# The Characteristics and Mechanisms of High-Intensity Sound in a High-Speed Multistage Compressor
## 4 Numerical Method Validation
### Calculation Parameter Settings  
In the calculation process of flow-induced sound in the rectangular cavity pipeline, the characteristics of the unsteady flow field are first calculated and analyzed in the present investigation, and then the sound source information is derived using the acoustic analogy method. The results, including acoustic modal frequency and distribution characteristics, are obtained. Firstly, the steady flow field calculation is carried out, and then the results of it are taken for the initialization of the transient flow. The RNG \(k-\varepsilon\) turbulence model is used for the steady flow and a standard wall is selected to process the flow near the wall. Transient flow uses the LES turbulence model. There are two steps of correction in the process of applying the Pressure-Implicit with Splitting of Operators (PISO) algorithm to solve the pressure equation. Firstly, the first velocity correction value is obtained by \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Shape** & **W/H** & **H/mm** & **W/mm** \\ \hline Square cavity & 1.0 & 254 & 254 \\ Rectangular cavity 1 & 0.95 & 254 & 241.3 \\ Rectangular cavity 2 & 0.9 & 254 & 228.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric dimensions of rectangular cavity. Figure 10: Schematic diagram of experimental device of rectangular cavity pipe model and sound pressure measurement positions. (**a**) is a schematic diagram of the cross-section of the device. (**b**) is a schematic diagram of the position of the microphone measuring point and the direction of the flow velocity (1,2,3 and 4 represent the position of the microphone). solving the momentum equation on the premise of a given assumed pressure field. Then, the continuous equation is solved to derive the pressure correction value which is used to correct the pressure equation. Secondly, the momentum equation is solved again with the corrected pressure equation to obtain the second velocity correction value. Consequently, the iterative equation satisfies both mass conservation and the implicit momentum conservation equation. The iterative calculation does not stop until the velocity field satisfies the continuity equation. The amount of computation of each iteration increases due to the repeated solving of the momentum equation. The PISO algorithm shows obvious advantages in dealing with unsteady flow problems or high-distortion flow field grid problems due to its effective and efficient convergence. Therefore, in the calculation of the flow field characteristics of the rectangular cavity pipeline, the Semi-Implicit-Method for Pressure Linked Equations Consistent (SIMPLEC) algorithm is used for the steady flow field calculation, the second-order upwind style is used for field variable interpolation, the Green-Gauss node-based calculation method is used for the gradient calculation, and the second-order interpolation method is used for surface pressure calculation. The PISO algorithm is taken for transient flow field calculations, the second-order upwind style is taken for field variable interpolation, the gradient and surface pressure calculations are also the same as the steady state calculations, and the transient equations are second-order implicit. The initialization of the flow field is realized with 200 iterations in the steady state calculation process. In the transient calculation, the transient calculation time step is set to 0.000167 s to capture the sound source information for 0-3000 Hz. The number of calculation time steps is set to 600 steps, and the iteration number of each time step is set to 20 steps. Total number of time steps is 600, with 20 iterations over one time step. The computational model is constructed according to the geometry of rectangular cavity 2 in Table 1. The middle rectangular cavity part of the model is the sound source extraction region, and the two ends are the inlet and outlet flow development sections of the cavity flow field, respectively.