# Simulation of Lid-Driven Cavity Flow with Internal Circular Obstacles
## 1 Introduction
In particular, the centerline velocity distributions (i.e., horizontal and vertical centerlines) in a square cavity have been compared with those of benchmark tests and often cited as a benchmark ([PERSON] et al. [2]). [PERSON] et al. [8] simulated a skew cavity flow for two distinct _Re_ values, i.e., 100 and 1000. [PERSON] et al. [9] studied the effect of compressibility on the lid-driven cavity flow using the Lattice Bhatnagar-Gross-Krook (LBGK) model under an _Re_ of less than 7500 and using the single relaxation time (SRT) method. They also attempted to investigate the mesh dependence with the objective of determining the time required to arrive at a steady state depending on the lattice size and to observe the Hopf bifurcation at an approximate _Re_ of 5000 in the cavity flow. In terms of further studies on cavity flows within complex geometries, [PERSON] et al. [10] compared the stability of the cavity flow for the SRT method with that of the multiple relaxation time (MRT) method with the LBGK model for various _Re_ values. [PERSON] et al. [11] incorporated the MRT-based LBM for computing lid-driven cavity flows at _Re_ values ranging from 100 to 7500 and found that it agreed well with existing data for the range of aspect ratios (i.e., the ratio of the depth and width) from 1 to 4. Similarly, GHIA also calculated lid-driven cavity flows with _Re_ values ranging from 100 to 10,000 using the MRT model. Interestingly, the majority of the existing results were consistent with those of GHIA, which have been used as reference data for lid-driven cavity flows. In terms of the internal obstacles in the cavity, [PERSON] et al. [12] studied the velocity vector plot of lid-driven cavities with multiple obstacles and steady velocity components along the centerline through the cavity for various grid sizes. [PERSON] and [PERSON] [13] and [PERSON] and [PERSON] [14] worked on the cavity flow in other _Re_ ranges based on a method comprising the stream function and vorticity formulation. [PERSON] et al. [1] designed an LBGK model without the consideration of compressible effects for simulating incompressible flows and that was called the incompressible LBGK model. [PERSON] et al. [15] simulated the lid-driven flow with an internal cylinder for a wide range of \(Re\) values. Recently, [PERSON] and [PERSON] [16] performed a simulation on the flow around a cylinder in a cavity for \(Re\) = 5000, 10,000, 15,000, and 1,330,000 and for a dimensionless rotational speed of the cylinder, \(0\leq\omega\leq 10\). In their study, two different methods--i.e., large eddy simulation and unsteady Reynolds-averaged Navier-Stokes (URANS)--were applied, and the study showed that the flow patterns were strongly influenced on increasing either the rotating cylinder speed or \(Re\). Considering the previous studies mentioned above, it can be concluded that there is already (to a great extent) a thorough understanding of lid-driven flows. Additionally, lid-driven cavity flow combined with LBM has been one of the most popular benchmark studies in incompressible fluid flow problems. Therefore, we have focused on a more complex and practical aspect of the lid-driven cavity flow in this study, related to internal obstacles. Although this has wide applicability in medical, thermo-fluid, and industrial sectors, such as heat exchanges, HVAC, and blood flow, we have not found any research on this particular topic in the existing open resources. Regarding the adopted method (i.e., the LBGK model), there are some advantages and limitations. One of its strongest points is that the method was designed to run efficiently on parallel clusters so that it enables complex physics and a sophisticated algorithm. Additionally, the method uses a uniform mesh spacing everywhere in the domain, so that it is usually very fast per time-step. On the other hand, despite the increasing popularity of LBM in simulating complex fluid systems, this method has some limitations.