## 1 Introduction  
Nowadays, turbomachinery plays a pivotal role in many fields of national economy [1; 2; 3]. As parts of turbomachinery components, centrifugal pumps have been widely applied for water supplying system, cooling system, agriculture irrigation system, and so on. However, the hydraulic performance and human health are affected by flow-induced noise during operation. Thus, the noise generation, distribution characteristics and noise reduction technology have attracted wide attention. In the past, experiments have been conducted for the centrifugal pump noise analysis, and valuable conclusions were obtained [4; 5; 6; 7]. However, it was difficult for the experimental method to identify the acoustic source accurately and analyze the coupling between flow and acoustic fields, by comparison, the numerical simulation approach was an ideal method to solve these issues. With the improvement of the Lighthill equation, a hybrid numerical method combining Lighthill acoustic analogy theory with Computational Fluid Dynamics (CFD) was widely used for flow-induced noise calculation [8; 9; 10; 11]. According to the hybrid method, when the flow field simulation was completed, the wall pressure fluctuation in time domain was extracted and transformed into dipole acoustic source, then the noise signal in frequency domain was obtained. Based on the hybrid method, scholars explored the noise changing trend under different operation conditions, noise distribution characteristics, and the effect of different pump structures on noise [12; 13; 14; 15]. [PERSON] et al. [16] also compared the influence of different dipole sources on acoustic calculation results. [PERSON] et al. [17], [PERSON] et al. [18] revealed the noise radiation directivity characteristics with different dipole sources, respectively. However, the Lighthill acoustic analogy theory assumed that the acoustic source generated by the flow was known beforehand, the flow and acoustic fields were separated, so the theory could not explain the sound wave generation and propagation mechanism, etc. [19]. With the development of research, [PERSON], [PERSON] [20; 21], proposed the vortex sound theory, which made up for this deficiency, and it was applied to acoustic research gradually [22; 23]. By using Green function, the analytical solution of the equation could be derived to solve the noise signal in time domain at a certain monitoring point. [PERSON] et al. [24], [PERSON] et al. [25] and [PERSON] et al. [26] applied the analytical solution to aerodynamic noise computation of fans and noise directivity characteristics analysis of submerged jet, respectively. By summarizing the current research results, it is seen that the main concern is the effect of operation condition and pump structure on the noise distribution characteristics, while the acoustic field propagation mechanism in centrifugal pumps is rarely involved. According to synergy principle of flow and acoustic fields [27], the flow and acoustic fields synergy could affect the acoustic field propagation characteristics. Analyzing the propagation characteristics of acoustic field of centrifugal pumps is beneficial for the research about the directional noise reduction. To analyze the propagation characteristics of sound wave in centrifugal pumps based on synergy principle by using CFD, the basic and crucial step is to solve the parameters of flow and acoustic fields simultaneously, and figure out the temporal distribution and evolution characteristics of acoustic field. However, the analytical solution of vortex sound equation based on Green function is used for noise calculation at a certain monitoring point, which is not suitable for noise calculation in the whole field. Obviously, the numerical approach based on discretizing and solving the vortex sound equation in the whole field [28] is the main method to figure out this issue. In current work, the effectiveness of the simulation approach is firstly verified by corresponding experiment.