## 2 Mathematical and Physical Formulation
### Cascaded Lattice Boltzmann Method (CLBM)
#### 2.2.1 CLBM for the Flow Field  
The D2Q9 lattice is used in this study to depict the flow field, which looks at two-dimensional (2D) problems [23]. The lattice space and time steps are represented by the variables \(\Delta x\) and \(\Delta t\), respectively, and the selected lattice speed is \(C=\frac{\Delta x}{\Delta t}=1,\Delta x=1,\Delta t=1\). The distinct speeds: \(e_{i}=\left[|e_{ix}>,|e_{iy}>\right.\) are defined by  
\[|e_{ix}>=[0,1,0,-1,0,1,-1,-1,1]^{tr} \tag{5}\]  
\[|e_{iy}>=[0,0,1,0,-1,1,1-1,-1]^{tr} \tag{6}\]  
where the transposition is indicated by the superscript \(tr\), and the column vector is indicated by \(i=0\ldots 8\),\(|\). \(>\). Raw moments and central moments for discrete distribution functions (DFs) \(f_{i}\) are provided to build the collision operator based on central moments [24]. \[K_{mn}=\left\langle f_{i}\middle|e_{ix}^{m}e_{iy}^{n}\right\rangle \tag{7}\]  
\[\overline{K}_{mn}=\left\langle f_{i}\middle|\left(e_{ix}-u_{x}\right)^{m} \left(e_{iy}-u_{y}\right)^{n}\right\rangle \tag{8}\]  
The discrete equilibrium distribution function (EDFs), \(f_{i}^{eq}\), is substituted for \(f_{i}\) to define the equilibrium values \(k_{mn}^{eq}\) and \(\overline{k}_{mn}^{eq}\) in an analogous manner. Numerous researchers have adopted the recombined raw moments, \(k_{20}+k_{02}\) and \(k_{20}\), \(k_{02}\), to treat the trace of the pressure tensor and the normal stress difference independently in literature [25; 26; 27; 28; 29; 30; 31; 32]. However, the practical use is time-consuming, especially when dealing with three-dimensional (3D) models. A simpler approach was presented in [33] to make practical implementation easier. It involved using the raw moments, \(k_{20}\) and \(k_{02}\), and making minor changes to the relaxation matrix. The simplified raw-moment set is used in this study,  
\[|\Gamma_{i}>=[K_{00},K_{10},K_{01},K_{20},K_{02},K_{11},K_{21},K_{12},K_{12},K_ {22}] \tag{9}\]and the united key Shifted moments also do \(\widetilde{\Gamma}_{i}\). More precisely, \(|\Gamma_{i}>=\)M\(|f_{i}>\) transforms the raw moments from _fi_ via a transformation matrix \(M\) and \(|\widetilde{\Gamma}_{i}>=\)N\(|f_{i}>\)[30; 34] separates the essential moments from the raw moments using shift matrix \(N\). The notation for \(M\) and \(N\) is as follows:  
\[M= \begin{bmatrix}1&1&1&1&1&1&1&1&1\\ 0&1&0&-1&0&1&-1&-1&1\\ 0&0&1&0&-1&1&1&-1&-1\\ 0&1&0&1&0&1&1&1&1\\ 0&0&1&0&1&1&1&1\\ 0&0&0&0&0&1&-1&1&-1\\ 0&0&0&0&0&1&1&-1&-1\\ 0&0&0&0&0&1&-1&-1\\ 0&0&0&0&0&1&1&1&1\end{bmatrix}  \tag{10}\]  
\[N= \begin{bmatrix}1&0&0&0&0&0&0&0\\ -u_{x}&1&0&0&0&0&0&0&0\\ -u_{y}&0&1&0&0&0&0&0\\ u_{x}^{2}&-2u_{y}&0&1&0&0&0&0&0\\ u_{y}^{2}&0&-2u_{y}&0&0&0&0&0\\ u_{x}u_{y}&-u_{y}&-u_{x}&0&0&0&0&0\\ -u_{x}^{2}u_{y}&2u_{x}u_{y}&u_{x}^{2}&-u_{y}&0&-2u_{x}&1&0&0\\ -u_{y}^{2}u_{x}&u_{y}^{2}&2u_{x}u_{y}&0&-u_{x}&-2u_{y}&0&1&0\\ u_{x}^{2}u_{y}^{2}&-2u_{x}u_{y}^{2}&-2u_{y}u_{x}^{2}&-u_{y}^{2}&u_{x}^{2}&4u_{ x}u_{y}&-2u_{y}&-2u_{x}&1\end{bmatrix}  \tag{11}\]  
The expressions used here are simpler and have the potential to lower processing costs when compared to the \(M\) and \(N\) expressions in [29]. The post-collision central moments are obtained by separately relaxing each central moment to its equilibrium equivalent, as follows:  
\[\left|\widetilde{\Gamma}_{i}^{*}>=(I-S)\right|\widetilde{\Gamma}_{i}>+S\left| \widetilde{\Gamma}_{i}^{eq}>+\left(I-\frac{S}{2}\right)\right|C_{i}> \tag{12}\]  
where \(C_{i}\) are the forcing source terms in central moment space, and the block-diagonal relation matrix is given by  
\[S=diag([s_{0},s_{1},s_{1}], \begin{bmatrix}s_{+}&s_{-}\\ s_{-}&s_{+}\end{bmatrix} ,[s_{v},s_{3},s_{3},s_{4}] \tag{13}\]  
with \(s_{+}=\frac{s_{0}+s_{v}}{2}\) and \(s_{-}=\frac{s_{0}-s_{v}}{2}\).