## 3 Results
### Flow Past a Cylinder between Parallel Plates  
In the third numerical example, laminar flow past a cylinder between two parallel plates was simulated using the proposed hybrid method, and the results were compared with those obtained by the reference method. The geometry of the problem was the same as that shown in Figure 9. The flow was driven by a body force \(\mathbf{F}\) along the horizontal direction. Periodic boundary conditions were applied in the horizontal direction. No-slip boundary conditions were imposed at the top and bottom boundaries and the wall of the cylinder. For solving the pressure Poisson equation in the hybrid method, it can be shown that, with the staggered grid and central difference scheme, \(\mathbf{u}^{*}\cdot\mathbf{n}\) at the top and bottom boundaries cancelled out in the final algebraic equations for the pressures at the internal grid points; therefore, their values can be effectively set to zero for the calculations in which  
Figure 12: Comparison of the time histories of the pressure at Monitoring Point 1 and the pressure difference between the two monitoring points 2 and 3 obtained by the reference method and the hybrid method for the second test case. they are involved. This justifies the use of the Neumann boundary condition specified by Equation (7) at the top and bottom wall boundaries, and consequently the application of the FFT-based solver. If needed, the pressures on the top and bottom boundaries can be calculated using Equation (8) with the values of \(\mathbf{u}^{*}\cdot\mathbf{n}\) as they are. It is worth mentioning that the IB force was not applied at a boundary where the Neumann boundary condition was imposed. The velocity field was initialised with zero values everywhere. The reference pressure is set to zero at the left-bottom corner. The geometric and physical parameters were the same as those of the second test case. With this setting, the maximum horizontal velocity \(U_{max}\) can reach about \(2.5\times 10^{-3}\) m s\({}^{-1}\) eventually. Based on \(U_{max}\), \(r\), and \(\
u\), the Reynolds number of the flow simulated here was about \(\mathrm{Re}=50\). The computational domain was discretised with 139,142 points in the flow region and 1200 boundary points at the top, bottom, and cylinder boundaries for the reference method, additional 19,756 virtual points inside the cylinder, and a staggered pressure grid of \(400\times 400\) points for the hybrid method. The average grid size was about \(L/400\). The time step was set to \(\Delta t=0.03\) s to ensure numerical stability, for which the corresponding maximal CFL number was about 0.3. The flow was simulated up to \(t_{max}=300\) s corresponding to 10,000 time steps. The tolerance for the IB force iteration was set to \(10^{-3}\). Figure 13 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 \(P\). Figures 14 and 15 show the time histories of \(U\), \(V\), and \(P\) at the selected monitoring points indicated in Figure 9. Like in the second test case, all these results showed excellent agreement between the solution of the hybrid method and that of the reference method. The relative differences for the flow variables shown here were below \(3.2\times 10^{-3}\). Furthermore, our tests showed that, for the hybrid method, the simulation results were not sensitive to the choice of the method (pseudo-spectra or finite difference) and grid (regular or staggered) for solving the pressure Poisson equation. Figure 13: Cont. The total wall clock time of the hybrid solver was about 840 s, and the total wall clock time of the reference solver was about 1622 s. The hybrid solver was about twice as fast as the reference solver. To have a further speedup, a coarser grid (e.g., \(256\times 256\)) can be used to solve the pressure Poisson equation without deteriorating the overall simulation accuracy.