## 2 Description and Numerical Model of the Foil under Waves  
### Description of the Foil under Waves  
In our study, the rigid or flexible foil shows heaving and pitching motions in regular waves. The numerical model is in two dimensions, as shown in Figure 1, in which \(h_{l}\) is the submergence depth of the foil's pitching center, \(c\) is the chord length of the foil, and \(h_{w}\) is the water depth of the fluid domain. Wave field is set as a regular wave with wavelength \(\lambda\) and wave height \(H=2a\), where \(a\) is wave amplitude. The origin point of the geodetic coordinate system \(xoy\) in Figure 1 is located at the interface between the water domain and the air domain, while the positive direction of the \(x\) axis is parallel to the wave propagating direction and the \(y\) axis point upwards to the air domain from the water domain. In numerical simulations, the foil is fixed in the direction of \(x\) axis and its horizontal average advancing velocity is transferred to the uniform water inflow velocity, shown as \(V_{0}\) in Figure 1. Similarly, the origin point of the local coordinate system \(x_{1}o_{1}y_{1}\) for the foil is located at its leading edge point, meanwhile this point is also the pitching center of the foil. \(L\) is the horizontal distance between the origins of the \(xoy\) and \(x_{1}o_{1}y_{1}\) coordinate systems.  
In order to capture the wave surface deformation, the volume of fluid (VOF) method is adopted and then the numerical wave tank (NWT) can be built, in which the regular progressive waveform can be expressed as:  
\[\zeta=a\cos(kx-\omega_{0}t) \tag{1}\]  
where \(k\) is the wave number, \(k=2\pi/\lambda\). \(\omega_{0}\) is the encounter frequency and for deep water conditions:  
\[\omega_{0}=\sqrt{gk}+kV_{0} \tag{2}\]  
where \(g\) is the gravitational acceleration. Imitating the motion of the caudal fin of fish, we simply define the foil movement in harmonic mode coupling in two degrees of freedom--heave and pitch. Based on the above assumption, the foil motion can be described in local coordinates as:  
\[y_{1}(t)=y_{0}\sin(2\pi ft+\psi) \tag{3}\]  
Figure 1: The scheme for an oscillating hydrofoil under a wave.  
\[\theta(t)=\theta_{0}\sin(2\pi ft+\psi+\varphi) \tag{4}\]  
where \(y_{1}(t)\) and \(\theta(t)\) represent the vertical and angular displacement of the foil in local coordinates, while \(y_{0}\) and \(\theta_{0}\) represent the heating amplitude and maximum pitching angle of the foil, respectively. \(f\) is the motion frequency of foil and \(\varphi\) defines the phase difference between heating and pitching motions of the foil. Meanwhile, \(\psi\) is the phase difference between the foil motion and the wave motion, and \(\psi\) decides the foil motion attitude when it is meeting a wave crest or trough.  
In addition, for the flexible foil, its deformation is defined along the chord direction. According to reference [33], the deforming offset \(\delta_{y}\) of the foil to the initial position versus time can be expressed as:  
\[\delta_{y}(t)=\delta_{c}\cdot\left(\frac{s}{s-1}x_{1}-\frac{1}{s-1}\right)^{ \varepsilon}\cdot\sin(2\pi ft+\psi+\varphi_{0}) \tag{5}\]  
where \(\delta_{c}\) is the deforming amplitude. \(f\) is the deforming frequency and assumed to be equal to the motion frequency of the foil in this paper. \(x_{1}\) is the distance between a location on the foil and its leading edge point, that is, the \(x_{1}\) axis value in the local coordinate system of Figure 1. \(s\) and \(\varepsilon\) are the controlling coefficients for deformation and \(s\), \(\varepsilon\)\(>\)1. According to Equation (5), a larger \(s\) makes the flexible deformation closer to the leading edge of the foil, while \(\varepsilon\) makes it closer to the trailing edge. \(\varphi_{0}\) is the phase difference between the deformation and the heating motion of the foil. In this paper, we take \(\delta_{c}/c=0.2\), \(s=100\), \(\varepsilon=2.0\), \(\varphi_{0}=-90^{\circ}\). Further details about the deformation of flexible foil can be found in [33].