# Simulation of Lid-Driven Cavity Flow with Internal Circular Obstacles
## 4 Conclusions  
In this study, the LBGK model has been adopted to simulate the flow inside a two-dimensional cavity and the effect of an internal obstacle under lid-driven cavity flow. For the LBM simulation of lid-driven cavity flow, Reynolds (\(Re\)) number varies 100 to 5000 and the aspect ratio of one (i.e., a square cavity, where \(K=1\)). Regarding the LBGK model, one of advantages is that the method was designed to run efficiently on parallel clusters so that it enables complex physics and sophisticated algorithm. In addition, the method uses a uniform mesh spacing everywhere in the domain so that it is usually very fast per time-step. Based on this condition, this study is focused on the LBM for  
Figure 17: Vector profiles for obstacles of various sizes.  
Figure 18: Flow speed around the obstacle for obstacle of various sizes at \(Re=5000\).  
calculating the flow inside a square cavity with and without an internal obstacle depending on \(Re\) number and the size of the internal obstacles. In order to validate the study, the results were thoroughly compared with existing data. The current LBM successfully simulated and used to visualize the precise secondary vortices in the cavity as well as the primary vortex inside the cavity.  
The results of this study can be summarized as follows:  
* 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 \(Re\) increases, the primary vortices inside the cavity without an obstacle move towards the center of the cavity, and the secondary vortices at the bottom corners increase in size. However, the core area in the central region is almost constant and fully developed in the range of \(Re\) = 3000-5000.
* In order to analyze the vortex variation inside a cavity, the stream function (symbol \(\psi\)) as well as the locations of the primary and secondary vortices were compared with existing data. The location of the vortex centers obtained in this study have some scatter depending on the location comparing with the previous work, but the deviation of location is, in the majority of cases, less than 20%.
* For the cavity with internal obstacles, as Re number increased, the secondary vortices close to the internal obstacles became smaller owing to the strong primary vortices. Depending on the obstacle size (\(R/L\) = 1/16, 1/6, 1/4, and 2/5), however, secondary vortices were induced at each corner of the cavity and remained stationary, but the secondary vortices close to the top region of the obstacle became larger as the obstacle size was increased.
* In order to analyze the velocity variation around the obstacle, speeds around obstacles of various sizes (\(R/L\) = 1/16, 1/6, 1/4, and 2/5) were compared at \(Re\) = 5000. Interestingly, depending on the size of the obstacle, the magnitude of the circumferential velocity showed 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 greater magnitude over \(u_{s}/U\) = 0.001, which tends to be similar in the other cases as well.  
Data curation, T.H.; Funding acquisition, H.-C.L.; Project administration, H.-C.L.; Writing original draft, T.H.; Writing review and editing, H.-C.L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
This work was supported by 'Human Resources Program in Energy Technology' of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), granted financial resource from the Ministry of Trade, Industry and Energy, Republic of Korea (no. 20184030202200). In addition, this work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (no. 2019 RI11A3A01058576).  
The authors declare no conflict of interest.