## 3 Results
### DMD Analysis  
As a newly proposed mixing technique, it can be found that the RAM generated by WJISW at a low frequency provides a significant degree of mixing enhancement, and the reveal of its mechanism contributes to the design optimization of the mixing augment device. In this paper, our cases are nonlinear dynamic systems, and the original flow field includes a large amount of information. Thus, it is not easy to grasp the focus of the data. It is worth noting that as an emerging reduced-order method applied in the flow field, DMD can realize the decomposition of the original data to obtain a specific flow field structure with a dominant frequency and growth rate [38]. This method can provide a convenient means for flow field dimensionality reduction and information acquisition, and it has achieved some application in the field of compressible flow [47; 48; 49; 50], especially in revealing the typical structural characteristics of the flow [51]. In this paper, 1200 snapshots of the flow field with a time interval of 1.5\(\upmu\)s are extracted from the transient flow field of three cases close to 8 circulation cycles for DMD analysis. The penalty terms \(\Upsilon\) of the three cases in our paper are chosen as \(1\times 10^{4}\), \(1\times 10^{5}\), and \(1\times 10^{4}\), respectively. After optimizing the amplitude of the modes by the DMD method, the number of non-zero modes of the three cases reduced from the original 1199 to 385, 423, and 401, respectively, which reflects the advantage of the DMD method in dimensionality reduction. To confirm the reliability of the DMD method, the original flow field and the reconstructed flow field employing DMD method are compared and analyzed, as shown in Figure 12. The flow fields are characterized utilizing the dimensionless vorticity \(\omega_{nor}\) and it can be defined as  
\[\omega_{nor}=\left|\frac{\omega_{z}\delta_{(0)}}{u_{2}-u_{1}}\right| \tag{39}\]  
\[\omega_{z}=\frac{\partial u}{\partial y}-\frac{\partial v}{\partial x} \tag{40}\]  
It can be seen that the DMD reconstructed flow field has basically restored the vortex behavior in the flow field, and the position and size also match with the original flow field. The roll-up, pairing, and merging of vortex structures can be clearly seen from the DMD reconstructed flow field, indicating that the vortex structure dynamics, which are important for the development of the mixing layer, can be captured by the DMD method. Figure 11: Pressure loss coefficient along streamwise direction for three cases. Furthermore, a quantitative comparison is performed by extracting the dimensionless vorticity on the centerline of the reconstructed flow field and the original flow field, as shown in Figure 13. From the figure, it can be more intuitively seen that the overall trend of the reconstructed values remains consistent with the original values, which also coincides with the results analyzed above. It can also be seen from the figure that the dimensionless vorticity on the centerline of the flow field of Case 1 fluctuates more, while Case 2 and Case 3 fluctuate less after x = 70 mm, which is due to the fact that the mixing layer impacted by WJISW will be deflected by the shock wave interaction, so the region of the most intense turbulent pulsation is not on the centerline of the flow field. Figure 14 shows the frequencies and energy proportion distribution of DMD modes in three cases. Except for the first mode, all the modes are conjugate modes. The conjugate mode refers to modes with opposite frequencies, but with the same growth rate and modal energy proportion. Thus, here only frequencies greater than 0 are shown. The black points represent the values with eigenvalues less than or equal to 1 or a modal growth rate less than or equal to 0 (the two concepts can be found equivalent by simple logarithmic calculation), which correspond to stable or periodic modes, while the red points represent the values with eigenvalues greater than 1 or a modal growth rate greater than 0, which corresponds to unstable modes.