## 3 Results
### Lid-Driven Polar Cavity  
In order to validate the proposed hybrid method, the lid-driven polar cavity flow was here simulated, which was first studied both experimentally and numerically by [PERSON] and [PERSON] [40]. As shown in Figure 20, the flow was in an annular region defined by \(R_{i}\leq r\leq R_{o}\) and \(-\alpha/2\leq\theta\leq\alpha/2\). No-slip boundary conditions were applied at the four boundaries of the flow domain. For the rotating wall at \(r=R_{i}\), an azimuthal velocity of \(U_{\theta}=U_{i}\) was prescribed. The dimensionless geometrical and physical parameters matching the experiments [40] were set as:  
\[R_{i}=1.0,R_{o}=2.0,\alpha=1,U_{i}=1,\mathrm{Re}=\frac{U_{i}R_{i}}{\
u}=280,3 40,380,\mathrm{and}\ 410. \tag{21}\]  
The flow domain was discretised with 59,501 velocity points and 1000 boundary points. In addition, inside the extended rectangular domain, but outside the four boundaries of the flow domain, 75,106 virtual points were generated. The pressure Poisson equation was solved in the extended rectangular domain using a regular grid of \(280\times 480\) points. It is important to point out that, unlike the previous test cases, enforcing the divergence-free constraint outside the flow domain could be problematic in this case because a part of the solid body is moving. Hence, we chose to introduce a source/sink term outside the flow domain, which effectively turned the right-hand side of the Poisson equation into zero for the pressure points outside the flow domain. The dimensionless grid size was about 0.005. To ensure numerical stability, the dimensionless time step was set to \(\Delta t=0.001\), and the corresponding maximal CFL number was about 0.2. An initial condition of zero velocity was applied in the interior of the cavity. The flow was simulated for a sufficiently long time to ensure that the steady-state was reached. Figure 21 shows the radial (\(U_{r}\)) and azimuthal (\(U_{\theta}\)) velocity profiles along four radial lines \(\theta=\pi/18,-\pi/18,\pi/9\), and \(-\pi/9\)  
Figure 19: Comparison of time histories of the pressure at the two monitoring points 1 and 2 obtained by the reference method and the hybrid method for the fourth test case.  
Figure 18: Comparison of time histories of the horizontal and vertical velocity components at Monitoring Point 1 obtained by the reference method and the hybrid method for the fourth test case.  
for \(\mathrm{Re}=410\), \(280\), \(380\), and \(340\), respectively, in comparison with the experimental data of [PERSON] and [PERSON] [PERSON] and [PERSON] (2013). In all four cases, the simulation results obtained by the hybrid method were in good agreement with the experimental results, which also validated the proposed hybrid algorithm. As a side note, the simulation results here were also consistent with those obtained by the other numerical methods (for example, see Figure 18 of ([PERSON], 2016)).  
Figure 21: _Cont._  
Figure 20: Schematic of the geometry for the lid-driven polar cavity flow. The flow is driven by the moving wall at \(r=R_{i}\), and the polar cavity is defined by \(R_{i}\leq r\leq R_{o}\) and \(-\alpha/2\leq\theta\leq\alpha/2\). The red dashed rectangle indicates the extended computational domain for solving the pressure Poisson equation.