## 2 Research Object and Methods
### Numerical Setups
#### 2.4.3 Computation setup of the acoustic field  
There are three main computational domains for acoustic calculations (as shown in Figure 6) named water flow domain, pump casing structure, and air radiation sound fields.  
The water flow domain and pump casing structure domain are coupled to process fluid-borne noise calculation. The inlet tube and rotating impeller are not included here, where sound source information calculated by CFD would be transferred to the volute import as a surface sound source. The volute water area is used as both acoustic generation area and the sound propagation area, and its exit is set to the outlet. The calculation of radiated noise has added package air domain and infinite element radiation surface. The pump casing structure domain is also used for modal calculations.  
The acoustic source information obtained from CFD is transformed into computational acoustics through meshes. Source information of inlet pipe and rotating impeller are loaded into the interface between impeller and volute as a surface acoustic source. Additionally, the sound source information in the volute area is loaded into the volute domain itself as a volume acoustic source. In general, the acoustic grid is coarser than the CFD grid. This paper takes the integration solution of the CFD results on the acoustic mesh to ensure all information is conserved in the information transfer process,as shown in Figure 7. This work is done in the software Actran's preprocessor iCFD. It provides the mean flow interpolation method when CFD nodes are localized in the acoustic mesh structure. Moreover, smoothing operations are available when the CFD mesh is coarser than the acoustic mesh at the boundary. In order to ensure the convergence of acoustic equations, it also regularizes the flow field and automatically removes the boundary layer computed in RANS simulation [28].  
The minimum wavelength of the acoustic finite element calculation should include at least 6 mesh cells, as follows:  
\[L_{\text{max}}<\frac{c}{6f_{\text{max}}} \tag{12}\]  
Figure 6: The sound field calculation: (**a**) water flow domain; (**b**) pump casing structure; (**c**) air radiation sound fields.  
Figure 7: Grid interpolation method: (**a**) integration solution; (**b**) solution at boundary.  
where \(L_{\text{max}}\)--maximum mesh size for acoustic calculation, \(c\)--speed of sound, \(f_{\text{max}}\)--maximum frequency identified by acoustic calculation. The maximum frequency covered by the unsteady flow field calculation is 6750 Hz, which identifies the maximum frequency of acoustic calculation to 6750 Hz. The speed of sound in the air is 346 m/s when the temperature reaches 25 \({}^{\circ}\)C, and in the water, it is 1497 m/s. Then, the grid size of the acoustic propagation region should be less than 8.54 mm in the air domain and 36.96 mm in the water domain. The maximum grid size of the computational domain used in this paper is 0.0014 m, which can meet the requirements of grid size in the acoustic finite element calculation.  
The calculation process of radiation noise is similar to that of the fluid-borne noise, but the interface between structure domain and air domain needs to be defined. The acoustic finite element calculation requires the setting of the boundary conditions. Modes of pipelines are added to the outlet pipe, and infinite space elements are added to model the infinite air surface during acoustic preprocessing settings so that the sound propagates from the inside to the outside without reflection. The pump casing structure represents an aluminum shell (PPS) of 3 mm thickness, and the material of the pump base plant is ADC. The properties of both mentioned materials are shown in Table 1. The calculated acoustic frequency range is 0-6757 Hz, and the frequency resolution is set to 11.3 Hz.