## 4 Finite Element Preprocessing
### Boundary Conditions  
This paper assumes that the gas inside the working chamber of the vortex compressor behaves as an ideal gas. Vortex compressors have a complex structure, and the gas flow during operation generates vortices. To achieve higher credibility and accuracy in a wide range of flows, the RNG k-\(\epsilon\) turbulence model is chosen. Considering the temperature changes in the gas during compression, the energy equation is enabled. With high-speed and high-pressure gas flow in the chamber and a finely resolved grid, a high-accuracy, fast-converging coupled solver is chosen. The fluid domain model of the vortex compressor is complex, and to quickly meet convergence criteria, coupled initialization is used, applying different solution methods for initial value assignment.  
Figure 4: Working cavity fluid domain grid. (**a**) Fluid domain grid of combined profile vortex compressor; (**b**) fluid domain grid of uniform wall thickness vortex compressor.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Numerical Value (D)** & **Numerical Value (B)** \\ \hline Base circle/mm & R\({}_{1}\) = 3.2 & R\({}_{1}\) = 3.2, R\({}_{2}\) = 6.4 \\ Rotation radius/mm & 6.67706433 & 6.67706433 \\ Radial clearance/um & 25 & 25 \\ Tooth height/mm & 40 & 40 \\ Inlet pressure/Pa & 101,325 & 101,325 \\ Inlet temperature/k & 300 & 300 \\ Outlet pressure/Pa & 101,325 \(\times\) 4 & 101,325 \(\times\) 4 \\ Speed/r-min\({}^{-1}\) & 3000 & 3000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of basic parameters of vortex compressor.  
The solution equations for the RNG k-\(\epsilon\) turbulence model are as follows:  
\[\frac{\partial}{\partial t}\int\limits_{\Omega(t)}\rho kd\Omega+\int\rho((\
u- \
u_{\sigma})\cdot\eta)kd\sigma=\int\limits_{\sigma}\left(\mu+\frac{\mu_{t}}{ \sigma_{k}}\right)(\
abla k\cdot n)d\sigma+\int\limits_{\Omega}\left(G_{t}-p \varepsilon\right)d\Omega \tag{11}\]  
\[\begin{array}{l}\frac{\partial}{\partial t}\int\limits_{\Omega(t)}\rho ed \Omega+\int\limits_{\sigma}\rho((\
u-\
u_{\sigma})\cdot n)\varepsilon d\sigma= \\ \int\limits_{\sigma}\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)(\
abla\epsilon \cdot n)d\sigma+\int\limits_{\Omega}\left(c_{1}G_{t}\frac{\epsilon}{k}-c_{2}( RNG)\cdot\rho\frac{\epsilon^{2}}{k}\right)d\Omega\\ \left\{\begin{array}{l}k=\frac{1}{2}(\
u^{\prime}\cdot\
u^{\prime}),\ \varepsilon= \frac{2\mu}{\rho}\big{(}S^{\prime}{}_{ij}\cdot S^{\prime}{}_{ij}\big{)},S^{ \prime}{}_{ij}=\frac{1}{2}\left(\frac{\partial\mu_{t}{}_{i}}{\partial x_{j}}+ \frac{\partial\
u^{\prime}{}_{i}}{\partial x_{i}}\right)\\ \mu_{t}=\rho C_{t}\frac{\kappa^{2}}{\varepsilon}\\ G_{t}=-\rho\mu^{\prime}{}_{i}^{\prime}{}_{i}^{\prime}\mu^{\prime}\frac{ \partial u_{i}}{\partial x_{j}}\\ c_{2}(RNG)=c_{2}+\frac{C_{2}\eta^{3}(1-\eta/\eta_{0})}{1+\beta^{3}},\eta=\frac {k}{\varepsilon}\sqrt{P}\end{array}\right.\end{array} \tag{12}\]  
where \(k\) is the turbulent kinetic energy, \(\sigma_{k}\) is the turbulent energy prandtl number, valued at 1, \(\varepsilon\) is the turbulence dissipation rate, \(\sigma_{\varepsilon}\) is turbulence dissipation rate prandtl number, valued at 1.3, \(\
u^{\prime}\) is turbulence fluctuation velocity, \(S^{\prime}{}_{ij}\) is the stress strain, \(\mu_{t}\) is the turbulent viscosity, \(G_{t}\) is the turbulent kinetic energy (k) production term, \(C_{2}(RNG)\) is the correction coefficient.