# Simulation of Lid-Driven Cavity Flow with Internal Circular Obstacles
## 3 Results and Discussion
### Flow Characteristics Inside a Cavity with an Internal Obstacle (\(R/L=1/6\))  
Figure 11 shows the variation of the flow inside a lid-driven square cavity with obstacles (i.e., \(R/L=1/6\)), with the objective of understanding the characteristics of the flow around obstacles in a lid-driven cavity. In order to perform consistent calculations, \(Re\) is varied from 100 to 5000. As \(Re\) increases, the corner vortex at the bottom gradually appears at the bottom corner of the cavity and becomes larger. In addition, the center of the internal primary vortex tends to move closer to the internal obstacle. However, when \(Re\) exceeds 3000, a corner vortex on the upper left appears and remains there. Interestingly, a large vortex appears around the obstacle at \(Re=100\) and becomes a primary vortex. However, as \(Re\) increases, the vortex becomes smaller. Figure 12 shows the vorticity contours for the varying \(Re\). As shown in the figure, the strength of the vorticity for a smaller \(Re\) stays in the region of the top corners. However, as \(Re\) increases, owing to the lid-driven flow, the strength of the vorticity moves gradually to the right and right bottom corner, and finally, at \(Re=5000\), the strength of the vorticity dominates all the surrounding edges (due to the thin shear layer close to the wall). In contrast, the value of central core region shows almost even such that it would have a nearly constant vorticity field. However, the internal obstacle has a substantial influence on the generation of a strong vorticity at a low \(Re\), but as \(Re\) increases, this influence almost disappears. Figure 13 presents the two-dimensional vector field inside the square cavity with an obstacle. The length of each vector indicates the flow speed. As shown in the figure, in the case of \(Re=100\), as the flow looks laminar (i.e., a low \(Re\) flow), the vector field in the cavity is slightly erratic, and a circulating flow is observed in the upper region above the obstacle. However, as \(Re\) increases, the upper circulating flow appears to disappear, and a smaller vortex remains at the upper right edge, close to the obstacle, and becomes almost one large circulating flow having a clockwise rotation around the obstacle. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Re**} & \multirow{2}{*}{**Result**} & \multicolumn{4}{c}{**Left Bottom Vortex**} & \multicolumn{4}{c}{**Primary Vortex**} & \multicolumn{4}{c}{**Right Bottom Vortex**} \\ \cline{3-10}  & & **x** & **y** & \(\boldsymbol{\psi_{max}}\) & **x** & **y** & \(\boldsymbol{\psi_{max}}\) & **x** & **y** & \(\boldsymbol{\psi_{max}}\) \\ \hline \multirow{3}{*}{100} & Current work & 0.034 & 0.037 & 2.06 \(\times\)\(10^{-6}\) & 0.61 & 0.74 & \(-\)0.12 & 0.95 & 0.065 & 1.2 \(\times\)\(10^{-5}\) \\  & GHIA & 0.031 & 0.04 & 1.80 \(\times\)\(10^{-6}\) & 0.62 & 0.73 & \(-\)0.11 & 0.94 & 0.062 & 1.3 \(\times\)\(10^{-5}\) \\  & Deviation [\%] & 8.8 & 8.1 & 12.6 & 1.6 & 1.4 & 8.3 & 1.1 & 4.6 & 8.3 \\ \hline \multirow{3}{*}{400} & Current work & 0.056 & 0.05 & 1.3 \(\times\)\(10^{-5}\) & 0.56 & 0.61 & \(-\)0.12 & 0.88 & 0.14 & 6.5 \(\times\)\(10^{-4}\) \\  & GHIA & 0.051 & 0.04 & 1.4 \(\times\)\(10^{-5}\) & 0.54 & 0.60 & \(-\)0.11 & 0.89 & 0.12 & 6.4 \(\times\Figure 11: Streamline distribution with variation in \(Re\). Figure 12: Vorticity contours for various \(Re\). Figure 14 shows the horizontal and vertical velocity profiles along the centerlines \(x\) = 0.5 and \(y\) = 0.5. Therefore, the velocity inside the obstacle should be zero in the range of \(L/3<x<2L/3\), and \(L/3<y<2L/3\). As shown in the Figure 14a, as \(Re\) increases, the magnitude of the negative horizontal velocity in the bottom region increases, and the peak velocity moves lower and close to the wall, whereas the velocity differences in the top region of the cavity are very small, except in the case of a low \(Re\) (i.e., \(Re\) = 100).