## 3 Numerical Examples
### Results and Discussion
When \(Re\) is greater than 3000, a new vortex appears on the upper left position of the cavity, and the corner vortex becomes larger with the increase in \(Re\). Figure 5 displays the variation of the flow in the lid-driven square cavity at different Reynolds numbers at the radius \(r=1/6\). As it can be seen in Figure 5, with the increase in \(Re\), the corner vortex at the bottom gradually grows in size. The center of the vortex at the upper right position of the cavity gradually approaches the boundary of the embedded obstacle and decreases. When \(Re\) is greater than 3000, an induced vortex appears on the upper left position of the cavity, and the induced vortex becomes larger with the increase in \(Re\). However, in the results obtained by [PERSON] et al. [13] using the lattice Boltzmann method, the corner vortex in the lower right corner of the square cavity is not observed at \(Re=100\). Owing to the higher-order properties of the method used here, the present spectral/hp element methods successfully capture the third vortex on the lower right corner of the cavity, and the vortex on the lower right corner is clearly visible. Figure 4: Streamlines of the lid-driven square cavity flow with an embedded circular obstacle at various Reynolds numbers at \(r=1/6\). Figure 6 shows the streamlines in the lid-driven square cavity at different Reynolds numbers \(Re\) at the radius \(r=1/5\). At this radius, the characteristics of the flow in the square cavity at various Reynolds numbers are essentially the same as at the radius \(r=1/6\). With the increase in \(Re\), the corner vortex at the two sides of bottom gradually expand, but the vortex in the upper right corner of the square cavity gradually becomes smaller. Compared to the streamlines at \(r=1/6\), the primary vortex at the upper position of the embedded obstacle in the cavity tends to elongate to some extent in the case of \(Re=100\). We continue to increase the radius \(r\); when \(r=1/4\), the primary vortex at the upper position of the internal obstacle in the cavity gradually splits into two vortices. However, with the increase in the Reynolds number, these two vortices merge again into a vortex gradually contracting towards the upper right position of the embedded obstacle. The induced vortices on the upper left side of the cavity also expand with the increase in \(Re\), as displayed in Figure 7. In the case of \(r=1/4\), \(Re=100\), the number of meshes is 4096 and the tolerance is set to \(10^{-10}\), the time required to arrive to a steady state of CPU (a single core) is 1652 s.  
When the radius of the embedded circular obstacle becomes larger due to the strengthening of the wall effect, at \(r=2/5,\;Re=100\), the upper primary vortex in the cavity transforms almost into two corner vortices, as shown in Figure 8. With the increase in \(Re\), the two vortices on the upper region of the square cavity do not show the merger of the vortices as at \(r=1/4\); instead, a small-scale vortex on the upper left side and a large-scale vortex on the upper right of the obstacle are formed by gradual contraction. Figure 5: Streamlines of the lid-driven square cavity flow with an embedded circular obstacle at different Reynolds numbers \(Re\) at \(r=1/6\). Figure 6: Streamlines of the lid-driven square cavity flow with an embedded circular obstacle at various Reynolds numbers at \(r=1/5\). Figure 7: Streamlines of the lid-driven square cavity flow with an embedded circular obstacle at different Reynolds numbers \(Re\) at \(r=1/4\). Two newly induced vortices are observed on the upper left and right sides of the circular obstacle, respectively. This result is different from the result in [PERSON] et al. [13]; the lattice Boltzmann method cannot capture these new vortices in [13]. This also shows the high-resolution characteristics of the spectral/hp element methods. Figure 9 displays the vorticity contours for the different radius \(r\) at \(Re=100\).