## 3 Results  
### Flow Structure Visualization  
The numerical schlieren technique proposed by [PERSON] [39] is employed here to obtain the instantaneous flow structures; it displays the flow field image based on density gradient. The strategy can be defined as  
\[S(x,y)=\beta\ exp\bigg{(}-\frac{\kappa|\
abla\rho|}{|\
abla\rho|_{max}}\bigg{)} \tag{30}\]  
\[|\
abla\rho|=\sqrt{\big{(}\frac{\partial\rho}{\partial x}\big{)}^{2}+\big{(} \frac{\partial\rho}{\partial y}\big{)}^{2}} \tag{31}\]  
Here, the parameter \(\beta\) determines the color corresponding to the zero gradient, the parameter \(\kappa\) governs the amplification degree of the small gradients, and the flow field is usually better displayed when the \(\beta\) and \(\kappa\) are taken as 0.8 and 15, respectively. Figure 6 shows the numerical schlieren contour for present three cases. Due to the Kelvin-Helmholtz(K-H) instability generated by different upper- and lower-layer speeds, the typical coherent structure called K-H vortices in the mixing layer rolling up can be seen in all three cases. The K-H vortices gradually grow, and the phenomenon of pairing and merging between vortices appears. [PERSON] [40] confirmed that the dynamic of pairing and merging between adjacent vortex structures is an important mechanism of free mixing layer growth. Apart from that, the shocklets in three cases are clearly visible. The shock waves generate by the local flow being impeded by the large-scale structure in the turbulent mixing layer when the local Mach number relative to the large-scale structure is greater than 1.0 [41]. Generally, shocklets appear in the high Mach number region of the flow field, corresponding to the upper flow area in our cases. It can be found that shocklets become more obvious due to the compression effect induced by jet shock in Case 2 and in Case 3. Figure 6: Instantaneous flow structures of present three cases: (**a**) Case 1; (**b**) Case 2; (**c**) Case 3. In Case 1, the mixing layer develops around the central region of the flow field. However, in Case 2 and Case 3, due to the introduction of jet shock, the mixing layer is deflected downward after the incident shock, then deflected upward after the reflection shock. Meanwhile, the jet shock is first refracted through the mixing layer, then reflected upward on the lower wall, and then again refracted on the mixing layer. Refraction shock interacts with shocklets and finally becomes segmented state. Relative to Case 1, a new phenomenon of vortices rolling up closer to the inlet can be seen in Case 2 and Case 3, indicating that there are subsonic regions at the shock/mixing interaction area. This leads disturbance to transmit backward and can promote the vortices to roll up. Compared Case 2 with Case 3, it can be found that incident shock will be segmented in Case 2, while relatively straight in Case 3. Fast jet frequency in Case 2 makes the shock wave propagation speed lower than the jet disturbance speed, eventually it causes the 'disconnection' phenomenon. Overall, the vortex structures belonging to the interaction region possess characteristics of coexistence of both large- and small- scale vortices, and this dynamic is of great importance for promote mixing. Thus, detailed analysis concerning the structure behaviors is essential. To better analyze the interaction process between jet shock and mixing layer in case 3, the detailed visualization around the interaction position is presented in Figure 7. One important phenomenon is that compared to the interaction process in case 2, in case 3 the interaction position is not fixed. In Figure 7a the interaction position located at x = 83 mm approximately, while in Figure 7b at about x = 62 mm. Indeed, by analyzing our calculated results it can be found that these two positions correspond to the farthest and nearest interaction locations in the flow field.