## 2 Incompressible Fluid Mode with D2Q9  
In this section, consider a 2D incompressible fluid with standard lattice Boltzmann method (LBM). The fluid domain is divided into regular lattices and the fluid molecular velocity of each lattice is restricted to nine specific velocity vectors (see Figure 1), which is \(e_{i}\) in Equation (1). \[e_{i}= \begin{cases}(0,0),&i=0\\ (\cos\theta_{i},\sin\theta_{i}),&\theta_{i}=\frac{(i-1)\pi}{2},i=1,2,3,4\\ \sqrt{2}(\cos\theta_{i},\sin\theta_{i}),&\theta_{i}=\frac{(i-5)\pi}{2}+\frac{ \pi}{4},i=5,6,7,8\end{cases}  \tag{1}\]  
In D2Q9 model standard LBM simulation, the lattice cell size is \(\Delta x=1\), the time step is \(\Delta t=1\), and the density distribution function is \(f_{i}\), which describes the total fluid molecular mass moving with specific velocity \(e_{i}\) per volume. By summation, \(f_{i}\) is recovered to macroscopic fluid variables, density \(\rho\) and momentum density \(\rho u\) with Equations (2) and (3). \[\rho=\sum_{i=0}^{8}f_{i} \tag{2}\]  
\[\rho u=\sum_{i=0}^{8}f_{i}e_{i} \tag{3}\]  
The time derivative of \(f_{i}\) undergoes two processes. The first is streaming, in which \(f_{i}\) with velocity \(e_{i}\) moves to ith directional neighbor lattice cell. The second is collision. Using the second law of thermodynamics, a new set of \(f_{i}\) collide towards the maximum entropy equilibrium distribution function \(f^{(0)}\), that is the Maxwell-Boltzmann distribution, which is discretized by second order hermit polynomials corresponding to nine velocity \(e_{i}\), in which the \(f^{(0)}\) expression is given by Equation (4). \[f_{i}^{(0)}=w_{i}\rho\Bigg{\{}1+\frac{3(e_{i}\cdot u)}{c^{2}}+\frac{9(e_{i} \cdot u)^{2}}{2c^{4}}-\frac{3u^{2}}{2c^{2}}\Bigg{\}} \tag{4}\]  
Figure 1: D2Q9 velocity sets with nine directional discretized velocity. \[w_{i}= \begin{cases}\frac{4}{9},&i=0\\ \frac{1}{9},&i=1,2,3,4\\ \frac{1}{36},&i=5,6,7,8\end{cases}  \tag{5}\]  
where \(c^{2}=\Delta x/\Delta t=1\). In collision process, distribution function transfers from chaos state to equilibrium state gradually. [PERSON] et al. [24] proposed the most popular BGK model with single relaxation time \(\tau\), in which \(f_{i}\) move towards equilibrium distribution function \(f_{i}^{(0)}\) linearly with time,  
\[f(x+e_{i}\Delta t,u,t+\Delta t)=f(x,u,t)-\frac{\Delta t}{\tau}\Big{[}f(x,u,t)- f^{(0)}(x,u,t)\Big{]} \tag{6}\]  
To obtain \(\tau\), distribution function \(f_{i}\) is transformed to the Navier-Stokes equation by Chapman-Enskog analysis, and \(\tau\) is given by Equation (7), \(\overline{v}\) is lattice dimensionless fluid viscosity, which is scaled from physical fluid viscosity. As necessary, viscosity must be non-negative for numerical stability, thereby \(\tau\) has to satisfy \(\tau>0.5\). \[\tau\ =\ 3\overline{v}\ +\ 0.5 \tag{7}\]  
In this paper, there are two lattice Boltzmann boundary conditions: (a) non-slip resting boundary and (b) velocity boundary. The non-slip boundary rests between the fluid lattice and solid lattice, and satisfies the bounce back rule in Equation (8). \(f_{i}\) bounces back to the opposite direction on the half way when streaming to ith directional neighbor lattice cell. In Equation (8), \(-i\) denotes the reverse direction of \(i\) while the \(+\) represents post collision, and \(\mathrm{x}_{b}\) is boundary lattice cells position. \[f_{-i}(\mathrm{x}_{b},t+\Delta t)=f_{i}^{+}(\mathrm{x}_{b},t) \tag{8}\]  
Boundary with specific velocity such as inlet and outlet (see Figure 2) lacks the streaming source. The half side \(f_{i}\) are unknown, namely; \(f_{1}\), \(f_{5}\), and \(f_{8}\). [PERSON] and [PERSON] [25] reconstructed the unknown distribution function from the channel velocity using Equations (9)-(11).