**1. Introduction**

Centrifugal pumps are of vital importance for a variety of commercial and industrial applications [1], which could handle a great quantity of liquids at relatively high pressures [2]. For those applications that require extremely high pressures, multistage centrifugal pumps, which usually have more than two or even up to dozens of stages, are used. Owing to the high reliability requirements for a long shaft, the hydraulic design of the multistage centrifugal pumps is more complex than that of the single-stage pumps [3]. Recently, many researchers have conducted both numerical simulations and experiments on the efficiency of multistage centrifugal pumps to enhance their hydraulic design [4–6].

The operation process of multi-stage centrifugal pumps is often accompanied by vibration and noise. When the pump produces static pressure, it also produces dynamic pressure, which is also named as pressure fluctuation along with the periodic changing features like AC signal. Pressure fluctuations are inevitable during the operation of a pump, having strong unsteady characteristics that may lead to strong vibration and noise under some special working conditions [7].

In previous studies, a series of numerical investigations of pressure fluctuations within a complete single-stage centrifugal pump were undertaken. Khalifa et al. [8] investigated the pressure fluctuations and vibration in a single-stage centrifugal pump experimentally under different flow conditions. They found that under different flow rates, the vibrations have a close relationship with the pressure fluctuations, and the vibration could be controlled by optimizing the gap between the volute tongues and the impeller. Guelich et al. [9] studied the physical mechanisms behind the pressure fluctuations under the design flow conditions. They found that the pressure fluctuations mainly depend on the geometry of the diffuser and the impeller by performing 36 tests with different pump designs. Spence et al. [10] used the multi-block, structured mesh CFD (Computational Fluid Dynamics) code TASC flow to study the pressure fluctuations in a centrifugal pump for three different flow rates. They found that the volute gap and the arrangement of vanes have the greatest effect on the different monitored positions. In a previous work of ours [11], we simulated the transient flow field of a centrifugal pump by employing ANSYS-CFX software based on the frozen rotor technique and utilizing the Shear Stress Transport (SST) *k*-ω turbulence model. The results shown that the pressure fluctuation is mainly affected by the number of impeller blades, but the influence decreased gradually when the water flowed into the diffuser. Pei et al. [12] focused on the numerical simulation of unsteady flow under part-load conditions, the results shown that the intensity of pressure fluctuation is larger than that of design flow condition because of leakage flow. Jiang et al. [13] studied the clocking effect and unsteady radial force in a centrifugal pump by numerical and experimental methods, they found that the unsteady flow and pump performance mainly effected by the relative position between the diffuser the volute. Wang et al. [14] carried out an experimental measurement to study the influence of rotating speed and flow rate on the pressure fluctuation in a double-suction centrifugal pump. They found that the pressure fluctuation under zero flow rate is two times that of the pressure fluctuation under the design flow rate. Posa and Lippolis [15] used Large Eddy Simulations (LES) to investigate the influence of diffuser setting angle and operating conditions on the pressure fluctuation and unsteady flow in a centrifugal pump. They found that the larger radial gaps between diffuser and impeller could lead to lower pressure fluctuations, but the lower flow rate could lead to higher pressure fluctuations.

While numerous studies have investigated the pressure fluctuation and unsteady characteristics in different kinds of pumps, which includes the effects of working conditions [14,16], inlet guide vanes [17], tip clearance [18], and volute curvature [19–21], but the influence of the number of diffuser vanes on the pressure fluctuation in the centrifugal pump is unclear and further investigation is needed. So, in this paper a numerical approach was adopted in a centrifugal pump with a different number of diffuser vanes, and the pressure fluctuations are analyzed and compared in detail. The results could supply a basis for further design improvement of the pump body structure and working reliability.
