## 4 Results and Discussion
### Experimental Results of Pressure Attenuation and Particles Dispersion
(**a**) at 0.5 ms; (**b**) at 2 ms; (**c**) at 4 ms; (**d**) at 6 ms.  
Figure 10: The simulated dispersion process of the quartz sand particles with an average diameter of 0.95 mm. (**a**) at 0.5 ms; (**b**) at 2 ms; (**c**) at 4 ms; (**d**) at 6 ms.  
Figure 9: The simulated dispersion process of the quartz sand particles with an average diameter of 2.5 mm. (**a**) at 0.5 ms; (**b**) at 2 ms; (**c**) at 4 ms; (**d**) at 6 ms.  
As can be seen from Figures 9-11, the velocity distribution of the quartz sand particles showed a clear sense of layering, and the velocity of the particles at the central axis position was significantly greater than that of the surrounding particles, which also explained the reason why the particles show a gradually opened \"umbrella\" type pattern. The smaller the particle size, the farther the dispersion distance, the larger the formed \"umbrella\", and the more uniform the particle distribution. Therefore, the simulation results were consistent with the shock wave experimental results, indicating the reliability of the numerical model. In addition, the details of the flow field were also clearly revealed through the CFD simulation. It can be seen from the positions of the shock wave front and the solid particles at 0.5 ms that the particles lagged significantly behind the shock wave during the dispersion process, which verified the theoretical analysis in Section 4.1. According to the velocity vector distribution of the flow field, an annular vortex was generated around the solid particles, which was mainly caused by the entrainment effect of the fluid. At the same time, the vortex continued to rise as the particles were dispersed upward. In order to further analyze the shape and formation mechanism of the vortex structure, the vortex generated in the interaction process between the shock wave with the 0.95 mm quartz sand particles was taken as the research object, and the evolution of the three-dimensional annular vortex at different times is shown in Figure 12.  
Figure 12: Evolution process of the three-dimensional annular vortex. (**a**) at 0.5 ms; (**b**) at 2 ms; (**c**) at 4 ms; (**d**) at 6 ms.  
As shown in Figure 12, the three-dimensional annular vortex was generated above the shock tube outlet, and the vortex gradually expanded and rose with time. After the shock wave passed through the quartz sand particles, a high-speed jet was formed at the central position above the shock tube. The jet could be divided into a jet core area and a jet boundary layer. The central area that maintained the velocity near the outlet of the shock tube was the jet core area, and the part between the jet core and the static air was the jet boundary layer. In the vertical direction, the transition section from the shock tube outlet to the jet core is called the \"initial section\" of the jet. The three-dimensional annular vortex observed in the simulation was mainly generated in the jet boundary layer. During the interaction process between the shock wave and the solid particles, the high-speed jet generated by the shock wave exchanged momentum and energy with the surrounding static air, and then the surrounding air was entrained and moved with the jet. As a result of entrainment and mixing effects, the flow field was gradually homogenized, the jet velocity decreased, and the jet region and the cross-sectional area of the vortex caused by the entrainment effect continued to increase.