# Large Eddy Simulation and Dynamic Mode Decomposition of Turbulent Mixing Layers
## 3 Results and Discussions
### Reduced Order Reconstruction and Prediction of Flow Field  
#### 3.3.1 Reduced Order Reconstruction and Prediction of Basic Case  
In this section, the flow field under basic case is reduced, and the first 150 orders are truncated (the sum of the first 150 singular values accounts for more than 90% of the total singular values). The reduced order reconstruction and prediction effect of DMD on the turbulent mixing layer flow field with strong nonlinearity are studied. The \(x\)-direction velocity \(U\), \(y\)-direction velocity \(V\) and vorticity \(\omega\) are reconstructed and predicted, respectively.  
Figure 11a shows the comparison between the reconstructed contours and the instantaneous contours. The physical computational domain (0-0.0254 m) is compared. According to the vorticity contours, the flow field is divided into three regions: the region before instability, the rolling up region and the vortex interaction region. In the upstream of the flow field, there is no obvious fluid exchange and vortex structure. The vorticity concentrates on the shear line, and its width increases gradually along the flow direction. We call this region the pre-instability region. In the middle of the flow field (i.e., downstream of the region before instability), there is obvious material exchange between the upper and lower flow fields. An obvious single vortex (a series of vortex structures) can be observed from the vorticity contours. The distance between these vortex structures is approximately proportional to their own scale, and gradually increases along the flow direction. We call this region the roll up region. In the downstream of the flow field (that is, the downstream of the roll up region), the interaction between the vortices of the flow field occurs. We call this region the vortex interaction region.  
It can be seen that in the basic case, whether it is velocity \(U\), \(V\) or vorticity \(\omega\), the reconstruction effect of the flow field is poor, and only the region before instability (about 0-0.006 m region) can be reproduced. For the roll up region (about 0.006-0.015 m region) and vortex interaction region (about 0.015-0.0254 m region), the reconstruction effect of DMD is not good because of the strong nonlinearity of the region.  
Six points (0.005,0), (0.008,0), (0.011,0), (0.014,0), (0.017,0) and (0.02,0) on the middle line of the physical computational domain are selected to further analyze the reconstruction and prediction of DMD.  
Figure 11b is the reduced order reconstruction and prediction relative error diagrams of perturbed velocity \(U\). Among them, 0.0035 s-0.0042 s is the time range within the sample, and 0.0042 s-0.0049 s is the predicted time range.  
The relative error \(\delta\) is defined as  
\[\delta=\frac{|a-A|}{A} \tag{34}\]  
Figure 11: 150 order reconstruction and prediction of basic case. (**a**) Comparison of reconstruction contours (right) and instantaneous contours (left 42,000 steps) with forcing; (**b**) relative error.  
where \(a\) is the reconstructed and predicted value of DMD and \(A\) is the actual value.  
As can be seen more clearly from Figure 11b, the error of reconstruction segment is obviously less than that of prediction segment, most of the errors of reconstruction segment are less than 10%, a considerable part of the errors of prediction segment are more than 20%, and individual errors are more than 40%. The effect of velocity \(V\) and vorticity \(\omega\) is similar, as such there is no additional illustration.