## 4 Partitioned Strong Coupling Formulation
Therefore, the time step required to obtain accurate numerical solutions is tolerably large. Moreover, the spectral ratio \(\rho_{\infty}\leq 1\) to meet the partitioned scheme's unconditional stability. Alongside the stability to provide the partitioned fluid-structure coupling, the force correction is much more compatible with strong partitioned coupling than the displacement correction in a predictor-corrector framework. ## 5 Numerical Setup and Verification

### The Net Panel

Two knotless square-mesh flexible net panels are examined in this study. The dimension of 0.25 m \(\times\) 0.25 m is applied to both of them. The nets are made of polyethylene terephthalate (PET) and this material is commonly used for offshore aquaculture cages. The roughness of the net twine could be overlooked by using this material; thus, the net twines are considered as smooth cylinders in this study. A physical model of its geometry and physical properties, which have been used for numerical simulations, is displayed in Figure 2.

The parameters of the net twines and solidities are illustrated in Table 1. The velocity of the steady flow ranges from 0.15 m/s to 0.75 m/s with an increment of 0.15 m/s. Meaning that the effect of turbulence is not considered in this study. The corresponding Reynolds numbers are from 300 to 1500 based on the net twine diameter and the flow velocity. The drag coefficient is defined as \(C_{D}=F_{D}/\left(0.5\rho^{\prime}U_{\infty}^{2}A_{p}\right),\) where \(F_{D}\) is the time-averaged drag force estimated from the numerical simulations, \(\rho^{\prime}\) is the fluid density, and \(A_{p}\) is the projected area of the flexible net panel. During the simulation, the top of the net is connected to a rigid rod which remains completely fixed. The rest of the net panel is set free and with no sinkers attached in the bottom, so that the effect of the sinkers can be eliminated. Figure 1: Schematic of predictor–corrector procedure for ALE fluid and net panel coupling: (**a**) Solve structural displacement; (**b**) transfer predicted structural displacement, where \(\mathbf{u}^{\prime}=\mathbf{u}^{s}=\mathbf{w}^{f}\) and update the fluid dynamic mesh; (**c**) solve ALE fluid equations and (**d**) forces corrections. The density of the net is 1375 kg/m\({}^{3}\) and the modulus of elasticity, \(E=2.95\times 10^{9}\) Pa. With these material properties, the net panels in the present study are much heavier and much stiffer than that used in the previous experimental studies [15; 45; 46]. ### Computational Domain and Boundary Conditions

The computational domain in this work uses the symmetry of the problem and is set at 1.5 m (length) \(\times\) 0.5 m (width) \(\times\) 0.5 m (height) for the model, as shown in Figure 3. The domain parameters are larger than those used by [PERSON] et al. [15] for the effect of the computational domain on the wake behind the nets. Considering the fluid-solid interface, the tetrahedral grids are used to mesh the domain separately. Considering this, Dirichlet-Neumann coupling can be achieved via the interface between the fluid and the solid domains. Depending on the Reynolds number, grids near the net twines are set from \(7\times 10^{-5}\) m to \(2.5\times 10^{-4}\) m to capture the flow transition. The total number of grids is approximately \(1.2\times 10^{7}\). As addressed in Section 3.1, a uniform streamwise velocity (i.e., \(U_{\infty}\)) is imposed at the upstream inlet boundary, while a pressure outlet boundary condition is applied at the downstream outlet boundary. The lateral sides of the computational domain are set as slip boundaries using symmetry conditions. In particular, the interface traction from the fluid solver is considered as a Neumann boundary condition for the solid domain, and the interface velocity is applied as a Dirichlet boundary condition for the fluid problem. \begin{table}
\begin{tabular}{c c c} \hline \(S_{n}\) & \(d_{w}\) (mm) & \(I_{w}\) (mm) \\ \hline
0.16 & 2 & 25 \\
0.33 & 2 & 12 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the net panels.