## 3 Results
### DMD Analysis
The modes with long wavelengths are favorable to induce large vortex structures, while the modes with short wavelengths are favorable to induce small vortex structures, resulting in the coexistence of large and small scales of vortex structures in the flow field. Entrainment and engulfment of large-scale structures allow surrounding fluids to enter the mixing region, and the nibbling effect of the small-scale structure facilitates the expansion of the surface area of scalar contact in the mixing layer. These dynamic behaviors accelerate the exchange of mass, momentum, and energy between the upper and lower layers of the flow [54], which is the reason why Case 3 has the best degree of mixing enhancement in the three cases. Comparing these three cases, Case 1 has a single vortex size, and Case 2 can induce different scales of vortex structures, but the largest vortex size is still small, and the large-scale entrainment effect is insufficient. Whereas, the interaction of large- and small-scale vortices in Case 3 leads to the fragmentation of the vortex structure in the flow field, which is consistent with the structure of the flow field in the numerical schlieren of Case 3. Figure 15: Dimensionless vorticity contour of the first eight modes in Case 1: (**a**) the 1 th mode; (**b**) the 2 nd mode; (**c**) the 3 rd mode; (**d**) the 4 th mode; (**e**) the 5 th mode; (**f**) the 6 th mode; (**g**) the 7 th mode; (**h**) the 8 th mode. Figure 16: Dimensionless vorticity contour of the first eight modes in Case 2: (**a**) the 1 st mode; (**b**) the 2 nd mode; (**c**) the 3 rd mode; (**d**) the 4 th mode; (**e**) the 5 th mode; (**f**) the 6 th mode; (**g**) the 7 th mode; (**h**) the 8 th mode. \[\omega_{nor}=\left|\frac{\omega_{z}\delta_{(0)}}{u_{2}-u_{1}}\right| \tag{39}\]0 _Cont._  
Figure 18 compares the evolution of the 2 nd-8 th order mode amplitudes with time in three cases. In the figure, the horizontal ordinate is the dimensionless time \(t_{nor}\). \[\omega_{nor}=\left|\frac{\omega_{z}\delta_{(0)}}{u_{2}-u_{1}}\right| \tag{39}\]1  
\[\omega_{nor}=\left|\frac{\omega_{z}\delta_{(0)}}{u_{2}-u_{1}}\right| \tag{39}\]2 Dimensionless vorticity contour of the first eight modes in Case 3: (**a**) the 1 st mode; (**b**) the 2 nd mode; (**c**) the 3 rd mode; (**d**) the 4 th mode; (**e**) the 5 th mode; (**f**) the 6 th mode; (**g**) the 7 th mode; (**h**) the 8 th mode. \[\omega_{nor}=\left|\frac{\omega_{z}\delta_{(0)}}{u_{2}-u_{1}}\right| \tag{39}\]3 Evolution of time coefficients among seven dominant DMD modes: (**a**) Case 1; (**b**) Case 2; (**c**) Case 3. From Figure 18a, it can be seen that there is not much difference between the modal amplitude and frequency of the first few order modes of Case 1 and the 2 nd mode, which has a lower initial amplitude and is ranked ahead of the 3 rd mode. This is because the DMD mode ranking criterion in this paper not only considers the initial amplitude, but also focuses on its evolution in the entire time history of available snapshots. The 3 rd mode shows a weak decay characteristic with a negative growth rate, and the amplitude decreases slowly with time, while the 2 nd mode is an unstable mode with a positive growth rate, and the amplitude increases gently with time. Therefore, the overall contribution of the 3 rd mode to the flow field is greater than that of the 2 nd mode, making the 2 nd mode ranked before the 3 rd mode, which also reflects the advantage of the modal ranking criterion in this paper. A large difference in the modal frequencies of the first few order modes of the flow field in Case  
From Table 7, we can see that the energy proportion of the first-order mode in Case 2 is 0.10638, which is lower than that in Case 1, which indicates that the perturbation energy proportion of Case 2 in the flow field is higher. Furthermore, the energy of the 2 nd order and the 3 rd order mode is much larger than the later modes, which indicates that these two modes of Case 2 are dominant in the flow field.