## 3 Numerical Simulations
### External Flows
#### 3.2.2 Two Cylinders  
Numerical simulations of the flow around two cylinders in tandem arrangement were performed by varying the spacing ratio \(s\) from 2 to 6 at \(\mathbf{Re}=100\). Discontinuous changes in the Strouhal number and hydrodynamic forces are expected when \(s\) is approximately 4 and 5 [40,41].  
The simulation conditions were the same as those of the single cylinder, except for the number of cylinders. The distributions of the computational points near a body (a) with and (b) without inactive points at \(s=2\) are exhibited in Figure 15. It can be observed that the irregularity of the distribution increases in the region between the two cylinders. The total simulation time was 1000 s when the spacing \(s\) was 4, but only 400 s for the other cases.  
Figures 16 and 17 show the contour maps of the velocity magnitude and pressure with streamlines, respectively, for different spacing ratios \(s\). When \(s=2\), the wake of the upstream cylinder reattaches to the downstream cylinder, forming a negative pressure that results in negative drag, as indicated in Figure 18. The flows between the cylinders are symmetric against their centerlines, and long vortices are shed. This explains the small \(\mathbf{St}\) only for the downstream cylinder in Figure 19, because the free shear layer from the upstream cylinder completely covers the downstream cylinder. When \(s=4\), vortex shedding starts to occur in both the downstream and upstream cylinders. The attachment of the shear layer to the downstream cylinder is marked by an increase in the pressure coefficient at the reattachment point on the downstream cylinder surface, which causes a sudden increase in the drag coefficients of both cylinders [33,40]. When \(s=6\), the primary vortices from the upstream cylinder do not directly reattach to the downstream cylinder, but the stretched and weakened vortices reach the downstream cylinder, which causes a decrease in the drag force acting on the downstream cylinder [40]. The drag coefficients of both cylinders and the Strouhal numbers estimated using the proposed method are compared with the results of other methods in Figures 18 and 19, respectively. A discontinuous change in the values is observed near the critical spacing ratio, but the overall result is in good agreement with those of the other numerical methods.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(\mathbf{C_{D}}\) & \(\mathbf{C_{L}}\) & **St** \\ \hline [PERSON] et al. [38] & 1.35 & 0.339 & 0.165 \\ [PERSON] et al. [39] & 1.33 & 0.332 & 0.165 \\ [PERSON] [39] & 1.33 & 0.320 & 0.165 \\ [PERSON] et al. [39] & 1.33 & 0.312 & 0.164 \\ [PERSON] et al. [38] & 1.35 & 0.333 & 0.166 \\ Proposed & 1.347 & 0.330 & 0.164 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of drag and lift coefficients and Strouhal number (\(\mathbf{Re}=100\)).  
Figure 15: Distribution of computational points near a body (**a**) with and (**b**) without inactive points at a spacing ratio of 2.  
Figure 16: Contour maps of vorticity magnitude around two circular cylinders with spacing ratios of (a) 2, (b) 4, and (c) 6 at t = 400 s (**Re** = 100).  
Figure 17: Contour maps of pressure and streamlines around two circular cylinders with spacing ratios of (a) 2, (b) 4, and (c) 6 at t = 400 s (**Re** = 100).  
Figure 18: Comparison of drag coefficients for two circular cylinders in tandem arrangement (**Re** = 100).