## 2 Methodology
### Mathematical Model
#### 2.2.1 Hydrodynamic Modeling  
Firstly, a suitable model must be selected for the fluid flow problem by determining the Re of the open channel under consideration. The Reynolds number for the open channel is as follows [25]:  
\[\text{Re}=\frac{\rho_{\text{f}}\text{R}_{\text{h}}\text{U}}{\mu_{\text{f}}}, \tag{1}\]  
where \(\text{R}_{\text{h}}\) is the hydraulic radius. This radius for the open channel is determined as follows:  
\[\text{R}_{\text{h}}=\frac{\text{w}_{\text{ch}}\text{h}_{\text{f}}}{\text{w}_{ \text{ch}}+2\text{h}_{\text{f}}}, \tag{2}\]  
where \(\text{h}_{\text{f}}\) is the sewage water height (m), \(\text{w}_{\text{ch}}\) is the sewage channel width (m), and U is the sewage inlet flow velocity (m/s). The flow behavior in the open channel has the Reynold's number range of approximately 81,000 to 240,000 and is therefore assumed to be turbulent [26]. The k-\(\omega\) turbulence model is an appropriate choice for turbulent flows in an open channel with relatively high Reynolds numbers [27; 28]. This model was, therefore, selected to solve the hydrodynamics of the present study. The following assumptions are provided, considering this model, to solve the hydrodynamics of the 3D model. 1. The flow throughout the open channel is considered Newtonian and incompressible [29]. 2. The flow conforms to the principles of continuum theory, in which the governing equations are represented by the Navier-Stokes equations [30]. 3. The Wilcox k-\(\omega\) model, along with the two-way coupling ALE approach, is further assumed to be a suitable choice for open-channel flows with a Reynold's number range of 81,000 to 240,000 [27; 28]. 4. The fluid slips at the channel wall and the interface at the screening mechanism are neglected. Given the above assumptions, the flow in the sewage channel is governed by the following equations [25]:  
* Continuity equation: \[\rho_{\rm f}(\
abla.\mathbf{u_{f}})=0,\] (3) where \(\rho_{\rm f}\) is the fluid density (kg/m\({}^{3}\)), and \(\mathbf{u_{f}}\) is the fluid velocity vector (m/s). * Momentum equation: \[\rho_{\rm f}\frac{\partial\mathbf{u_{f}}}{\partial\mathbf{t}}+\rho_{\rm f}( \mathbf{u_{f}}\cdot\
abla)\mathbf{u_{f}}+\
abla\cdot[-\mathrm{pI}+\mu\Big{(} \
abla\mathbf{u_{f}}+(\
abla\mathbf{u_{f}})^{\rm T}\Big{)}+\mathbf{f_{f}},\] (4) where p denotes the fluid pressure (Pa), \(\mathbf{f_{f}}\) denotes the body force (N/m\({}^{3}\)), and \(\mathbf{I}\) denotes the identity matrix. These equations provide a time evolution of the flow. Reynolds-averaged Navier-Stokes equations are obtained by manipulating the aforementioned equations. The vector representing velocity, denoted as \(\mathbf{u_{f}}\) in Equations (3) and (4), shall be substituted with \(\mathbf{U}+\mathbf{u^{\prime}}\). Here, \(\mathbf{U}\) signifies the time-averaged velocity field in meters per second (m/s), while \(\mathbf{u^{\prime}}\) symbolizes the perturbative velocity component, characterized by fluctuations in meters per second (m/s). The k-\(\omega\) turbulence model is dictated by a pair of equations governing the turbulent kinetic energy and the specific dissipation rate. This model demonstrates efficacy, particularly within scenarios involving flow separation, thus yielding accurate solutions. The equations dictating the behavior of turbulent kinetic energy (k) are stated as [31]  
\[\frac{\partial\mathrm{k}}{\partial\mathrm{t}}+\mathbf{u_{j}}\frac{\partial \mathrm{k}}{\partial\mathrm{x_{j}}}=\tau_{\rm ij}\frac{\partial\mathrm{u_{i}} }{\partial\mathrm{x_{j}}}-\beta^{*}\mathrm{k}\omega+\frac{\partial}{\partial \mathrm{x_{j}}}\Bigg{[}(\
u+\sigma_{\mathrm{k}}\
u_{\rm T})\frac{\partial \mathrm{k}}{\partial\mathrm{x_{j}}}\Bigg{]}. \tag{5}\]  
The symbols \(\sigma_{\mathrm{k}}\) and \(\sigma_{\omega}\) represent the turbulent Prandtl numbers pertaining to kinetic energy and dissipation rate, respectively. k denotes turbulent kinetic energy (m\({}^{2}\)/s\({}^{2}\)), while \(\omega\) signifies the dissipation rate measured in reciprocal seconds (1/s).