**What is the Lattice Boltzmann method (LBM)?**

Lattice Boltzmann method is a class of computational fluid dynamics (CFD) methods for fluid flow simulation. Instead of solving the Navier-Stokes equations, LBM solves the discrete Boltzmann equation to simulate the flow of a Newtonian fluid with collision models such as Bhatnagar-Gross-Krook (BGK).

A D2Q9 lattice Boltzmann model on a lattice showing collision and streaming process.

#### Algorithm with BGKW approximation

It is nearly impossible to solve the Boltzmann equation because its collision term is extremely complicated. However, the outcome of two-body collisions is not likely to significantly influence the values of many measured quantities (Cercignani, 1990). Hence, it is possible to approximate the collision term with a simple operator without introducing significant error to the solution. Bhatnagar, Gross, and Krook (BGK) in 1954 introduced a simplified model for collision operator. At the same time Welander (1954), independently, introduced similar operator. The collision operator is replaced as:

By introducing BGKW approximation, the Boltzmann equation (without external forces) can be approximated as:

In the computer algorithm, the collision and streaming step are defined as follows:

**Two-dimensional characteristic boundary conditions for open boundaries in the lattice Boltzmann methods**

In the LBM, most recent researches on the boundary conditions have focused on developing high-order boundary schemes for various configurations, while little attention was paid on the treatment of spurious reflections at the open. The two-dimensional (2-D) characteristic boundary conditions (CBC) based on the characteristic analysis are formulated for the lattice Boltzmann methods LBM. In this approach, the classical locally-one dimensional inviscid (LODI) relations are improved by recovering multi-dimensional effects on flows at open boundaries. The 2-D CBC are extended to a general subsonic flow configuration in the LBM and the effects of the transverse terms are clarified.

Vortex-shedding over a square block with different characteristic boundary conditions (CBC), which was simulated with 2-D isothermal multi-relaxation time lattice Boltzmann method (MRT-LBM): Case 1-Conventional CBC (Poinsot and Lele), Case 2-Thompson’s CBC, and Case 3-Improved CBC.