## 4 Partitioned Strong Coupling Formulation
This \(\mathbf{TrW}\) state can be divided into two phases, and they are defined as \(\mathbf{TrW_{1}}\) and \(\mathbf{TrW_{2}}\). In the second phase \(\mathbf{TrW_{2}}\), the irregular eddy transition happens during its formation, where the Reynolds number, \(Re\in[220\sim 250,\ 350\sim 400]\). Figure 6: Velocity distributions of the flow at the angles of attack between \(\pi/4\) and \(\pi/3\) for net solidities \(Sn=0.16\) (**left** column) and \(Sn=0.33\) (**right** column): (**a,b**) \(U_{\infty}=0.60\) m/s; (**c,d**) \(U_{\infty}=0.75\) m/s. In the _TrSL_ state, or subcritical state, the second transition occurs along the free shear layers while the boundary layers remain fully laminar. There are three phases of transition along the free shear layers, namely _TrSL\({}_{1}\)_, _TrSL\({}_{2}\)_ and _TrSL\({}_{3}\)_. In the first phase _TrSL\({}_{1}\)_, the transition eddy formation happens, where the Reynolds number \(Re\in[350\sim 400,\ 1000\sim 2000]\). Depending on the \(Re\), this paper examines the interactions between the fluid and the flexible net panels within the _TrW\({}_{2}\)_ and _TrSL\({}_{1}\)_ regimes. In the _TrW\({}_{2}\)_ regime, the transition of the irregular eddy happens during its formation. These irregular eddies facilitate the velocity reduction in the wake, causing greater velocity reduction at lower Reynolds numbers (e.g., \(Re<600\)). The loss of velocity in the wake occurs due to a loss of momentum [48]. In the _TrSL\({}_{1}\)_ regime, the flow in the transition zone is free shear flow and the boundary layers remain laminar. This produces a moderate velocity reduction as the Reynolds number rises. [PERSON] et al. [50] also presented the velocity distribution across the wake to the outer region, showing experimental evidence of how the velocity was distributed in the transition zone. Figure 7 exhibits the magnitude of the flow velocity \(u\) along the centerline of the net panel at different velocities. The overall trends of these data demonstrate that the velocity reduction exists due to the net panel's shielding effect, and that the velocity reduction increases alongside the net solidity. The average velocity reductions for these two nets are \(7.5\%\) and \(12\%\) with regard to \(Sn=0.16\) and \(Sn=0.33\), respectively. As the attack angle decreases, the velocity reduction for \(Sn=0.16\) drops from \(9\%\) to \(7\%\). In comparison, these values for \(Sn=0.33\) slide from \(15\%\) to \(10\%\). To estimate the velocity reduction caused by the net's shielding effect, [PERSON] [48] proposed a linear correlation between the velocity reduction factor and the drag coefficient. That is, \(r=1-0.46C_{D}\), where \(r\) is the velocity reduction factor. This linear correlation indicates that the velocity reduction decreases linearly with the drag coefficient. In our simulations, the trends remain the same when \(Sn=0.16\), and the linear correlation is \(r=1-0.77C_{D}\). However, the correlation becomes \(r=0.688+0.386C_{D}\) when \(Sn=0.33\). This is because [PERSON] [48] does not consider the interactions of the net twines, which could affect the drag coefficient of the net and the velocity distribution. Aside from the linear correlation, several values for the net panel/cage's velocity reductions are summarized by Xu and Qin [1]. Most of them favor the value of \(10\%\)[12; 13; 14; 15; 51; 45]. It is better to compare these values in specific conditions regarding the net solidity, attack angle and flow regimes (i.e., \(Re\)). Moreover, to avoid underestimating the velocity reduction, a value of \(20\%\) has been recommended [1]. Figure 7: The magnitude of the flow velocity \(u\) along the centerline of the net panel at different velocities (the 0.0 m along the x-axis denotes the center-line of the net). The origin drifts along the \(x\)-direction due to the deformation of the net (see Figure 7). The offset increases alongside the flow velocity; however, a slight difference exists between these two nets with different solidities.