# A Parallel Algorithm Based on Regularized Lattice Boltzmann Method for Multi-Layer Grids
## 4 Numerical Simulations
### Flow over Two Circular Cylinders in a 2D Case  
To demonstrate the precision of the parallel algorithm of RLBM based on multi-layer grids, the flow over two circular cylinders at different spacing ratios was simulated. Two cylinders of the same size are placed in the channel. The diameter of the cylinder is defined as \(D\). The Reynolds number was set the same in all scenarios of this experiment at \(Re=200\). The length of the channel is \(80D\) and the width of the channel is \(60D\). The channel's inlet velocity is \(U\) (\(U=0.1\)). The front cylinder is located \(20D\) from the left boundary. The back cylinder is positioned at a distance \(L\) from the front cylinder. This experiment investigates the impact on the flow field as the position of the rear cylinder changes. The distance \(L\) between two cylinders is taken as \(1.5D\), \(2D\), \(3D\), and \(4D\). A schematic of the simulation scenario is shown in Figure 9.  
The fluid region grid is set up as follows: The size of the refined two-layer grid area is \(50D\times 35D\). and \(20D\times 15D\)., respectively. Each layer's grid size is half that of the preceding layer. Boundary grid refinement was performed around the two cylinders to capture flow details. The three-layer grid structure is shown in Figure 10.  
The flow field is analyzed for different spacing ratios as follows: As the evolutionary steps increase, the vortices between the cylinders gradually stabilize. The streamlines of the upstream cylinders closely follow the downstream cylinders. A symmetrical vortex pair was created between the two cylinders. The vortex of the downstream cylinder could not form a stable vortex pair due to the interference of the upstream cylinder, as shown in Figures 11-14. The different colors of velocity contours represent different ranges of velocity values. Figure 9: Schematic diagram of two-dimensional flow over two circular cylinders. Figure 11: Velocity contours and streamlines diagram at \(L/D=1.5\). Figure 12: Velocity contours and streamlines diagram at \(L/D=2\). Figure 10: Three-layer grid structure diagram. Figure 15-18 represent different spacings of the two cylinders. The cyclic variation of the lift coefficients \(C_{L}\) and drag coefficients \(C_{D}\) of the fluid flowing through the cylinders with time iteration steps. In Figure 19, the number of grids and CPU runtime are compared for a single-layer grid and a three-layer grid. The CPU time for a three-layer grid is only \(1/8\)th of the original, while the number of grids to be computed is \(1/4\)th of the original. The results demonstrate that the multi-layer grids regularized lattice Boltzmann method (RLBM) exhibits high accuracy and stability. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **[PERSON] et al. [42]** & **[PERSON] et al. [41]** & **[PERSON] et al. [43]** & **Present Work** \\ \hline \(L\)/\(D\) & \(\overline{C_{D2}}\) & \(C_{L2}\) & \(\overline{C_{D2}}\) & \(C_{L2}\) & \(\overline{C_{D2}}\) & \(C_{L2}\) & \(\overline{C_{D2}}\) & \(C_{L2}\) \\ \hline \(1.5\) & \(-0.201\) & \(0.041\) & \(-\) & \(-\) & \(-0.08\) & \(-\) & \(-0.211\) & \(0.056\) \\ \hline \(2\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-0.17\) & \(-\) & \(-0.231\) & \(0.189\) \\ \hline \(3\) & \(-0.130\) & \(0.209\) & \(-0.130\) & \(0.212\) & \(-0.08\) & \(-\) & \(-0.132\) & \(0.267\) \\ \hline \(4\) & \(0.482\) & \(1.210\) & \(0.360\) & \(1.117\) & \(0.380\) & \(-\) & \(0.380\) & \(1.117\) \\ \hline \end{tabular}
\end{table} 
Table 2: \(C_{L}\) and \(\overline{C_{D}}\) of the rear cylinder when \(Re=200\). Figure 16: Cylindrical bypass: \(C_{D}\) and \(C_{L}\) when \(Re=200\) and \(L/D=2\). (**a**) Front cylinder; (**b**) rear cylinder. Figure 15: Cylindrical bypass: \(C_{D}\) and \(C_{L}\) when \(Re=200\) and \(L/D=1.5\). (**a**) Front cylinder; (**b**) rear cylinder. Figure 19: Comparison of the running overhead of single-layer grid and multi-layer grid. Figure 17: Cylindrical bypass: \(C_{D}\) and \(C_{L}\) when \(Re\) = 200 and \(L/D=3\). (**a**) Front cylinder; (**b**) rear cylinder.