# Large Eddy Simulation and Dynamic Mode Decomposition of Turbulent Mixing Layers
## 2 Large Eddy Simulation of Turbulent Mixing Layer and Dynamic Mode Decomposition
### Dynamic Mode Decomposition Method  
Here, the standard DMD algorithm is briefly introduced. The \(n\) matrix snapshots \(\{x_{1},x_{2},x_{3},\cdots,x_{n}\}\) obtained by experiment or numerical simulation can be written into a snapshot sequence matrix \(X\) and \(Y\). the time interval between any two snapshots is \(\triangle t\). \[X=[x_{1},x_{2},\cdots,x_{n-1}],Y=[x_{2},x_{3},\cdots,x_{n}]\]  
It is assumed that the flow field \(xi\)+1 can be expressed by the linear mapping of flow field \(x_{i}\)  
\[x_{i+1}=Ax_{i} \tag{13}\]  
where \(A\) is the system matrix of high dimensional flow field. If the dynamic system itself is nonlinear, then the process is a linear estimation process. According to the assumed linear mapping relation, matrix \(A\) can reflect the dynamic characteristics of the system. Due to the high dimension of \(A\), it is necessary to calculate a from the data sequence by order reduction. Therefore  
\[Y=[Ax_{1},Ax_{2},\cdots,Ax_{n-1}]=AX \tag{14}\]  
For matrix \(X\), a matrix \(\widetilde{A}\) can be provided to replace the high dimensional matrix \(A\), and the two matrices are similar. In order to find the orthogonal subspace of similarity transformation, the singular value decomposition of \(X\) is used to obtain the following:  
\[X=U\Sigma V^{H} \tag{15}\]  
\[A=U\widetilde{A}U^{H} \tag{16}\]  
The matrix \(\Sigma\) is a diagonal matrix, and the diagonal elements contain \(r\) singular values. In the process of singular value decomposition, only \(r\) main singular values can be retained and the remaining small singular values can be truncated, so as to reduce the numerical noise. The unitary matrices \(U\) and \(V\) obtained by SVD satisfy \(U^{H}U=I\) and \(V^{H}V=I\). The calculation process of matrix \(\widetilde{A}\) can be regarded as the minimization problem of Frobenius norm  
\[\min_{\widetilde{A}}\lVert Y-U\widetilde{A}\Sigma V^{H}\rVert_{F}^{2} \tag{17}\]  
Then we can approximate \(A\) by  
\[A=\widetilde{A}=U^{H}YV\Sigma^{-1} \tag{18}\]  
Since the matrix \(\widetilde{A}\) is a similar transformation of \(A\), the matrix \(\widetilde{A}\) contains the main eigenvalues of \(A\). The \(j\)th eigenvalue is \(\lambda_{j}\) and the eigenvector is \(w_{j}\). Then the \(j\)th DMD mode is defined as  
\[\Phi_{j}=Uw_{j} \tag{19}\]  
The growth rate \(g_{j}\) and frequency \(\omega_{j}\) corresponding to the \(j\)th mode are defined as follows:  
\[g_{j}=\text{Re}\big{(}lg(\lambda_{j})/\Delta t\big{)} \tag{20}\]  
\[\omega_{j}=Im\big{(}lg(\lambda_{j})/\Delta t\big{)} \tag{21}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(Re_{\delta_{A}(0)}\) & \(\frac{\sigma_{xx}}{\Delta t}\) & \(\frac{\sigma_{yy}}{\Delta t}\) & \(\frac{\sigma_{xx}}{\Delta t}\) & \(\frac{1\sigma_{yy}}{\Delta t}\) & \(\frac{1}{\eta}\frac{\delta_{A}(0)}{\delta x}\) & **Reference** \\ \hline
720 & 0.053 & 0.076 & - & 0.024 & 0.173 & Present 2-D LES, double precision & Result 3-D LES, double precision \\
720 & 0.043 & 0.047 & 0.021 & 0.020 & 0.186 & Result 3-D LES, double precision \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of 2-D and 3-D computational results. When judging the stability, if the growth rate is positive, the corresponding mode is unstable; if the growth rate is negative, the corresponding mode is stable; if the growth rate is zero, the corresponding mode is periodic; if the eigenvalue falls within the unit circle, it means the stable mode, and vice versa. The dynamic modes of the flow field can be extracted by the above-mentioned DMD method, and the evolution process of the flow field can be further estimated according to the reduced order matrix \(\widetilde{A}\).