## 2 Numerical Model
### Time-Domain Numerical Simulation  
The numerical model of the stow net is designed by using a commercial software, OrcaFlex. This software is able to perform fully coupled analysis of floaters and line elements with consideration of nonlinear effects [21]. The net, rope, and mooring line are modelled by line and 3D buoy models. The line model is comprised of a series of nodes and segments. In the node, weight, buoyancy, drag, and properties related to force are lumped, and stiffness properties, i.e., axial, bending, and torsional stiffness, are modelled in the massless segment. The 3D buoy model is utilized for not only the connection of two or three lines considered as the dummy, rigid body causing negligible influences on entire dynamics but also the location buoy, weight, and other buoys. The equation of motion for the entire model can be describe as Equation (1):  
\[\mathbf{M(p,a)+C(p,v)+K(p)=F(p,v,}t) \tag{1}\]  
where \(\mathbf{M(p,a)}\) is the inertia force matrix, \(\mathbf{C(p,v)}\) is the damping force matrix, \(\mathbf{K(p)}\) is the stiffness force matrix, and \(\mathbf{F(p,v,}t)\) is the external force vector that is hydrodynamic force in this study. The symbols \(\mathbf{p}\), \(\mathbf{v}\), \(\mathbf{a}\), and \(t\) represent position, velocity, acceleration, and time, respectively. The hydrodynamic force from current and wave is calculated by the Morison equation for a moving object. The Morison equation is made up of linear inertia force and nonlinear drag force. In case of a cylindrical object, hydrodynamic force per unit length can be expressed as follows:  
\[\mathbf{F_{W}}=C_{m}\rho\frac{\pi D^{2}}{4}\dot{\mathbf{u}}-C_{a}\rho\frac{ \pi D^{2}}{4}\mathbf{a}+\frac{1}{2}\rho C_{d}D|\mathbf{u}-\mathbf{v}|(\mathbf{ u}-\mathbf{v}) \tag{2}\]  
where \(C_{m}\)(= 1 + \(C_{d}\)), \(C_{a}\), and \(C_{d}\) represent the inertia, added mass, and drag coefficients. \(D\) and \(\rho\) denote an outer diameter of a cylinder and density of seawater. \(\dot{\mathbf{u}}\) and \(\mathbf{u}\) are acceleration and velocity of fluid particles. \(C_{a}\) is fixed to be 1 for the entire stow net [22]. In addition, \(C_{d}\) of the net is chosen by the formulation suggested by [PERSON], et al. [23]. The formulation is revised from [PERSON] and Casarella [24] with expanded Reynold number up to \(10^{7}\), which is given in Equation (3):  
\[C_{d}=\left( \begin{array}{ll}\frac{8\pi r_{n}}{\mathrm{Re}_{n}s}\big{(}1-0. 87s^{-2}\big{)},&0<\mathrm{Re}_{n}<1\\ 1.45+8.55\mathrm{Re}_{n}^{-0.9},&1<\mathrm{Re}_{n}\leq 30\\ 1.1+4\mathrm{Re}_{n}^{-0.5},&30<\mathrm{Re}_{n}\leq 2.33\times 10^{5}\\ -3.41\times 10^{-6}\big{(}\mathrm{Re}_{n}-5.78\times 10^{5}\big{)},&2.33 \times 10^{5}<\mathrm{Re}_{n}\leq 4.92\times 10^{5}\\ 0.401\big{(}1-e^{-\mathrm{Re}_{n}/5.99\times 10^{5}}\big{)},&4.92\times 10^{5}< \mathrm{Re}_{n}\leq 10^{7}\\ s=-0.077215665+\ln(8/\mathrm{Re}_{n})&\end{array} \right), \tag{3}\]  
where \(\mathrm{Re}_{n}\) (= \(\rho V_{n}D/\mu\)) is Reynold number, and \(V_{n}\) denotes relative velocity between the fluid particle and the object acting on the normal direction. In case of mooring lines and ropes, \(C_{d}\) is fixed to be 1.2 [25]. Developing a numerically equivalent net model is required since it is unfeasible to make actual net model with a large number of threads. The entire procedures are well described in authors' previous publications (e.g., [PERSON] and [PERSON] [11]), and their numerical results have been well matched against various experimental results. The equivalent net model should be generated so that wet mass, projected area, and axial stiffness have to be equivalent between the actual physical net and numerical net. In this case, wet mass is adjusted by adjusting mass of 3D buoy installed between threads in the net setting the BWR of the net to be 1.01. The projected area is matched based on the solidity ratio that is defined as a ratio of total area of threads to the projected area perpendicular to wave and current directions, and solidity ratio is fixed to be 0.1 here.