## 4 Conclusions  
In this paper, we considered the effect of non-sinusoidal motion on the energy extraction performance of parallel foils. The energy extraction processes of two oscillation motions (the combined non-sinusoidal plunging and sinusoidal pitching motion, as well as the combined non-sinusoidal pitching and sinusoidal plunging motion) with \(Re=1100\) are simulated. Five non-sinusoidal motions are discussed and the optimal motion, the optimal frequency and the average power coefficient for these two combined oscillation motions are obtained. The contribution of plunging oscillation and pitching oscillation to energy extraction performance of foil is analyzed and the relationship between the lift force and the vortex structure is investigated.  
The results show that both the non-sinusoidal plunging motion and the non-sinusoidal pitching motion have an obvious effect on the energy extraction characteristics of foil. When the foils oscillate as the combined non-sinusoidal plunging and sinusoidal pitching motion, the optimal extraction performance of foil appears when \(K_{h}\) is \(-0.5\); the highest power output \(C_{Pm}\) and extraction efficiency \(\eta\) are 0.375 and 0.188, respectively. When \(k<1.0\), the combined motion with \(K_{h}=-0.5\) shows a better extraction performance, while the better performance appears at the combined motion with \(K_{h}=1\) when \(k>1.0\). At the same \(k\), the more the plunging motion approximates to the sinusoidal motion, the more energy is extracted by the parallel foils.  
When the motion is the combined non-sinusoidal pitching and sinusoidal plunging, the peak values of \(C_{Pm}\) under four different motions (except \(K_{\theta}=-1\)) are gained at \(k=0.8\). This means that the optimal frequency which can gain the best extraction performance of foil in most motions is 0.8. The optimal \(C_{Pm}\) and \(\eta\) are achieved at \(K_{\theta}=2\), which are 0.623 and 0.312, respectively. At a high frequency, the effect of the non-sinusoidal pitching motion on the extraction performance is obvious. With the increase in \(k\), the non-sinusoidal pitching motion at \(K_{\theta}=1\) and \(K_{\theta}=2\) cause a high consumption of energy. That is to say, the more the non-sinusoidal pitching motion is similar to the square wave, the worse extraction performance gained.  
According to the vortex fields under different non-sinusoidal pitching motions, it was found that the vortex structure at \(K_{\theta}=2\) is much bigger than that at \(K_{\theta}=1\) at every moment. The lift force of foil is higher at \(K_{\theta}=2\), and the synchronicity between \(C_{l}\) and \(V_{y}\) at these two motions are similar. As a result, the oscillation process of the foils extracts more energy when \(K_{\theta}\) is 2.  
[PERSON] and [PERSON] established the analysis method of the energy extraction performance of parallel foils; [PERSON] performed the numerical investigation and obtained the conclusions. [PERSON] established the computational model, discussed the results and wrote the paper.  
This research was funded by the National Natural Science Foundation of China, project number is 11872294.  
The authors declare no conflict of interest.