## 4 Partitioned Strong Coupling Formulation
The vortices generated by flow separation and recirculation exhibit distinct characteristics. These are: (1) a single vortex appears behind the net twine when the solidity \(Sn=0.16\), whereas when \(Sn=0.33\), there is a pair of vortices behind the net twine; (2) vortices distribute both in the shear layers and the wake when \(Sn=0.16\), but when \(Sn=0.33\), vortices distribute mostly in the wake rather than in the shear layers; (3) vortices decay faster in the top section than those in the middle and bottom sections; (4) when the flow transfers from \(\mathbf{TrW_{2}}\) to \(\mathbf{TrSL_{1}}\), the recirculation width increases when \(Sn=0.16\), and it decreases when \(Sn=0.33\). This is related to the transition of the wake and shear layers; (5) When \(Sn=0.16\), the reattachment of the shear layers occurs in the \(\mathbf{TrSL_{1}}\) regime. The reattachment moves upwards from one side to the other of the net twine in the middle and bottom sections. There is no significant change for the attachment in the top section. We would expect in-phase vortex structures since there is no mutual interference between net twines if the net solidity is \(0.16\). However, due to the net deformation, the single vortex exhibits a distinct formation even at the same frequency in different net sections. As the Reynolds number increases and the flow is in the \(\mathbf{TrSL_{1}}\) phase, vortices form in the shear layers. This could be the reason that the velocity fluctuation happens along the \(y\)-direction in the case that \(Sn=0.16\). The pair of vortices behind the net twines for \(Sn=0.33\) are more likely wake interactions. This finding differs from the predictions in Loland [48] because the author assumes that the interactions between the net twines can be overlooked if the net solidity is less than \(0.33\). In fact, the distance between net twines significantly affects the shear layers at \(Sn=0.33\), causing the shear layers to be more unstable. Figure 9: The selected sections on the net panels are used to monitor vortex formation behind vertical net twines. Here, these two deformed net panels are under a flow velocity of \(0.75\) m/s. We note that vortices decay faster in the top section than those in the middle and bottom sections. This indicates that, as the inclination angle increases, the shielding effect becomes weak, resulting in less energy loss. Thus, the recirculation bubble size (width) in the middle/bottom section is greater than those in the top section, except for \(Re=1500\). The statistical evidence is shown in Figure 11. In the case that \(Sn=0.16\), when the flow switches from \(\textbf{TrW_{2}}\) to \(\textbf{TrSL_{1}}\), the shear layers are supposed to reattach to the body surface, which they do, but this rarely happens in the top section of the net. The transition of the altered shear layers produces larger recirculation bubbles in the middle and bottom sections. Furthermore, the reattachment of the altered shear layers on the net twines should be responsible for the drag reduction. In the case that \(Sn=0.33\), where the shear layers have been disturbed, the shrink of the recirculation bubble in the top section is most likely attributed to the energy loss. However, this prediction needs a power spectral analysis and will not be addressed in the present study. Figure 11 presents the dependence of the non-dimensional recirculation width (\(W_{r}^{*}\)) on \(Re\). To compare the magnitudes of the recirculation size, we normalize the recirculation width with the diameter of the net twine \(d_{w}\). Furthermore, the non-dimensional recirculation width (\(W_{r}^{*}\)) is measured from a single vortex since most pairs of vortices when \(Sn=0.33\) seem symmetrical. Figure 10: Two-dimensional instantaneous streamlines of three selected sections at different Reynolds numbers in the same time frame. (The top, middle and bottom are three typical sections on the net panels, as shown in Figure 9).