# 3D Numerical Simulation and Performance Analysis of CO2 Vortex Tubes
## 3 Results and Discussion
### Contour Graphs of Axial and Radial Velocity Distribution  
The compressed gas enters the vortex chamber tangentially from the swirl flow pipe inlet, the gas expansion performs work, and part of the pressure potential energy is converted into kinetic energy at the inlet. The kinetic energy forms a high-speed vortex flow in the pipe (Figure 10), increasing the temperature at the wall of the pipe, and produce a pressure gradient from the outer edge of the tube to the axial area. When the inlet pressure reaches 1 MPa, the inlet of the swirl flow pipe is depressurized, expanded, and accelerated to a critical state at the minimum cross-section. The depressurization continues to reach a supersonic state in the vortex chamber. From the numerical simulation results, it can be seen that the supersonic fluid will appear only when the pressure at the swirl flow pipe inlet is high enough. The pressure gradient between the hot-end valve and the vortex chamber is caused by the action of the hot-end valve, which in turn leads to backflow in the axial region. The high-speed airflow is in a cyclonic state while moving toward the hot-end valve, so the vortex phenomenon occurs in the swirl flow pipe (Figure 11). Figure 12 shows the distribution curve of the axial velocity of the swirl flow pipe. From Figure 12, it can be seen that the vortex velocity gradually decreases along with increased axial distance in the direction of the hot-end tube. The velocity variation in the axial region are low, and the energy separation is precisely due to the interaction between the positive and counter-rotating fluids. Figure 13 shows a three-dimensional streamline diagram of the swirl flow pipe, from which it can be seen that the axial velocity gradually decreases from the swirl flow pipe inlet to the direction of the hot-end valve and gradually increases as the inlet pressure increases. A long hot-end tube facilitates an increase in axial velocity.  
Therefore, the axial velocity is a direct factor affecting the separation of hot and cold airflow in the swirl flow pipe. The interaction between the axially-oriented downstream and counter flows is a large contributor to the energy exchange in the swirl flow pipe, so it is vital to explore axial velocity. As the inlet pressure increases, the inlet velocity also increases, as shown in the blue part of Figure 11. As there is only one inlet nozzle for the swirl flow pipe, the sidewall of the vortex chamber near the inlet has the highest air velocity. To improve the durability of the swirl flow pipe, the vortex chamber can be designed with stronger materials, or the number of inlet nozzles can be increased to reduce erosion of the sidewall of the vortex chamber.  
Figures 14 and 15 show the axial circumferential velocity distribution of two different lengths of swirl flow pipes. The fluid entering the tube tangentially from the nozzle spontaneously generates free vortex flow along the outer edge of the swirl flow pipe (due to the restraining effect of the outer edge of the tube), while the counterflow fluid region away from the outer edge of the tube forms a forced vortex flow (due to the interaction between the inner and outer fluid).  
Figure 9: Effect of inlet pressure and hot-end tube length on radial temperature distribution.  
Figure 11: Axial velocity distribution of the swirl flow pipes at different inlet pressures.  
Figure 12: Axial velocity distribution curve of swirl flow pipes under different inlet pressures.  
Figure 10: Radial velocity distribution of swirl flow pipes at different pressures.  
Figure 14: Circumferential velocity distribution of 125 mm swirl flow pipe.  
Figure 15: Circumferential velocity distribution of 175 mm swirl flow pipe.