motions (except \(K_{\theta}=-1\)) ascends with the increasing \(k\). This is due to the increasing frequency lifts the plunging velocity \(V_{y}\) of foil, more energy is extracted by plunging motion as a result. When the frequency is relatively low, the different \(K_{\theta}\) has little influence on the power consumption of the pitching motion. Some non-sinusoidal pitching motions obviously cause the energy consumption with the increase in \(k\), like \(K_{\theta}=1\) and \(K_{\theta}=2\). This indicates that the more the non-sinusoidal pitching motion is similar to the square wave, the worse the extraction performance that is obtained. This is because the more the pitching motion is like a square wave, the higher the average pitching velocity in an oscillation cycle will be, causing higher energy consumption. That is the reason that causes the sharp decline of \(C_{Pm}\) at high frequencies in the case of \(K_{\theta}=2\) (see Figure 9). Figure 11 shows the \(C_{l}\) curves of the upper foil under two typical non-sinusoidal pitching motions \(K_{\theta}=1\) and \(K_{\theta}=2\) at \(k=0.8\). In these two cases, the plunging oscillation of the foil moves as with the same sinusoidal plunging motion. However, because of the effect from the non-sinusoidal pitching motion, the flow field around the foil is different, resulting in different forces in the \(y\)-direction. That is to say, the different pitching motions that cause different \(C_{l}\) values in the plunging motions are  
Figure 10: The \(C_{Plim}\) and \(C_{Plim}\) of parallel foils at different \(K_{\theta}\): (**a**) \(C_{Plim}\) curves under different \(k\); (**b**) \(C_{Plim}\) curves under different \(k\). Figure 9: The \(C_{Pm}\) and \(\eta\) of parallel foils at different \(K_{\theta}\): (**a**) \(C_{Pm}\) curves under different \(k\); (**b**) \(\eta\) curves under different \(k\). the same. This explains the reason for different \(C_{Plm}\) at the same \(k\) in Figure 10. From Figure 11, it is observed that the synchronicity of \(C_{l}\) and \(V_{y}\) at these two motions is similar. When \(K_{\theta}\) is 2, \(C_{l}\) fluctuates obviously near the plunging limited position, the peak value of \(C_{l}\) is higher.