## 2 Numerical Model and Method
### Kinematics  
The frequency of the parallel foils applied in this paper is the reduced frequency, which is the non-dimensional frequency. It is defined as follows:  
\[k=\frac{2\pi fc}{U_{\infty}} \tag{1}\]  
where \(f\) represents the frequency, \(c\) represents the chord length of foil, \(U_{\infty}\) represents the velocity of the free stream. Based on the research of [PERSON] et al. [17], the power coefficient shows a similar trend under the different nominal angle of attack \(\alpha_{0}\). \(\alpha_{0}\) is set at \(15^{\circ}\) in this paper, the corresponding pitching amplitude \(\theta_{0}\) can be calculated as follows:  
\[\theta_{0}=\arctan\left(\frac{2\pi fH_{0}c}{U_{\infty}}\right)+\alpha_{0} \tag{2}\]  
where \(H_{0}\) is the dimensionless plunging amplitude and it is set at 1. Our previous study focused on parallel foils with combined sinusoidal plunging and sinusoidal pitching motion [20]. To discuss the effect of non-sinusoidal motion, the energy extraction characteristics of parallel foils under the combined non-sinusoidal plunging and sinusoidal pitching motion, as well as the combined non-sinusoidal pitching and sinusoidal plunging motion, are both investigated. Sinusoidal pitching oscillation and sinusoidal plunging oscillation are defined as Equations (3) and (4):  
\[\alpha(t)=\theta_{0}\sin(2\pi ft+\beta) \tag{3}\]  
Figure 3: The mesh strategy. Figure 2: The computation domains arrangement. \[h(t)=H_{0}c\sin(2\pi ft+\phi+\beta) \tag{4}\]  
where \(\beta\) for the upper foil is set at 180\({}^{\circ}\) and \(\beta\) for the lower foil is set at 0\({}^{\circ}\). \(\varphi\) represents the phase difference between the plunging motion and pitching motion; \(\varphi\) is set as 90\({}^{\circ}\)[10]. The non-sinusoidal pitching motion is calculated based on the control parameter of pitching motion \(K_{\theta}\):  
\[\alpha(t)=\left\{ \begin{array}{ll}\frac{\theta_{0}\sin^{-1}[-K_{\theta}\sin( 2\pi ft+\beta)]}{\sin^{-1}(-K_{\theta})},&-1\leq K_{\theta}<0\\ \theta_{0}\sin(2\pi ft+\beta),&K_{\theta}=0\\ \frac{\theta_{0}\tanh[K_{\theta}\sin(2\pi ft+\beta)]}{\tanh K_{\theta}},&K_{ \theta}>0\end{array} \right. \tag{5}\]  
In a similar way, non-sinusoidal plunging motion is calculated based on the control parameter of plunging motion \(K_{h}\):  
\[h(t)=\left\{ \begin{array}{ll}\frac{H_{0}c\sin^{-1}[-K_{\theta}\sin(2\pi ft+ \phi+\beta)]}{\sin^{-1}(-K_{h})},&-1\leq K_{h}<0\\ H_{0}c\sin(2\pi ft+\phi+\beta),&K_{h}=0\\ \frac{H_{0}c\tanh[K_{h}\sin(2\pi ft+\phi+\beta)]}{\tanh K_{h}},&K_{h}>0\end{array} \right. \tag{6}\]  
In this paper, different non-sinusoidal plunging motions and non-sinusoidal pitching motions are considered, including the cases under \(K_{h}\) = \(-\)1, \(-\)0.5, 0, 1 and 2 as well as the cases under \(K_{\theta}\) = \(-\)1, \(-\)0.5, 0, 1 and 2. The curves of the five non-sinusoidal plunging motions and non-sinusoidal pitching motions at \(k\) = 0.8 are presented in Figure 4a,b, respectively. The energy extraction process of the foil over one cycle consists of two parts, the plunging oscillation and the pitching oscillation, the power output \(P\) is defined as follows:  
\[P=F_{y}(t)V_{y}(t)+M(t)\omega(t) \tag{7}\]  
where \(F_{y}(t)\) is the lift force in the direction of the \(y\)-axis, \(M(t)\) is the torque, \(\omega(t)\) is the velocity of the pitching oscillation. The transient power coefficient \(C_{P}\) can be computed by Equation (8):  
\[C_{P}=\frac{P}{\frac{1}{2}\rho U_{\infty}^{3}c} \tag{8}\]  
Figure 4: The curves under the different non-sinusoidal motions at \(k\) = 0.8: (**a**) non-sinusoidal plunging motion; (**b**) non-sinusoidal pitching motion.