# Regular Wave Effects on the Hydrodynamic Performance of Fine-Mesh Nettings in Sampling Nets
## 2 Material and Methods
### Numerical Simulation of the Netting Models
#### 2.3.2 Porous Media Model  
The porous media fluid model is a theoretical approach used to represent the flow resistance induced by netting by introducing appropriate porous resistance coefficients. Outside the porous media region, the source term \(S_{i}\) in the momentum equa  
Figure 2: Schematic diagram of the PIV measurement experiment setup in the wave tank.  
tion is set to zero. Within the porous media region, however, \(S_{i}\) is calculated using the following expression:  
\[S_{i}=-\left(\sum_{j=1}^{2}D_{ij}\vec{u_{j}}+\sum_{j=1}^{2}C_{ij}\frac{1}{2} \rho\Big{|}\vec{u_{j}}\Big{|}\vec{u_{j}}\Big{|}\vec{u_{j}}\right) \tag{10}\]  
\[D_{ij}=\left(\begin{array}{ccc}D_{n}&0&0\\ 0&D_{t}&0\\ 0&0&D_{t}\end{array}\right) \tag{11}\]  
\[C_{ij}=\left(\begin{array}{ccc}C_{n}&0&0\\ 0&C_{t}&0\\ 0&0&C_{t}\end{array}\right) \tag{12}\]  
where \(D_{ij}\) and \(C_{ij}\) are material matrices representing the viscous and inertial resistance coefficients, respectively. In these matrices, \(D_{n}\) and \(C_{n}\) are the normal resistance coefficients, while \(D_{t}\) and \(C_{t}\) denote the tangential resistance coefficients. The subscripts \(i\) and \(j\) correspond to the spatial directions \(x\) and \(y\), respectively. This formulation incorporates both a linear term (viscous resistance) and a quadratic term (inertial resistance), consistent with [PERSON]'s law extended for porous media flows. The total hydrodynamic force acting on the netting consists of drag and lift components. The drag force (\(F_{D}\)) acting on the netting panel is aligned with the water particle velocity in the \(x\)-axis direction, while the lift force (\(F_{L}\)) acts perpendicular to it, along the \(y\)-axis. These forces are calculated using the following expressions:  
\[F_{D}=\frac{1}{2}\rho C_{D}Au_{0}^{2} \tag{13}\]  
\[F_{L}=\frac{1}{2}\rho C_{L}Au_{0}^{2} \tag{14}\]  
where \(C_{D}\) and \(C_{L}\) are the drag and lift coefficients, respectively; \(A\) is the projected area of the netting; and \(u_{0}\) is the velocity of the upstream water particles.  
Previous studies have demonstrated that, under wave conditions, the vertical force exerted on fishing nets is relatively minor compared to other hydrodynamic forces and can, therefore, be reasonably neglected [29]. When modeling the netting as a porous medium, the hydrodynamic force acting on it can be expressed as:  
\[F=S_{i}\lambda A \tag{15}\]  
By substituting Equations (10) and (13) into this expression, the drag force acting on the porous medium can be derived:  
\[F_{D}=\left(D_{n}\mu u+\frac{1}{2}C_{n}\rho|u|u\right)\lambda A \tag{16}\]  
where \(\lambda\) is the thickness of the porous media, and a thickness of 20 mm was chosen in this study [17], and \(u\) is the water particles' velocity inside the porous media. In this study, the porous media resistance coefficients were calculated based on the wave force data obtained during the wave tank experiments (Section 2.1) for each netting. These coefficients were derived using a polynomial regression method, which fits the measured wave forces from the experiments to the corresponding water particle velocities, as described by Equations (3), (5), and (16) (\(R^{2}>0.9\)) (See Table 3).