## 2 Governing Fluid-Structure Equations  
### Governing Equations for the Fluid  
We consider a three-dimensional flexible net panel \(\Omega^{s}\) interacting with an incompressible steady flow \(\Omega^{f}(t)\). The Navier-Stokes equations for an isothermal, incompressible, viscous fluid flow on a deformable spatial domain \(\Omega^{f}(t)\) in an arbitrary Lagrangian-Eulerian (ALE) formulation are:  
\[\rho^{f}\frac{\partial\mathbf{w}^{f}}{\partial t}+\rho^{f}(\mathbf{w}^{f}-\mathbf{w})\cdot \
abla\mathbf{w}^{f}=\
abla\cdot\mathbf{\sigma}^{f}+\mathbf{f}^{f}\text{ in }\Omega^{f}(t) \tag{1}\]  
\[\
abla\cdot\mathbf{w}^{f}=0\text{ in }\Omega^{f}(t) \tag{2}\]  
where \(\rho^{f}\) is the fluid density; \(\mathbf{u}^{f}=\mathbf{u}^{f}(\mathbf{x},t)\) and \(\mathbf{w}=\mathbf{w}(\mathbf{x},t)\) represent the fluid and mesh velocities at each defined spatial point \(\mathbf{x}\in\Omega^{f}(t)\), respectively; \(\mathbf{f}^{f}\) is the fluid body force; and \(\mathbf{\sigma}^{f}\) is the Cauchy-stress tensor for a Newtonian fluid, written as:  
\[\mathbf{\sigma}^{f}=-p\mathbf{I}+2\mathbf{\mu}^{f}\mathbf{e}^{f}\left(\mathbf{u}^{f}\right),\ \mathbf{e}^{f}\left(\mathbf{u}^{f}\right)=\frac{1}{2}\bigg{[}\
abla\mathbf{u}^{f}+\left( \
abla\mathbf{u}^{f}\right)^{T}\bigg{]} \tag{3}\]  
where \(p\) is the fluid pressure; \(\mathbf{I}\) denotes the second-order identity tensor; \(\mathbf{\mu}^{f}\) is the dynamic viscosity of the fluid; \(\mathbf{e}^{f}\) is the rate-of-strain tensor; and superscript \(T\) means transpose.  
The ALE formulated Navier-Stokes Equations (1) and (2) in the weak form can be written as:  
\[\begin{array}{l}\int_{\Omega^{f}(t)}\rho^{f}\Big{(}\frac{\partial\mathbf{w}^{f} }{\partial t}+\left(\mathbf{u}^{f}-\mathbf{w}\right)\cdot\
abla\mathbf{u}^{f}\Big{)}\cdot \mathbf{\phi}^{f}d\Omega+\int_{\Omega^{f}(t)}\mathbf{\sigma}^{f}:\
abla\mathbf{\phi}^{f}d \Omega\\ =\int_{\Omega^{f}(t)}\mathbf{f}^{f}\cdot\mathbf{\phi}^{f}d\Omega+\int_{\Gamma^{f}_\mathbf{{x}}(t)}\mathbf{T}^{f}\cdot\mathbf{\phi}^{f}d\Gamma+\int_{\Gamma(t)}\left(\mathbf{\sigma}^ {f}(\mathbf{x},t)\cdot\mathbf{u}^{f}\right)\cdot\mathbf{\phi}^{f}(x)d\Gamma\end{array} \tag{4}\]  
\[\int_{\Omega^{f}(t)}q\
abla\cdot\mathbf{u}^{f}d\Omega\ =0 \tag{5}\]  
Here, \(\mathbf{\phi}^{f}\) and \(q\) are the smooth test functions for the fluid velocity and pressure, respectively; \(\Gamma^{f}_{\mathbf{n}}(t)\) represents the fluid Neumann boundary, where \(\mathbf{\sigma}^{f}(\mathbf{x},t)\cdot\mathbf{u}^{f}=\mathbf{T}^{f}\); \(\mathbf{n}^{f}\) is the normal to the fluid boundary. The update to the deformable fluid subdomain is enforced by the ALE formulation (35; 36).  
The Reynolds-averaged equation is not closed, so it must be solved in combination with the turbulence model equation. In this paper, the shear stress transmission \(k-\omega\) turbulence model (SST \(k-\omega\)) is selected, and the expression is as follows:  
\[\rho\frac{\partial k}{\partial t}+\rho\frac{\partial}{\partial x_{i}}(ku_{i}) =\frac{\partial}{\partial x_{j}}\Bigg{(}\Gamma_{k}\frac{\partial k}{ \partial x_{j}}\Bigg{)}+G_{k}-Y_{k}+S_{k} \tag{6}\]  
\[\rho\frac{\partial\omega}{\partial t}+\rho\frac{\partial}{\partial x_{i}}( \omega u_{i})=\frac{\partial}{\partial x_{j}}(\Gamma_{\omega}\frac{\partial \omega}{\partial x_{j}})+G_{\omega}-Y_{\omega}+D_{\omega}+S_{\omega} \tag{7}\]Here, \(k\) is the turbulent kinetic energy; \(\omega\) is the turbulent dissipation rate; \(G_{k}\) and \(G_{\omega}\) are the generation terms of turbulent kinetic energy; and \(\Gamma_{k}\) and \(\Gamma_{\omega}\) represent the effective diffusivity of \(k\) and \(\omega\). \(Y_{k}\) and \(Y_{\omega}\) represent the dissipation terms of \(k\) and \(\omega\), respectively; \(D_{\omega}\) is the cross-diffusion term; and \(S_{k}\) and \(S_{\omega}\) are the user-defined source terms.