## 3 Method
### OrcaFlex Drag Force Calculation  
In this section, the question of how to apply the OF calculation capability, described in Section 2.2 to a fishing net, is answered. The used approach is to convert the net into equivalent horizontal cylinders or lines called net equivalent lines, running along the system wings, segmented based on a user-defined slender length (\(l_{s}\)), and vertically connected by links. Figure 9 presents a section of the wings modeled in OrcaFlex. The drag contribution of each segment is added along the wing to obtain the total drag force on that line. Since the wing underwater shape also influences the drag, it is necessary at least to have two drag equivalent lines vertically distributed to obtain a better representation of the real-world data (cf. the two orange lines in Figure 9). \begin{table}
\begin{tabular}{c c c c} \hline
**Span Type** & **Span (m)** & **System Ratio** & **System Status** \\ \hline Narrow & \(\approx\)16 & 0.02 &
\begin{tabular}{c} Towing behind \\ one vessel \\ Optimal performance \\ in operation \\ Maximize \\ plastic catch \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 1: Span values for \(SL\) = 800 m.  
Figure 8: Sketch of the system with basic definitions. Figure 9: Wing section model in OF. The normal drag coefficients of the net equivalent lines (\(C^{*}_{Dx,\text{OF}}\) and \(C^{*}_{Dy,\text{OF}}\)) were assumed equal to the normal drag coefficient derived from the semi-empirical formula proposed by [PERSON] et al. (2011), which reads:  
\[C^{*}_{D,n}= \begin{cases}19.4\bigg{(}\dfrac{2\xi^{*}_{s}}{\mathcal{R}e_{t}^{*}} \bigg{)}^{0.36},&\mathcal{R}e_{t}^{*}<400\xi^{*},\\ 9.3\bigg{(}\dfrac{2\xi^{*}_{s}}{\mathcal{R}e_{t}^{*}}\bigg{)}^{0.22},&400\xi^{* }\leq\mathcal{R}e_{t}^{*}<\mathcal{R}e_{c}^{*},\\ 1.41+1.7\xi^{*},&\mathcal{R}e_{t}^{*}\geq\mathcal{R}e_{c}^{*}.\end{cases}  \tag{22}\]  
with  
\[\xi^{*} =\dfrac{d_{t}}{s_{m}\prod\limits_{i=0}^{n}|h_{i}|}\quad\text{ defined in }\mathbb{R}^{n},\,n=2 \tag{23}\] \[\mathcal{R}e_{c}^{*} =2\xi^{*}\bigg{(}\dfrac{9.33}{1.41+1.7\xi^{*}}\bigg{)}^{4.63} \tag{24}\]  
where \(d_{t},\,d_{m},\,\mathbf{h}_{i},\) and \(\mathcal{R}e_{c}^{*},\) respectively, denote the twine diameter, the mesh size or twine distance, the directional hanging ratios, and the critical Reynolds number related to the transition to a non-linear drag formulation. The OF axial drag coefficient (\(C^{*}_{D,\text{OF},\mu}=C^{*}_{Dz,\text{OF}}\)) was initially approximated as \(1/(3\pi)\) of the normal drag coefficient, as this approximation is conservative. As an example, in the work of [PERSON] et al. (1996), the tangential or axial drag coefficient was estimated to be 1-2% of the normal drag coefficient. The OF model had the following three important limitations that triggered the need for a second validation cycle. They also determined the optimization method for the OF axial drag coefficient:  
1. Due to the way the net is modeled, since potential flow and one-way coupling were considered, there was no option to directly include or capture what is called the shielding effect between the twines of a submerged net. In contrast, the drag calculation in the software AquaSim and Basilisk does include this effect. 2. Due to the way the net was modeled, it was not possible to calculate the drag twine-by-twine and actually differentiate between horizontal and vertical twines. In contrast, the drag calculation in the software AquaSim does include this option. 3. The effect of the underwater cross-section shape of the net has on the drag force was not included in the model. Both OF normal drag coefficients were considered equal. That is, \(C^{*}_{Dx,\text{OF}}=C^{*}_{Dy,\text{OF}}=C^{*}_{D,\text{OF},n}=C^{*}_{D,n}\). This effect can be included by considering the torsion of the drag equivalent line, but that considerably increases the computation time.