# Simulation of Lid-Driven Cavity Flow with Internal Circular Obstacles
## 3 Results and Discussion
### Lid-Driven Square-Cavity Flow without Obstacles  
This section describes the variation of the flow inside a lid-driven square cavity without obstacles with the objective of understanding the detailed flow structure inside the cavity. As mentioned earlier, the square-cavity flow would be a well-known benchmark example for validating existing data ([PERSON] et al. [11]). In order to observe the effect of \(Re\) on the flow inside the lid-driven square cavity, the magnitude of the viscosity in \(Re\) is increased by a factor of 50 in order to vary \(Re\) from 100 to 5000, as shown in Figure 7. As \(Re\) increases, the corner vortices gradually appear at the bottom corner of the cavity and become larger. In addition, the center of the primary vortex tends to move to the center of the square cavity. Interestingly, when \(Re\) increases beyond 3000, an induced corner vortex in the upper left corner appears and becomes clearer. Figure 8 presents the vorticity contours for the changing \(Re\) number. For the figure, the definition of vorticity can be expressed as \(\mathbf{\Omega}=\
abla\times\mathbf{u}\), where \(\
abla\) is the gradient operator ([PERSON] and [PERSON] [14]). In order to observe the variation of the vorticity strength depending on \(Re\), in general, the contour line is a curve connecting points where a specific line has the same value. When the lines are close together, the magnitude of the gradient is large, i.e., the variation is steep. As shown in the figure, the strength of the vorticity is evenly distributed inside the cavity except in the region of both top corners at a low \(Re\) number, i.e., \(Re=100\) and 400. However, as \(Re\) increases, because of the velocity magnitude of the lid-driven flow, the vorticity center at the top corner moves gradually to the right and right bottom corner, and finally, for \(Re=5000\), the strength of the vorticity spreads and dominates all the surrounding edges (due to the thin shear layer close to the wall). In contrast, as \(Re\) increases, the core area in the central region becomes almost constant and fully developed in the range of \(Re=3000\)-5000. Figure 9 shows the two-dimensional vector field inside a square cavity. The length of each vector indicates the flow speed. As shown in Figure 9, in the case of \(Re=100\), as the flow looks laminar (i.e., low Reynolds number flow), the center of the primary vortex in the cavity is located in region that is slightly higher, such that the dominant flow would be in the upper region. However, as \(Re\) increases, the primary vortex moves down towards the central region of the cavity and becomes almost axisymmetric in shape. Figure 7: Streamline distributions with the changing \(Re\) number. Figure 10 shows the vertical and horizontal velocity profiles along the centerline of the cavity with the objective of comparing the velocity components precisely with the existing results. In order to observe the effect of \(Re\) on the flow inside the cavity, the \(Re\) was varied from 100 to 5000. In Figure 10a, the velocity distribution shows a positive value at the top region, but a negative-value region below. Figure 8: Vorticity contours for various Re. Figure 9: Vector fields for the changing \(Re\) number. This indicates that the lid-driven flow naturally exhibits a maximum speed at the lid, which decreases gradually towards the bottom, and changes into a negative velocity in the bottom region due to the circulation of the flow inside the cavity. Interestingly, owing to the no-slip condition at the surface, the maximum negative peak occurs close to the wall, but not at the wall. In addition, the maximum horizontal velocity \(U\) varies substantially depending on the \(Re\) number. As \(Re\) increases, the magnitude of the maximum horizontal velocity increases, and the location of the maximum peak point moves towards the wall. As shown in Figure 10, as \(Re\) increases, the magnitude of the maximum horizontal velocity peak also increases, and the location of the peak point also moves towards the wall.