## 2 Numerical Methods
### Hydrodynamic and Turbulence Module  
In this model, the renormalization-group (RNG) \(k\)-\(\varepsilon\) turbulence model was implemented as the turbulence closure, which has been successfully employed to investigate the turbulence field around the structure and the whole computational domain under the interaction between structure and water [29; 30]. The volume of fluid (VOF) method was used to capture the interface between the air and water [31; 32; 33], and the pressure and velocity was coupled by the generalized minimal residual methods (GMRES) pressure-velocity solvers. In the free surface flow, the mass and momentum conservation equations in the flow can be expressed as  
\[\
abla\cdot(S\mathbf{V})=0 \tag{1}\]  
\[\frac{\partial\mathbf{V}}{\partial t}+\frac{1}{F}\
abla\cdot(S\mathbf{V})=- \frac{1}{\rho}\
abla\cdot p+\mathbf{A}_{\text{object}}+\mathbf{A}_{\text{flow}} \tag{2}\]  
where \(\
abla\) is a gradient operator; \(\mathbf{V}\) is the velocity vector; \(S\) is the area fraction of the fluid in the system grid in Cartesian coordinate; \(F\) is the volume fraction of the fluid (the percentage of the fluid volume to the volume of a single cell). \(\rho\) is the fluid density, \(p\) is the pressure, \(\mathbf{A}_{\text{object}}\) and \(\mathbf{A}_{\text{flow}}\) are acceleration vector of object and viscous acceleration vector of flow, respectively. According to the volume of fluid (VOF) method, the transportation equation of the free surface between water and air is formulated as  
\[\frac{\partial f}{\partial t}+\frac{1}{F}\
abla\cdot(fS\mathbf{V})=0 \tag{3}\]  
where \(f\) is the water volume fraction in the cell of a free surface. \(f=1\), \(0<\)\(f<1\), and \(f=0\) represent the different phases of water, interface and air, respectively. The viscous acceleration vector of flow \(\mathbf{A}_{\text{flow}}\) in the three directions of the Cartesian coordinate can be written as  
\[\rho FA_{\text{flow},x}=str_{x}-\left[\frac{\partial}{\partial x}(S_{x}\tau_{ xx})+\frac{\partial}{\partial y}\Big{(}S_{y}\tau_{xy}\Big{)}+\frac{\partial}{ \partial z}(S_{z}\tau_{xz})\right] \tag{4}\]  
\[\rho FA_{\text{flow},y}=str_{y}-\left[\frac{\partial}{\partial x}\Big{(}S_{x} \tau_{yx}\Big{)}+\frac{\partial}{\partial y}\Big{(}S_{y}\tau_{yy}\Big{)}+ \frac{\partial}{\partial z}\Big{(}S_{z}\tau_{yz}\Big{)}\right] \tag{5}\]  
\[\rho FA_{\text{flow},z}=str_{z}-\left[\frac{\partial}{\partial x}(S_{x}\tau_{ zx})+\frac{\partial}{\partial y}\Big{(}S_{y}\tau_{zy}\Big{)}+\frac{\partial}{ \partial z}(S_{z}\tau_{zz})\right] \tag{6}\]  
where \(str\) (\(l=x\), \(y\), \(z\)) is the wall stress in three directions and included in an implicit way to avoid the possible numerical instabilities arising in cells with large wall areas and small flow volumes. The basic approach for the \(Vz\) (\(z\) direction velocity) equation, for example, is as follows. Wall shears influencing\(Vz\) can arise from the wall areas located on x or y cell faces surrounding \(Vz\). For any one of these faces, if the fractional flow area \(S\) is less than a unity, the remaining area fraction \((1-S)\) is considered to be a wall on which a stress is generated. On an \(x\) face to the right of \(Vz\), for instance, the acceleration due to wall shear in laminar flow, \(str_{z}\), is an approximation to \(\partial/\partial x\) (\(\mu\) (\(\partial Vz/\partial x\))):  
\[str_{z}=-\frac{2\mu}{S_{z}\delta x^{2}}(1-S_{x})(V_{z}-V_{z0}) \tag{7}\]  
where \(S_{x}\), \(\delta x\) (the cell size) and \(\mu\) (the viscosity at the cell center) are evaluated in the cell in which \(Vz\) is located. The area fraction \(S_{z}\) is estimated at the same control volume face where \(Vz\) is located. The velocity \(Vz_{0}\) is either zero or is equal to the z direction tangential velocity at the moving component and mesh boundaries. Because \(Vz\) is on the boundary between the two cells, the \(S_{x}\) is an average for these cells. Similar stresses are evaluated at each of the four surrounding cell walls, and their sum is taken as the total stress.