## 3 Establishment of Numerical Model
### Flow Field Model Verification for Authenticity  
In order to validate the feasibility of the interaction between the flow field and particles as described above, a simulated experiment of particle free settling within the fluid is conducted to verify the accuracy of the bidirectional fluid-particle coupling method.  
In the simulated experiment of underwater particle free settling, three particles with different radii but consistent density and centroid positions are placed within the flow field to allow for their unconstrained settling. The vertical velocities of these particles are monitored during the experiment. The data utilized in the experiment are provided in Table 3.  
Given the premise of a particle undergoing unimpeded sedimentation within a fluid medium, the initial velocity of the particle relative to the fluid is zero. As the particle begins to move, according to [PERSON] law, the viscous drag force acting on the particle by the fluid is given by [38]:  
\[F=6\pi\eta\mu r \tag{19}\]  
In Equation (19), \(\eta\) represents the viscosity coefficient of the fluid, \(\mu\) denotes the settling velocity of particles, and \(r\) stands for the particle radius.  
The gravitational force acting on the particle is given by:  
\[G=\frac{4}{3}\pi r^{3}\rho_{p}g \tag{20}\]  
The buoyant force acting on the particle is given by:  
\[F_{f}=\frac{4}{3}\pi r^{3}\rho_{f}g \tag{21}\]  
Initially, the particle undergoes accelerated descent, and the viscous drag force increases. When the particle reaches a certain value of acceleration, the sum of the viscous drag force and the buoyant force equals the gravitational force. At this point, the particle experiences a net external force of zero, and its velocity becomes constant. The equilibrium equation at this stage is as follows:  
\[G=F_{f}+F \tag{22}\]  
\[\frac{4}{3}\pi r^{3}\rho_{p}g=\frac{4}{3}\pi r^{3}\rho_{f}g+6\pi\eta\mu r \tag{23}\]  
In a state of equilibrium, the particle's descending velocity can be calculated as follows:  
\[\mu=\frac{2r^{2}(\rho_{p}-\rho_{f})g}{9\eta} \tag{24}\]  
By substituting the data from Table 3 into the Equation (24), the free settling velocities for the three types of particles are determined as follows: \(-1.45\) mm/s, \(-3.27\) mm/s, and \(-5.81\) mm/s.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter Name** & **Symbol** & **Unit** & **Numerical Value** \\ \hline particle radius & \(r\) & m & \(0.001\backslash 0.0015\backslash 0.002\) \\ particle number & \(id\) & / & \(1\backslash 2\backslash 3\) \\ Particle density & \(\rho_{p}\) & g/cm\({}^{3}\) & 2000 \\ flow field density & \(\rho_{f}\) & g/cm\({}^{3}\) & 1000 \\ viscosity coefficient & \(\mu\) & 1 & 1.5 \\ gravitational acceleration & \(g\) & m/s\({}^{2}\) & 9.81 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The mesoscopic parameters utilized in simulation experiments of particle settling in water.  
The settling velocities of the three types of particles obtained through the PFC-CFD method are shown in Figure 9. Due to variations in particle sizes, they attain different free settling velocities. Based on the measured data, the final settling velocities for the three particles are \(-\)1.51 mm/s, \(-\)3.35 mm/s, and \(-\)5.86 mm/s, respectively. The simulated results closely agree with the calculated values based on Stokes' law.  
Consequently, it is inferred that the PFC-CFD computational approach employed in this research exhibits a notable degree of precision.