## 1 Introduction  
As the most widely applied type of pump equipment, centrifugal pumps perform work on liquid media by relying on the high-speed rotation of the impeller, thus achieving the transportation and pressurization of the liquid. Operational economy and stability are the two most crucial considerations in the design and manufacturing process of centrifugal pumps [1, 2]. With the development of modern industry, the requirements for pump performance in various fields are constantly increasing. The design of modern centrifugal pumps is continuously evolving toward higher rotational speeds, larger flow rates, and higher heads [3, 4, 5]. Curved blades, accompanied by the high-speed rotation of the main shaft, generate more significant multi-wall shear rotational turbulence. This intensifies the rotational, transient, and nonlinear characteristics of the flow within the pump, leading to increasingly prominent complex unsteady flow phenomena such as the interaction between the rotating and stationary parts, flow separation, wake-jet flow, and secondary flow inside the centrifugal pump [6; 7; 8; 9; 10]. The unsteady flow within the pump under high energy density conditions generates significant radial forces which primarily result from circumferential pressure asymmetry induced by impeller-volute interactions. These forces drive shaft deflection, accelerate bearing wear, and elevate vibration and noise levels through fluid-structure interactions. This leads to an increase in the deflection of the main shaft, an acceleration of bearing wear, and an elevation in the levels of vibration and noise. In severe cases, it may even cause the blades to break, thus having a highly detrimental impact on the operational stability of the pump unit [11; 12; 13; 14]. Additionally, the periodic hydraulic impact of unsteady flow on dynamic and static sealing components such as the sealing rings may disrupt the original installation and fitting relationships, increase mechanical friction and sealing clearances, and exacerbate the internal leakage of the fluid medium. Ultimately, this results in a decrease in volumetric efficiency and mechanical efficiency and an increase in the energy consumption level during the operation of the pump [15; 16; 17; 18]. Therefore, a large number of researchers have initiated studies on unsteady flow phenomena inside centrifugal pumps, analyzing the formation mechanism and dynamic characteristics of the hydraulic induction effect on the impeller, so as to provide effective guidance for the weakening and suppression of the hydraulic induction effect. The hydraulic load acting on the impeller inside a centrifugal pump consists of two parts: the axial force and the radial force. The axial force can be transmitted via thrust bearings and actively balanced through methods such as balance discs and balance holes [19; 20; 21]. However, the radial force cannot be simply balanced, but rather weakened and suppressed through the optimized design of the stator geometry and other key components in the hydraulic model [22]. A large number of scholars have conducted research on the variation laws of the hydraulic radial force on the impeller and its influencing factors, thereby providing theoretical guidance for the design and operation of centrifugal pumps. [PERSON] et al. [23; 24] showed, through experimental measurements and numerical simulations, that the hydraulic radial force acting on the impeller mainly originates from the pressure pulsation and uneven pressure factors within its internal flow field. It has the characteristic of periodic variation with constantly changing magnitude and direction and is closely related to the operating conditions, such as the flow rate. In recent years, a large amount of research has focused on optimizing the geometric structure of the hydraulic model to effectively suppress the hydraulic radial force. The fluid domains inside a centrifugal pump can generally be divided into three parts: the suction part, the volute, and the impeller.