# Simulation of Lid-Driven Cavity Flow with Internal Circular Obstacles
## 1 Introduction  
In recent years, the Lattice Boltzmann method (LBM) ([PERSON] et al. [1]) has played an important role in resolving a variety of computational engineering problems, particularly incompressible flows ([PERSON] et al. [2], hereafter GHIA), porous-media flows ([PERSON] et al. [3]), magneto-hydrodynamics ([PERSON] et al. [4]), and single-phase and multi-phase fluid flows ([PERSON] et al. [5]). LBM is a new approach in the area of computational fluid dynamics (CFD), in which the Reynolds-averaged Navier-Stokes equations (RANS) was originally used for simulating macroscopic quantities of flows ([PERSON] et al. [2]). LBM comprises the use of the kinetic model with some parameters, and one of them, called the distribution function ([PERSON] et al. [1]), describes the detailed movement and relocation of the fluid particles. These particles are finally located at the lattice nodes through the continuous streaming and collision processes. Furthermore, LBM possesses several distinct merits: it is a simple based on high parallelism, and produces a solution with the complicated boundary condition relatively fast and without transforming the mesh grid ([PERSON] et al. [6]). Moreover, the macroscopic pressure can also bedirectly solved using the state equation, which provides stable results with a second-order accuracy over a wide range of Reynolds numbers (Re). The LBM can be divided into two folds, namely the Boltzmann Bhatnagar-Gross-Krook (BGK) model of kinetic theory and the Chapman-Enskog expansion theory. The LBGK method serves as a method for solving continuum problems on a kinetic basis. This method takes into consideration the discrete molecular velocity distribution functions for uniform Cartesian lattices with additional diagonal links. In recent years, the LBGK model has achieved great success in modeling physics in fluids and in viscous flow simulations in a square, two-dimensional cavity driven by a shear layer due to one moving wall for a laminar flow at an _Re_ up to 10,000. The Chapman-Enskog expansion provides a condition under which hydrodynamics equations for a fluid can be formulated from the Boltzmann equation. This technique justifies the use of various transport coefficients such as thermal conductivity and viscosity, which are important for the transition from a microscopic, particle-based description to a continuum hydro-dynamical and momentum condition. However, the individual molecules are no longer important, and the state of the fluid is described in terms of some average properties of the fluid elements containing billions or much larger numbers of molecules. Therefore, the LBGK model, which was applied in this study, would present more advantages in terms of applicability and domain geometry (i.e., a reasonable running time and boundary condition in complicated geometries). However, there are some limitations to this model. For instance, when the LBGK model runs at a high Re, it runs into the issue of numerical instabilities such that an additional model (such as the MRT model) would be required in some cases. Lid-driven cavity flow combined with LBM has been one of the benchmark studies in incompressible fluid flow problems. Owing to the simplicity of the geometry under consideration, the reliability and robustness of the governing equations with the incorporated models of the LBM have been thoroughly evaluated in the CFD. However, although it has a relatively simple shape, the internal flow driven by a constant velocity on the lid becomes complicated with the variation of Re, and it forms almost all the phenomena that could possibly exist in incompressible flows: eddies, secondary flows, instabilities, chaotic particle motions, transitions, and turbulence ([PERSON] and [PERSON] [7]). Therefore, this type of flow calculation has been considered as a classic problem and used extensively for the validation of various numerical simulations.