## 3 Characteristics of the Aerodynamic Noise
### CFD Method for Exploring the Source of Aerodynamic Noise  
Here, CFD modeling is carried out for the two-stage centrifugal air compressor. The impeller speed and boundary conditions in the model are set according to the operating parameters under rated conditions in Table 2. The inlet boundary is the pressure inlet (101.3 kPa) and the outlet boundary is the pressure outlet (257.0 kPa). The wall surfaces are all non-slip walls. The y+ of the mesh is set to 100, and the wall distance is 3.2 \(\times\) 10\({}^{-4}\) m. The mesh in the impeller domain and diffuser domain is the structured hexahedron mesh. The mesh in the volute domain is the unstructured tetrahedral mesh. The minimum elements' quality is above 0.2. The interface models between the moving and stationary parts are the frozen rotor for steady-state simulation and the transient rotor-stator for transient simulation. The boundary conditions for turbulence fractional intensity are 0.05 at both the inlet and the outlet. The Eddy length scales are 0.0386 m and 0.0394 m at the inlet and the outlet, respectively. The CFD calculation adopts ANSYS CFX for the transient solution. The time step is 1.852 \(\times\) 10\({}^{-6}\) s, and the corresponding impeller rotates 1\({}^{\circ}\) at each time step under the rated speed [9]. According to the actual structure of the air compressor, the model is established, as shown in Figure 7.  
As the three operating conditions shown in Table 3 are all near the design condition, the Reynold-averaged Navier-Stokes (RANS) equation is used as the governing equation of the model [17], and the shear stress transport (SST) \(k\)-\(\omega\) model is used as the turbulence model. Six sets of grids with different element numbers are set for the two-stage centrifugal air compressor, and the mass flow of the air compressor is selected as the basis for judging the grid independence. The curve is shown in Figure 8. It can be seen from the figure that the mass flow calculation difference is less than 0.5% after increasing the number of elements in Case 3, and the number of elements in Case 3 meets the calculation requirements. Figure 7: CFD model of the compressor. The results of CFD calculation are verified by the thermal performance obtained from the test. The pressure ratio (\(PR\)) of the compressor is for two stages and calculated as follows:  
\[PR=P_{\text{out}}/P_{\text{in}} \tag{1}\]  
where \(P_{\text{out}}\) is the total pressure of the outlet of the second solute and \(P_{\text{in}}\) is the total pressure of the inlet of the first impeller. As shown in Table 5, the error between the mass flow rate and pressure ratio of the air compressor calculated by CFD and the test results is within 0.5%. As a result, the model calculation is accurate and reliable. The flow separation will not only bring power loss and efficiency reduction but also induce aerodynamic noise. The change in static entropy can provide a reference for the study of flow separation. Figures 9 and 10 are the B-B static entropy contours of the first and second stage impellers at 90% and 50% blade heights, respectively. There is flow separation caused by the tip clearance flow in both the first and second stage impellers. There is obvious leading-edge flow separation in the second stage impeller. There is the trailing edge flow separation phenomenon in first and second stage impellers, of which the first stage impeller is more violent. Figure 8: Grid independence verification of the CFD model. Figure 9: Static entropy contours of 90% heights of impellers. (**a**) First stage. (**b**) Second stage. \begin{table}
\begin{tabular}{c c c c} \hline  & **Test Results** & **Calculation Results** & **Error** \\ \hline Pressure ratio & 2.51 & 2.512 & 0.080\% \\ \hline Mass flow rate [g\(\cdot\)s\({}^{-1}\)] & 125.36 & 125.687 & 0.26\% \\ \hline \end{tabular}
\end{table} 
Table 5: Error of the CFD results. Figure 11 shows the static pressure contours of the first and second stage diffusers and solutes.