## 2 Numerical Modeling and Validation  
### Description of the Oscillating Foil Model  
The paper details an inquiry into an oscillating foil model with a commonly used NACA0015 airfoil as its baseline configuration. A novel model is proposed, featuring a cylindrical spoiler structure located at the trailing edge, illustrated in Figure 1a. The parameters of the trailing edge cylindrical spoiler structure that are scrutinized in this study are expounded upon as follows: the diameter (\(D\)) of the cylinder, the transverse distance (\(X\)) from the leftmost point of the cylinder to the trailing edge of the foil, along the mid-chord direction, and the vertical distance (\(Y\)) from the center of the cylinder to the mid-chord of the foil. Presently, studies concerning energy harvesting from oscillating foils predominantly employ an actively coupled motion mode that involves both pitching and heaving. This motion pattern involves a rotational pitching motion around the axis of rotation, accompanied by a vertical heaving oscillation, as depicted in Figure 1b. These two motions are self-contained and are characterized by the classic sinusoidal waveform expressed as follows:  
\[\theta(t)=\theta_{0}\sin(2\pi ft) \tag{1}\]  
\[h(t)=H_{0}\sin(2\pi ft+\phi) \tag{2}\]  
where \(\theta(t)\) is the instantaneous pitching amplitude/rad; \(h(t)\) is the instantaneous heaving position/m; \(\theta_{0}\) is the pitching amplitude/rad; \(H_{0}\) is the heaving amplitude/m; \(f\) is the  
Figure 1: Schematics of the oscillating foil with a cylindrical spoiler structure at the trailing edge. (**a**) Oscillating foil with a cylindrical spoiler structure; (**b**) motion pattern of the oscillating foil. oscillating frequency/Hz; \(t\) is the motion time/s; and \(\phi\) is the phase difference between the coupled heaving and pitching motions, which is fixed at 90\({}^{\circ}\) in the study of this paper. The non-dimensional frequency, denoted as \(f^{*}\), is defined as follows:  
\[f^{*}=\frac{fc}{U_{\infty}} \tag{3}\]  
where \(c\) denotes the chord length of the foil/m, and \(U_{\infty}\) represents the freestream velocity far upstream of the oscillating foil. The power extracted from the flow field can be decomposed into the following two components: one from the heaving process and the other one from the pitching process. The expressions of the instantaneous total power are as follows:  
\[P(t)=F_{y}(t)\cdot V_{y}(t)+M(t)\cdot\omega(t) \tag{4}\]  
where \(P(t)\) is the instantaneous total power/W; \(F_{y}(t)\) is the instantaneous aerodynamic lift in the y-direction/N; \(V_{y}(t)\) is the instantaneous velocity of the foil during the heaving motion/m\(\cdot\)s\({}^{-1}\); \(M(t)\) is the instantaneous torque of the foil's pitching center/N\(\cdot\)m\({}^{-1}\); and \(\omega(t)\) is the instantaneous rotational angular velocity of the pitching motion/rad\(\cdot\)s\({}^{-1}\). The corresponding instantaneous total power coefficient \(C_{P}(t)\) and the average total power coefficient \(C_{Pm}\) of the oscillating foil during one cycle are the following:  
\[C_{P}(t)=\frac{P(t)}{\frac{1}{2}\rho U_{\infty}^{3}cL} \tag{5}\]  
\[C_{Pm}=\frac{1}{T}\int_{0}^{T}\frac{P(t)}{\frac{1}{2}\rho U_{\infty}^{3}cL}dt \tag{6}\]  
where \(T\) is the motion period of oscillation/s; \(\rho\) is the density of the incoming fluid/kg\(\cdot\)m\({}^{-3}\); and \(L\) is the spreading length of the foil/m. In this paper, the latter is considered to be a unit length for the two-dimensional flow investigation discussed.