## 2 Materials and Methods
### Governing Equations and Parameters  
#### 2.2.1 Governing Equation  
The flow was controlled by the continuity equation and momentum equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}\big{(} \rho u_{i}u_{j}\big{)}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\Bigg{[}\mu\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac {\partial u_{j}}{\partial x_{j}}\Bigg{)}\Bigg{]}+\frac{\partial}{\partial x_{ j}}\Big{(}-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\Big{)}+S_{i} \tag{2}\]  
Figure 1: The structure diagram of each zone of the flow channel of the membrane pool, a: inlet, b1: outlet1; b2: outlet2 c: porous media semipermeable membrane; d: outlet.  
where \(S_{i}\) is the addition of a momentum source term.  
The turbulent \(k\)-\(\varepsilon\) model was used to take turbulence into account.  
\[\rho\frac{\mathrm{d}k}{\mathrm{d}t}=\frac{\partial}{\partial x_{i}}\bigg{[} \bigg{(}\mu+\frac{\mu_{\mathrm{t}}}{\sigma_{\mathrm{k}}}\bigg{)}\frac{\partial k }{\partial x_{i}}\bigg{]}+G_{\mathrm{k}}+G_{\mathrm{b}}-\rho\varepsilon-Y_{ \mathrm{M}} \tag{3}\]  
\[\rho\frac{\mathrm{d}\varepsilon}{\mathrm{d}t}=\frac{\partial}{\partial x_{i}} \bigg{[}\bigg{(}\mu+\frac{\mu_{\mathrm{t}}}{\sigma_{\mathrm{k}}}\bigg{)}\frac{ \partial\varepsilon}{\partial x_{i}}\bigg{]}+C_{1\varepsilon}\frac{ \varepsilon}{k}(G_{\mathrm{k}}+C_{3\varepsilon}G_{\mathrm{b}})-C_{2 \varepsilon}\rho\frac{\varepsilon^{2}}{k} \tag{4}\]  
where \(\mu\) is dynamic viscosity, \(\mu_{t}\) is the turbulent viscosity coefficient, \(G_{k}\) is the turbulent kinetic energy generation due to the mean velocity gradient, \(G_{b}\) is the turbulent kinetic energy generation with floating effects, \(e\) is the turbulent dissipation rate, \(k\) is the turbulent kinetic energy, \(Y_{M}\) is the influence of compressible turbulent pulsating expansion on total dissipation rate, and \(C_{1\varepsilon}\), \(C_{2\varepsilon}\), \(C_{3\varepsilon}\) are model parameters.  
The standard wall function, based on Launder, B.E. and Spalding, D.B. [24], was used in this numerical calculation. Furthermore, by setting the boundary layer close to the wall, the mesh near the wall was refined, and the \(y+\) of the first layer near the wall was set to about 30.  
The particle's motion is controlled by the equation:  
\[\frac{d\overrightarrow{u}_{p}}{dt}=\frac{\overrightarrow{u}-\overrightarrow{ u}_{p}}{\tau_{r}}+\frac{\overrightarrow{s}(\rho_{p}-\rho)}{\rho_{p}}+ \overrightarrow{F} \tag{5}\]  
\[\tau_{r}=\frac{\rho_{p}d_{p}}{18\mu}\frac{24}{C_{\mathrm{d}}Re} \tag{6}\]  
\[Re\equiv\frac{\rho d_{p}\left|\overrightarrow{u}_{p}-\overrightarrow{u} \right|}{\mu} \tag{7}\]  
here \(\tau_{r}\) is the particle relaxation time, \(\overrightarrow{u}\) is the fluid phase velocity, \(\overrightarrow{u}_{p}\) is the particle velocity, \(\mu\) is the molecular viscosity of the fluid, \(\rho\) is the fluid density, \(\rho_{p}\) is the density of the particle, \(d_{p}\) is the particle diameter, \(Re\) is the relative Reynolds number, and \(\overrightarrow{F}\) is the virtual quality force.  
[PERSON]'s law describes the linear relationship between the seepage velocity and hydraulic gradient of water in saturated soil, also known as the linear seepage law. Porous media were modeled by the addition of a momentum source term to the standard fluid flow equations. The source term is composed of two parts: a viscous loss term and an inertial loss term.  
\[\
abla p=-\frac{\mu}{\alpha}\overrightarrow{v} \tag{8}\]  
\[\
abla p=-\sum_{j=1}^{3}C_{2\psi}\left(\frac{1}{2}\rho v_{j}|v|\right) \tag{9}\]  
where \(C_{2}\) is the inertia resistance coefficient, and \(\frac{1}{\alpha}\) is the viscous resistance coefficient. The parameters \(\frac{1}{\alpha}\) and \(C_{2}\) are determined by experimental results, as shown later.