## 3 Results
### Flow around a Cylinder  
The second test case considered herein was the flow around a cylinder with periodic boundary conditions in both horizontal and vertical directions. This is equivalent to the flow through a periodic lattice of cylinders, which has been studied extensively as a simple model of flow through fibrous porous media. The geometry of the problem is shown in Figure 9, which consists of a single circular cylinder of radius \(r\) and its associated volume within the square lattice of side length \(L\). Figure 8: Temporal error convergence for laminar flow over a square simulated by the reference (**left**) and hybrid (**right**) methods. Figure 7: Spatial error convergence for laminar flow over a square simulated by the reference (**left**) and hybrid (**right**) methods. The flow is driven by a body force \(\mathbf{F}\) along the horizontal direction. On the wall of the cylinder, the no-slip boundary condition was imposed. The velocity field was initialised with zero values everywhere. The reference pressure was set to zero at the left-bottom corner. For the geometric parameters, we had \(L=0.1\) m and \(r=0.02\) m. The kinematic viscosity was set to \(\
u=10^{-6}\) m\({}^{2}\)s\({}^{-1}\), and the body force was set to \(F=1.5\times 10^{-5}\) ms\({}^{-2}\). The maximum horizontal velocity \(U_{max}\) can reach \(5\times 10^{-3}\) ms\({}^{-1}\) eventually. Based on \(U_{max}\), \(r\), and \(\
u\), the Reynolds number of the flow simulated here was \(\mathrm{Re}=100\). For the reference method, the computational domain was discretised with 139,552 velocity/pressure points in the flow region and 400 boundary points on the wall of the cylinder. For the hybrid method, the computational domain was discretised with 139,552 velocity points in the flow region, 400 boundary points on the wall of the cylinder, and 19,720 virtual points inside the cylinder. In addition, a regular grid of \(400\times 400\) points was used for the pressure. For both methods, the average grid size was about \(L/400\). The time step was set to \(\Delta t=0.03\) s, for which the corresponding maximal CFL number was about 0.6. The flow was simulated up to \(t_{max}=300\) s, corresponding to 10,000 time steps. Figure 10 compares the numerical solutions at \(t_{max}\) obtained by the two methods for the horizontal velocity component \(U\), the vertical velocity component \(V\), and the pressure \(P\). All these results showed excellent agreement between the solutions of the hybrid method and the reference method. The relative differences for \(U\), \(V\), and \(P\) were \(5.2\times 10^{-5}\), \(8.8\times 10^{-4}\), and \(9.6\times 10^{-4}\), respectively. The time histories of \(U\) and \(V\) at Monitoring Point 1 are shown in Figure 11. The time histories of \(P\) at Monitoring Point 1 and the pressure difference between the two monitoring points 2 and 3 are shown in Figure 12. It turned out that the agreement between the numerical solutions from the two methods was also excellent with regard to the time evolution of these flow variables. The relative differences for \(U_{1}\), \(V_{1}\), \(P_{1}\), and \(P_{2}-P_{3}\) were \(5.1\times 10^{-5}\), \(1.5\times 10^{-4}\), \(9.9\times 10^{-4}\), and \(2.8\times 10^{-3}\), respectively. For the hybrid method, the results presented in Figures 10-12 were obtained with the pressure Poisson equation solved by the central finite-difference approximation on a regular grid and the tolerance for the IB force iteration set to \(10^{-3}\). To check the grid convergence of numerical solutions, we performed a simulation using the hybrid method with the grid resolution increased by a factor of 1.5 and the time step reduced to \(\Delta t=0.01\) s. The results from this simulation matched very well with those from the two presented simulations of coarser resolutions.