# Large Eddy Simulation and Dynamic Mode Decomposition of Turbulent Mixing Layers
## 3 Results and Discussions
### Reduced Order Reconstruction and Prediction of Flow Field
#### 3.3.3 Influence of Inlet Disturbance on Reduced Order Reconstruction and Prediction  
In this section, method 2 is used to select 150 modes for reconstruction and prediction to analyze the influence of disturbance existence (the difference between the undisturbed condition and the disturbed condition) and disturbance form (the difference between random disturbance and periodic disturbance).  
Figure 16a shows the comparison between the undisturbed reconstructed contours (right side) and the instantaneous contours (left side). Figure 17a shows the comparison between the reconstructed contours (right side) and the instantaneous contours (left side). From top to bottom are \(U\), \(V\), and \(\omega\). Under the undisturbed condition, the flow field in the computational domain belongs to the region before instability, and the reconstructed contours are almost the same as the instantaneous contours. Compared with Figure 17a, it can be seen that the reconstruction effect of undisturbed DMD is better than that of perturbed DMD.  
As before, the same six points for reconstruction and prediction were selected. The relative error is shown in Figures 16b and 17b. The fluctuation of each undisturbed point value with time is much smaller. In particular, the actual velocity of the points (0.005,0) and (0.008,0) near the entrance is almost unchanged. The relative error is within 1%. In general, the reconstruction and prediction of undisturbed DMD are better than that of perturbed DMD.  
Figure 16: 150 order reconstruction and prediction of no forcing case. (**a**) Comparison of reconstruction (**right**) and instantaneous contours (**left**); (**b**) relative error.  
Figure 17: 150 order reconstruction and prediction of random forcing case. (**a**) Comparison of reconstruction (**right**) and instantaneous contours (**left**); (**b**) relative error.  
The influence of disturbance form is studied below. The disturbance conditions in the previous sections are all random disturbances. We added a group of periodic disturbance conditions for comparative study. The computational settings are consistent with that in Section 2. The sampling method is also consistent with random disturbance, and 351 snapshots are obtained. The periodic disturbance formula is as follows:  
\[v(y)=\alpha U_{c}\exp(-y^{2}/\Delta y_{0}{}^{2})\sin(6.28\times 14,900\times t) \tag{37}\]  
where \(\alpha\), \(\Delta y_{0}\) consistent with Section 2.2.2, \(t\) is the time variable.  
Figure 18a shows the comparison between the reconstructed contours of 150 modes selected by method 2 and the instantaneous contours. It is obvious that the reconstruction effect is ideal, and the reconstructed contours are almost the same as the instantaneous contours. The shape of the whole flow field, the size and the position of the vortex are almost the same. From the comparison of the instantaneous contours in Figures 17a and 18a, it can be seen that the flow field under the condition of random disturbance develops faster than that under the condition of periodic disturbance. The results show that the flow field under the random disturbance condition consists of the pre-instability region, the rolling up region, the vortex interaction region, and the rolling up the position of the vortex is about 0.007 m in the \(x\)-direction. However, there is no obvious vortex interaction zone in the flow field under the condition of periodic disturbance, and the vortex curling position is about 0.017 m in the \(x\)-direction.  
Similarly, the six selected points are reconstructed and predicted, and the relative error is shown in Figure 18b. Except for the point (0.02, 0), the error of other points is less than 5%. Compared with the results of random disturbance in Figure 17, it can be seen that DMD has poor reconstruction and prediction effect for highly nonlinear and aperiodic conditions, but good reconstruction and prediction effect for periodic disturbance conditions.