## 3 Numerical Examples
### Results and Discussion  
Figure 2 displays the comparison of vorticity contours and streamlines with [PERSON] et al. [8] at radius \(r=0.2\), Reynolds number \(Re=1000\). As we can see from Figure 2, three vortices appeared in the lid-driven square cavity. One is at the upper right corner of the  
Figure 1: The lid-driven square cavity flow with an embedded circular obstacle. (**a**) Model (**b**) Meshes. circular obstacle and the other two vortices locate at the bottom lower left and right corners. These results are similar to those of [8]. The only difference is that the immersed boundary method identifies the internal circular obstacle as a fluid. The center coordinates of the three vortices are also listed in Table 1. Compared with the data in [PERSON] et al. [8], the relative errors of this paper are less than 1.8%. It is worth noting that the total mesh number of 200 \(\times\) 200 is used in [8] applying the immersed boundary method, but the total number of meshes used in the spectral/hp element methods is 4864, which shows that the present method is accurate and effective for the numerical investigation of the lid-driven square cavity flow with an embedded circular obstacle, and with lesser preprocess costs. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Upper Right Vortex**} & \multicolumn{2}{c}{**Lower Left Vortex**} & \multicolumn{2}{c}{**Lower Right Vortex**} \\ \hline  & \(x\) & \(y\) & \(x\) & \(y\) & \(x\) & \(y\) \\ \hline Present & 0.7044 & 0.6800 & 0.0762 & 0.0715 & 0.8890 & 0.1072 \\ [PERSON] et al. [8] & 0.6942 & 0.6881 & 0.0789 & 0.0720 & 0.8852 & 0.1063 \\ Error/\% & 1.14 & \(-1.18\) & \(-1.75\) & \(-1.53\) & 0.43 & 0.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of vortice center coordinates with the moving immersed boundary method (at \(r=0.2\), \(Re=1000\)). Figure 2: Vorticity contours and streamlines at \(r=0.2\), \(Re=1000\). (**a**) Vorticity (Present) (**b**) Vorticity ([PERSON] et al. [8]) (**c**) Streamlines (Present) (**d**) Streamlines ([PERSON] et al. [8]). Figure 3 shows the comparison of the horizontal and vertical velocity of the lid-driven square cavity flow with an embedded circular obstacle at different Reynolds number with [PERSON] et al. [13]. The results are similar to those of [13]. In order to evaluate the characteristics of flow around obstacles in a lid-driven square cavity with an embedded circular obstacle, the variations of the flow are numerically simulated at radius \(r=1/16\), \(1/6\), \(1/5\), \(1/4\), \(2/5\) and Reynolds number \(Re=100\), \(400\), \(1000\),\(3000\), \(5000\), respectively. The numbers of meshes used in the simulation are \(7168\), \(4096\), \(4864\), \(6376\) and \(1536\) corresponding to radius \(r=1/16\), \(1/6\),\(1/5\), \(1/4\), \(2/5\), respectively. Since the computational domain varies with the cylinder radius, the number of meshes also varies with the different division of meshes. The order of interpolated polynomial on individual grid is different when using the adaptive hierarchical techniques; because the upper and lower limits of tolerance are the same for all cases, the computational accuracy is guaranteed. Figure 3: Comparison of the profiles of u-velocity and v-velocity in the lid-driven square cavity flow with an embedded circular obstacle at various Reynolds numbers. (**a**) The profiles of u-velocity and v-velocity alone \(x=0.5\) and \(y=0.5\) ([PERSON] et al. [13]). (**b**) The profiles of u-velocity and v-velocity alone \(x=0.5\) and \(y=0.5\) (Present). Figure 4 displays the variation of the flow in the lid-driven square cavity for different Reynolds numbers at the radius \(r=1/16\). As we can see from Figure 4, with the increase in \(Re\), the vortex at the bottom of the square cavity gradually grows in size. The center of the vortex at the upper right side of the cavity gradually approaches to the boundary of the internal obstacle and decreases.