# Large Eddy Simulation and Dynamic Mode Decomposition of Turbulent Mixing Layers
## 3 Results and Discussions
### Decomposition of Flow Field
#### 3.2.2 DMD Analysis of Inlet Disturbance Characteristics  
DMD analysis is carried out for different inlet disturbance conditions and the influence of disturbance amplitude is studied. The previous basic case is random disturbance condition \(\alpha\) = 0.0045 (1 forcing). Here, two groups of random disturbance condition \(\alpha\) = 0.045 (10 forcing), 0.45 (100 forcing) and one group of no forcing condition \(\alpha\) = 0 are added. DMD analysis was carried out on the \(x\)-direction velocity \(U\) of the three groups of working conditions, and 351 snapshots were also sampled. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**DMD Mode** & **Singular Value** & **Growth Rate** & **Frequency (Hz)** \\ \hline
1 & 31,611 & 1756 & 386,176 \\
2 & 31,266 & 1756 & \(-\)386,176 \\
3 & 24,985 & 2084 & 323,111 \\
4 & 24,734 & 2084 & \(-\)323,111 \\
5 & 21,943 & 1054 & 314,746 \\
6 & 21,916 & 1054 & \(-\)314,746 \\
7 & 19,192 & \(-\)1027 & 304,800 \\
8 & 19,113 & \(-\)1027 & \(-\)304,800 \\
9 & 16,153 & \(-\)1970 & 296,576 \\
10 & 16,107 & \(-\)1970 & \(-\)296,576 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Singular values and growth rates of the first 10 DMD modes. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**DMD Mode** & **Singular Value** & **Growth Rate** & **Frequency (Hz)** \\ \hline
1 & \(1.05\times 10^{6}\) & 0 & 0 \\
2 & 20,410 & \(-\)1296 & 10,303 \\
3 & 20,286 & \(-\)1296 & \(-\)10,303 \\
4 & 18,143 & 412 & 18,042 \\
5 & 17,992 & 412 & \(-\)18,042 \\
6 & 16,831 & 33  
Figure 8: DMD analysis of \(y\)-direction velocity \(Y\). (**a**) Singular value distribution; (**b**) modal growth rate versus frequency; (**c**) modal eigenvalues distribution; (**d**) modal amplitude versus frequency (**e**) first 10 modes. Figure 9: DMD analysis of vorticity \(\omega\). (**a**) Singular value distribution; (**b**) modal growth rate versus frequency; (**c**) modal eigenvalues distribution; (**d**) modal amplitude versus frequency (**e**) first 10 modes. Figure 10a shows the order 2\(\sim\)350 singular value arrangement under four disturbance conditions. The first-order singular values of no forcing, forcing, 10 forcing and 100 forcing are 1,040,287.5, 1,045,597.5, 1,046,331.7 and 1,046,807.3, respectively. It can be seen that the larger the disturbance is, the larger the first-order singular value is. However, the difference between the three perturbed singular values is smaller than that between the perturbed and undisturbed singular values. As can be seen from Figure 10a, the first-order singular value is more than one order of magnitude larger than other singular values. The singular value of DMD with forcing is obviously larger than that without forcing. There is no significant difference in the singular values of the three disturbed conditions. It demonstrates that the composition of the flow field of the three kinds of conditions of disturbance is similar. Figure 10b shows the relationship between the growth rate of the four disturbances and the frequency. Since the growth rate is symmetrical about the frequency, Figure 10b only shows the right half for easy observation. It can be seen that the growth rates of the four disturbances are mainly concentrated in the region of \(-\)5000\(-\)0, indicating that most of the modes are stable, and there is a point at the origin under each condition, and the corresponding dynamic mode at this point corresponds to the mean flow structure. In general, the larger the disturbance is, the farther away the points are from the origin. It shows that the nonlinear intensity of the disturbed flow field is directly related to the magnitude of the disturbance.