# Large Eddy Simulation and Dynamic Mode Decomposition of Turbulent Mixing Layers
## 3 Results and Discussions
### Reduced Order Reconstruction and Prediction of Flow Field
#### 3.3.2 Influence of Dominant Mode Selection Method on Reduced Order Reconstruction and Prediction  
After the flow field is decomposed, the selection of the dominant mode is the key step of flow field reduction reconstruction and prediction. At present, there is no specific method to select the dominant mode of the flow field after DMD decomposition. For different flow characteristics, the most suitable mode selection method may be different. In this section, four modal selection methods are compared: (1) according to the size of singular value; (2) according to the size of modal amplitude; (3) according to the size of modal energy [63]; (4) according to the size of the integral modal amplitude of sampling time [64], to investigate the mode selection method of the turbulent mixing layer suitable for this study. Here, four methods are briefly introduced. The first and second methods are easier to understand. The expression of the third method is as follows:  
\[\|\varphi_{i}\|=\sqrt{\sum_{i=1}^{n}{\lvert\varphi_{i}\rvert}^{2}} \tag{35}\]  
where \(\|\varphi_{i}\|\) is the energy contained in the \(i\)th mode and \(\varphi_{i}\) is the flow velocity of each node in the \(i\)th mode. In the fourth method, the integral modal amplitude is defined as [64]:  
\[I_{j}=\sum_{i=1}^{n}{\lvert\alpha_{j}(\mu_{j})^{i-1}\rvert}{\lVert\varphi_{j} \rVert}_{F}^{2}\times\Delta t \tag{36}\]  
DMD analysis is carried out for the stochastically perturbed velocity \(U\). Figure 12a shows the relationship between modal energy and frequency, from which it can be seen that high modal energy (greater than 0.7) is mainly concentrated near low frequency and high frequency, and the modal energy of middle frequency is almost about 0.7. The first three modes with the largest energy are shown in Figure 12a, which is named mode 1, mode A and mode B. The modal energy of mode 1 is the largest, and the frequency is 0 Hz. Mode 1 is the first-order mode arranged according to the eigenvalue, which also shows that the first-order mode is the most important mode. Mode A is the second-highest energy mode with a frequency of 1,567,540 Hz. Mode B is the third-highest energy mode with a frequency of 1,570,796 Hz. Modes A and B are high-frequency modes in the flow field. Figure 12b shows the relationship between the integral mode amplitude and frequency, and marks the three modes with the largest amplitude of integral mode which are also mode 1, mode A and mode B. Different from modal energy, modal B is larger than modal A. This shows that different methods extract the dominant mode differently. Compared with Figure 7d, we can see that the morphology has changed significantly. The integral amplitude of the first-order mode is still the largest, which shows once again that the first-order mode is the most important. Figure 13 shows the contours of mode A and mode B. compared with modes 1-10 in Figure 7, it can be seen that the high-frequency modes present row upon row of small-scale structures. The low-frequency modes present large-scale structures. In order to ensure a certain accuracy and reduce the order of the flow field, four methods are used to select 150 modes to reconstruct and predict the velocity \(U\) flow field. The comparison results are shown in Figures 14 and 15. It can be seen from Figure 14 that the reconstructed contours by methods 2 and 3 are close to the instantaneous contours and can reconstruct a smaller scale structure. The reconstructed contours by methods 1 and 4 are not ideal, only some large structures can be reconstructed, and the shape difference is large, which is very different from the instantaneous flow field. Figure 15 shows the comparison between the predicted contours and the instantaneous contours. It can be seen that the predicted results of all methods have some deviation. The prediction results of method 1 and method 3 are almost the same, both of which overestimate the flow field. The result of method 4 is still not optimal.