## 3 Results
### Collision of Multiple Particles with a Flat Wall  
As a practical application, the present flow solver was applied to the flow around multiple particles colliding with a flat wall. Given the particle diameter and the relative velocity between the freestream and the particle after its impact with the wall, the Reynolds number was set to 400. The grid size was fixed at 0.05\(D\). The particle was assumed to be made of hard steel and have same diameter \(D\). Adiabatic conditions and non-slip boundary conditions were imposed on the particle surfaces. The particles were assigned random initial positions outside the computational domain. From the start of the simulation, many particles entered into the computational domain and were defined by  
Figure 16: Vorticity distributions during a particle–particle collision: (**a**) \(t^{*}=0.0\), (**b**) \(t^{*}=0.1\), (**c**) \(t^{*}=1.0\). Figure 15: Vorticity distributions during a particle–particle collision: (**a**) coarse mesh, (**b**) fine mesh. the level set function. The velocity of each particle was constant outside the computational domain but varied under gravitational and aerodynamics forces in the domain. The initial angular velocity of the particle was zero. The number of particles \(N_{\mathrm{p}}\) was set to 50, 100, 200 or 500. The size of the computational domain was 25.6\(D\)\(\times\) 25.6\(D\)\(\times\) 25.6\(D\) (Figure 17), and the wall was fixed at \(z=1D\). Dirichlet boundary conditions for the velocity and pressure were set on the upper and outer side boundaries, whereas Neumann boundary conditions are set on the upper boundary for the pressure and side boundaries for the velocity. The initial flow field was a steady flow without particles. Two coupling schemes were compared in this simulation: a one-way (A) and a two-way (B) scheme. The computational conditions are summarized in Table 2. In scheme A, the flow field was a steady flow and the aerodynamic forces for particle were calculated based on this steady flow. In scheme B, one the other hand, the flow field was influenced by the moving particle and the aerodynamic forces were calculated based on the flow conditions. In this simulation, the lubrication force [25] was not considered, because the difference between the one-way and two-way scheme was focused on. The CPU times are shown in Appendix A. The instantaneous wake structure of the particle in scheme B was visualized by the second invariant of the velocity tensors (Q-criterion). The results for 50, 100, 200 and 500 particles are displayed in Figures 18-21, respectively. The nondimensional time \(t^{\star}\) was defined by the particle diameter and freestream values. At \(t^{\star}=40\), multiple particles collided with the flat wall and rebound (panels (a) in Figures 18-21). At \(t^{\star}=50\), they rebounded and collided with other particles (panels (b) in Figures 18-21). After collision with the wall, a wake structure from the particle was observed as a hairpin vortex and developed with the change in relative velocity. When the number of particles became large, the vortices frequently interfered with the particles and other vortices. Moreover, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scheme** & **Number of Particles** & **Case** \\ \hline \multirow{3}{*}{one–way} & 50 & A050 \\  & 100 & A100 \\  & 200 & A200 \\  & 500 & A500 \\ \hline \multirow{3}{*}{two–way} & 50 & B050 \\  & 100 & B100 \\ \cline{1-1}  & 200 & B200 \\ \cline{1-1}  & 500 & B500 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test cases in the multiple particle–wall collision simulation. Figure 17: Computational domain used for the multiple-particle collisions with a flat wall (gray slab). Observation points are \(z=5D\), \(10D\) and \(15D\). many particles moved not only in the direction of the falling particles, but also in all other directions, being scattered by the freestream and in collisions with particle-particle. Figure 22 shows the instantaneous distributions of the velocity magnitudes at the center of the computational domain.