# Simulation of Lid-Driven Cavity Flow with Internal Circular Obstacles
## 3 Results and Discussion
### Flow Characteristics Inside a Cavity with an Internal Obstacle (\(R/L=1/6\))
However, as \(Re\) increases (see Figure 14b), the magnitude of the negative velocity becomes larger, and the peak velocity moves towards the left sidewall, whereas the positive velocity also becomes larger, and the peak velocity moves towards the right sidewall. Figure 14: Vertical (**a**) and horizontal (**b**) velocity profiles depending on the \(Re\) number. Figure 13: Vector profiles with the changing \(Re\). Flow Characteristics Inside a Cavity with Varying Sizes of Internal Obstacles (\(R/L=1/16\), \(1/6\), \(1/4\), and \(2/5\))  
Figure 15 shows the variation of the flow inside the lid-driven square cavity for \(Re=5000\) with the radius of the obstacle varying as \(R/L=1/16\), \(1/6\), \(1/4\), and \(2/5\). As shown in Figure 15, as the size of the obstacle increases, a vortex appears around the obstacles, and it splits into two vortices of different size around the largest obstacle (i.e., \(R/L=2/5\)), which would result in a viscous effect between the obstacle and surface wall of the cavity. In the case of the largest obstacle (i.e., \(R/L=2/5\)), the viscous effect results in an induced vortex in the upper left region around the obstacle. Figure 16 also shows the strength of the vorticity, and depending on the size of the obstacle, dominant vorticity can be shown around the obstacle. The vortex strength is almost similar in all the cases close to the surface wall around the cavity, but the size of the internal obstacle could have a significant effect on the surrounding vorticity. Figure 15: Streamline distributions for an obstacle of various sizes. Figure 17 shows the two-dimensional vector field inside the square cavity with an \(Re\) of 5000 with obstacles of various sizes (\(R/L=1/16\), \(1/6\), \(1/4\), and \(2/5\)). As shown in the figure, as the size of the obstacle increases, the vector field does have some effect on the flow around the surface of the cavity wall, but the velocity field in the corner region is only slightly disturbed by the obstacle. In order to analyze the velocity variation around the obstacle, Figure 18 shows the speed against the azimuthal angle around the obstacle (\(R/L=1/16\), \(1/6\), \(1/4\), and \(2/5\)) for the obstacles of various sizes at \(Re=5000\) (see Figure 4 for the azimuthal angle). Interestingly, depending on the size of the obstacle, the magnitude of the circumferential velocity shows a significant variation. In the case of \(R/L=1/16\), the magnitude of the surface velocity (\(u_{s}/U\)) is less than 0.001 in the range of \(0^{\circ}<\theta<180^{\circ}\), whereas the other half has a larger magnitude over \(u_{s}/U=0.001\), which tends to have similar magnitudes in other cases. However, as the size of the obstacle increases, the variation of the circumferential flow varies significantly in the rage of \(180^{\circ}<\theta<360^{\circ}\), which could be because the lid-driven flow moves substantially around the obstacle, with a wobbly variation in the lower half of the domain but a smooth flow in the upper half of the domain. Figure 16: Vorticity contours for obstacle of various sizes.