## 5 Simulation Results and Analysis
### Temperature Field  
Figure 10a shows the variation in temperature at a fixed detection point at the radial gap over one cycle of the main shaft's movement. As the main shaft rotates, when the radial engagement point approaches the monitoring point, the temperature at the radial gap shows a decreasing gradient, transitioning from a high temperature zone above to a lower temperature zone below. However, due to the pressure difference between adjacent working chambers causing tangential leakage, the gas in the high-pressure chamber flows into the low-pressure chamber, leading to temperature fluctuations at the engagement points. Moreover, the temperatures at symmetric engagement points are not equal, with the combined profile showing greater unevenness in temperature at symmetric engagement points, and a more noticeable temperature difference between adjacent compression chambers.  
As shown in Figure 10b, the exhaust temperature exhibits periodic changes as the main shaft rotates. Within a single cycle, the exhaust temperature gradually increases and tends to stabilize. The combined profile vortex compressor reaches thermal equilibrium with the external environment in the 6-th cycle, with an average temperature tending towards 577 K. Meanwhile, the uniform wall thickness vortex compressor shows periodic changes in outlet temperature after 12 cycles, with an average temperature approaching 600 K. After the vortex compressor stabilizes in operation, at any angle of the main shaft rotation, the outlet temperature of the uniform wall thickness vortex compressor is slightly higher than that of the combined profile vortex compressor. Uniform thickness vortex compressors need to consume more energy to compensate for efficiency losses caused by thermal leakage, directly leading to a decrease in the overall efficiency of vortex compressors.  
Figure 9: Mass flow comparison. (**a**) Comparison of inlet and outlet mass flow; (**b**) comparison of mass flow per unit return angle.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Numerical Value (B)** & **Numerical Value (D)** & **Percentage Increase** \\ \hline Maximum outlet & & & \\ mass flow/kg.s\({}^{-1}\) & & & \\ Maximum inlet mass & & & \\ flow/kg.s\({}^{-1}\) & & & \\ Average outlet mass & & & \\ flow/kg.s\({}^{-1}\) & & & \\ Average inlet mass & & & \\ flow/kg.s\({}^{-1}\) & & & \\ Unit return angle & & & \\ outlet mass & & & \\ flow/kg.s\({}^{-1}\) & & & \\ Unit return angle inlet & & & \\ mass flow/kg.s\({}^{-1}\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of mass flow values of different profiles.  
Figure 11 shows the distribution of temperature field cloud maps at different rotational angles after the working fluid in the working chamber stabilizes. It can be seen that the temperature gradually increases from the suction chamber to the central compression chamber. Due to tangential leakage caused by the radial gap, the temperature distribution within a single working chamber is uneven both radially and axially. Furthermore, since the exhaust area is not located at the center of the working chamber, the temperature distribution in the symmetric compression chambers is also uneven. As the dynamic vortex disk moves with the main shaft, the gas in each chamber continuously moves towards the center. The pressure difference between adjacent compression chambers increases, leading to more severe tangential leakage closer to the central chamber, which results in a more pronounced uneven distribution within the chamber. The gas brought by the leakage flow might have been compressed and heated upstream, causing a local temperature increase when mixed with the gas in the low-pressure chamber.  
Figure 11: Temperature nephogram distribution at different corner positions. (**a**) Temperature nephogram of combined profile; (**b**) temperature nephogram of constant wall thickness profile.  
Figure 10: Temperature contrast chart. (**a**) Temperature curve of marking point; (**b**) outlet temperature.