## 3 Numerical Simulations
### External Flows
#### 3.2.1 Single Circular Cylinder  
The diameter (D) of the cylinder was 1 m, and the rectangular computational domain was set to \(-10\)D to \(30\)D and \(-10\)D to \(10\)D in the \(x\) and \(y\) directions, respectively. As for the arrangement of the calculation points, \(401\) and \(201\) points of equal intervals in the \(x\) and \(y\) directions were placed. For convenience, these points are hereinafter referred to as background points. Then, \(101\) points in the circumferential direction and \(21\) points concentrated around the cylinder surface in the radial direction were added. Simply put, the distributions are the same as the combination of H- and O-type structured grid systems for the entire computational domain and near the cylindrical body, respectively. Finally, the background points inside the body and those that are extremely close to the points that are not included in the background points were flagged and inactivated during the cloud construction and simulations. This method of distributing points weakens the regularity, but the authors want to show that this irregularity does not cause problems during the simulations using the proposed method. At this stage, the points in one or two layers near the boundaries were matched with the boundary points, as explained in Section 2.5. The flagging technique was used because it is effective in dealing with problems involving single or multiple moving bodies. The cloud construction near the body was not changed, and operations were performed only for the flag-changed points and those within their effective radius. Figure 7 shows the distribution of the computational points near a body (a) with and (b) without inactive points. As boundary conditions for velocity, \(1.0\) and \(0.0\) m/s of velocity components \(u\) and \(v\) were applied to the inflow boundary, and a first-order Neumann condition was employed to other open boundaries. The reference pressure was zero for the top and bottom boundaries, and the Neumann condition was applied for the rest. A no-slip condition was imposed on the cylinder surface for the velocity. For the pressure boundary condition on the surface, Equation (23) was implemented. The **Re** value was set to \(40\), where vortices attached to the cylinder were observed, and \(100\), where periodic vortex shedding occurred. The time  
Figure 7: Distribution of the computational points near a body (**a**) with and (**b**) without inactive points (flag values of \(-1\), \(0\), and \(1\) indicate inactive, fluid, and boundary points, respectively). interval was chosen such that the Courant number is 0.5. The total simulation time was 400 s.  
Figure 8 shows the contour maps of the velocity, pressure, and vorticity magnitude in the \(x\) and \(y\) directions when \(\mathbf{Re}=40\) at \(t=400\) s. Symmetric flow patterns against the \(y\) centerline of the cylinder were well reconstructed. The velocity vector fields and streamlines around the cylinder are depicted in Figure 9. A pair of attached symmetric vortices is observed, and the attachment length of the vortex is approximately 2.3, which is in the range reported by other studies [34,35]. Figure 10 presents a comparison of the pressure coefficients \(C_{P}\) on the cylinder surface with the results of the methods in [36], where good agreement can be observed. The time  
Figure 8: Contour maps of (**a**) \(u\), (**b**) \(v\), (**c**) \(p\), and (**d**) vorticity magnitude around a circular cylinder (\(\mathbf{Re}=100\), \(t=400\) s). Figure 9: (**a**) Velocity vector fields and (**b**) streamlines around a circular cylinder (\(\mathbf{Re}=100\), \(t=400\) s). series of the coefficients of the hydrodynamic forces (drag and lift) are shown in Figure 11. The coefficients converge after 50 s and the viscous forces are smaller than the pressure forces, but their magnitudes are not negligible.