## 3 Results
### DMD Analysis
Moreover, blue triangles correspond to the first eight modes with a higher energy proportion, which play dominant roles in the flow field and will be mainly  
Figure 12: Comparison of original flow field and DMD reconstructed flow field: (**a**) original flow field of case 1; (**b**) constructed flow field of case 1. Figure 13: Comparison of original value and DMD reconstruction value of flow field center line: (**a**) case 1; (**b**) case 2; (**c**) case 3.  
analyzed below. It can be found that the frequency distribution of the first eight modes in Case 1 is concentrated around 100 kHz, while the frequency distribution of the first eight modes of Case 2 and Case 3 is more scattered, ranging from 50 kHz to 240 kHz and 6 kHz to 180 kHz, respectively. Meanwhile, it can also be seen that the frequencies of the first few modes in Case 3 are much smaller than those in Case 1 and Case 2. Figures 15-17 show the first eight orders of spatial mode contours in three cases, respectively. The 1 st modes show the time-averaged solution of dimensionless vorticity [52], while the rest of the modes reflect the oscillation characteristics generated by the flow field pulsation. The pulsation of the flow field in Case 1 mainly comes from the mixing of the fluid above and below the mixing layer, that is to say, the roll-up of K-H vortices. The upper and lower boundaries of the topology extracted from each mode can be regarded as the upper and lower boundaries of the mixing layer, and its width increases with the distance in streamwise x-direction, indicating that the thickness of the mixing layer grows in the streamwise x-direction. The analysis of the 2 nd to 8 th order modes shows that the topological structure, with alternating positive and negative values of dimensionless vorticity, appears only in the core region of the flow field. This structure can be considered as the spatial form of the wave, and the distance between the centers of positive and negative values can be regarded as half a wavelength [53]. The values in other regions of the flow field are zero, indicating that pulsations occur mainly in the mixing layer. Since the original flow field can be seen as a superposition of the DMD modes obtained from its decomposition, the vortices' behavior can be seen in the superposition image. However, in the decomposed modes of different frequencies, only the wave structure can be seen in the flow field image. The above findings together converge to a conclusion that the vortex in the flow field is essentially a superposition of waves of different frequencies. It can be seen from Figure 16 that the first few order modes in Case 2 cover a larger wavelength range than Case 1, and the 2 nd, 3 rd, 4 th, and 5 th order spatial modes of the flow field have significant shock effects, while the 6 th, 7 th, and 8 th order spatial modes do not. This indicates that the 2 nd, 3 rd, 4 th, and 5 th order spatial modes of the flow field are shock-induced modes, and the energy of these modes accounts for a higher proportion of the total flow field energy compared to the later modes that are not shock-induced, i.e., K-H instability-induced modes. Therefore, the jet shock-induced modes play a dominant role in the flow field. From Figure 17, we can see that the topology of the first few orders modes in Case 3 are very huge, especially the wavelength of the 2 nd order mode has exceeded half of the flow field area. The 2 nd, 3 rd, 4 th, and 6 th order spatial modes can clearly see the effect of the shock wave, indicating that they are modes induced by jet shock wave, and the topological stratification can be seen more clearly than the other modes, induced only by K-H instability. Moreover, it can be clearly seen that the wavelengths of the modes induced by the shock are longer than those of the first two cases, but the wavelengths of these  
Figure 14: Distribution of mode frequency and energy: (**a**) Case 1; (**b**) Case 2; (**c**) Case 3.  
modes are varying in length.