## 2 Method Overview
### Governing Equations
#### 2.1.3 Coupling  
In the coupling calculation, the analysis of fluid and particle governing equations is included in the program of the discrete phase model. Before solving the equation, the void fraction is calculated according to the particle position and element geometry of the finite volume grid, and then the particle momentum equation is solved. The source term caused by the volume fluid particle interaction is the key to solve the coupling. The source term is calculated and stored in the user-defined memory to avoid additional loops, because the user-defined source function is invoked by the solver at the cell level. Then, the governing equation of the fluid is solved, and the collision dynamics is calculated. After updating the fluid region, the next time step is entered. On this basis, [PERSON] et al. completed a reasonable calculation of the discrete phase model in dense conditions by improving the UDF and proposed a multifaceted numerical strategy to achieve high computational efficiency and mass conservation [9; 10; 11] (Figure 1).  
In terms of coupling mode, when the influence of sufficiently sparse particles on turbulence can be ignored, the interaction between particles and turbulence is called one-way coupling. This means that, in this case, the particle dispersion depends on the state of turbulence, but since the particle concentration is negligible, the momentum transfer from particle to turbulence has little effect on the flow. When the momentum transfer of particles is large enough to change the turbulent structure, this interaction is called two-way coupling. The exchange of mass, momentum and velocity information occurs between the particle phase and the fluid phase. On the basis of two-way coupling, particle collision occurs due to the increase in particle load. It is necessary to further consider  
Figure 1: Flowchart of CFD-DPM coupling technology [9; 10; 11].  
the collision between particles, which is called four-way coupling [12]. In the numerical method, the one-way coupling is solved via the steady-state method, and the particle trajectory completely follows the turbulence, which is convenient to calculate. The two-way coupling is solved in transient mode, and the solvers of CFD and DPM run in parallel, which can significantly reduce the simulation time. However, particles larger than CFD cells cannot be processed. Due to ignoring the effect of particles on turbulence, one-way coupling is sometimes easily distorted. The accuracy of two-way coupling is often higher than that of one-way coupling [13]. The above two methods can only be used in diluted conditions. For dense conditions, the simulation technology of four-way coupling is also realized by parallel computing. Considering the complexity of particle turbulence and particle-particle interaction, four-way coupling needs a more accurate and efficient numerical strategy [14; 15].  
In the continuous method of two phases, the discrete phase model in computational fluid dynamics is typically used for dilute suspensions or sprays where the volume fraction of the dispersed phase is relatively low, typically up to about 10-15%. At higher volume fractions, the continuous phase is still treated as a fluid with its own set of conservation equations (for mass, momentum and energy), while the dispersed phase is now treated as a continuous medium rather than as individual particles. The transition from a DPM to a continuous method for the dispersed phase often involves the use of models such as the Eulerian-Eulerian approach, where both phases are treated as interpenetrating continua, or the Eulerian-Lagrangian approach, which combines the continuous treatment of the fluid phase with a discrete representation of the particle phase using a population balance model (PBM) or a similar technique.