## 1 Introduction  
When designing a pump impeller, two main targets need special considerations. Firstly, operation efficiency at design load is important. Secondly, reliability at off-design load is also crucial, because the pump usually operates off from the design condition [1]. The puzzle of how to improve the pump efficiency at design load has been solved with many typical methods in the past decades [2]. The current works are applying these methods for specific cases [3; 4; 5]. On the contrary, flow inside pump impeller under off-design conditions are treated as uncontrollable. The internal flow becomes complex at off-design load, especially at partial-load, with noise, vibration, and other undesirable phenomena [6; 7]. These phenomena would affect the stability and security of the pump unit, and even cause faults. Under the design flow rate condition \(Q=Q_{d}\), flow is along the impeller blade geometry with high efficiency. From design load to partial load, the incidence angle becomes higher and higher and causes undesirable flow on the blade suction side [8]. Typically, the undesirable flow develops in three steps. Firstly, small-scale flow separation occurs on the blade suction side [9]. Secondly, backflow occurs at both the inlet and outlet of the impeller. Passage blockage happens and brings vibration and efficiency drop [10]. Thirdly, the inlet and outlet backflow structures expand and block the entire passage [11]. Among them, the second stage, which may cover the flow rate from 0.3 to 0.8 \(Q_{d}\), is complex and important, because pumps usually operate in this flow rate range with partial passage blockage. [PERSON] [12] used \"rotating stall\" to describe this complex changing blockage and distinguish it against \"surge\". In actual pumps and compressors, impeller eccentricity or asymmetricity may exist due to installation, manufacturing, and instantaneous flow destabilization [13]. Affected by the eccentricity or asymmetricity, the blocked flow turns into neighboring passages, as shown in Figure 1a. In the neighboring passage rotational, turned flow strikes the suction surface and diminishes part of the stall cell. In the neighboring passage, counter-rotational, turned flow strikes the pressure surface and increases the scale of the stall cell. The asymmetric distribution of stall cell moves counter-rotationally during impeller rotation. It makes the flow regime more complicated. Many researchers studied the rotation stall mechanism and influences [14; 15; 16; 17]. Because of the compressibility of the fluid medium, people mainly focused on the rotating stall in aerodynamic turbomachines like compressors. [PERSON] [18] studied the mechanism of rotating stall and explained the partial passage blockage as an adaptability of small flow rate. [PERSON] et al. [19] experimentally tested the rotating stall in axial-flow compressors. The stall cell was found starting from the trailing-edge and developing to the leading-edge. Experiences show that the rotating stall is relative to the incidence angle at the leading-edge and the flow distortion at inlet [20; 21]. The counter-rotational movement of the stall cell [12] is found relative to the impeller rotation. The stall frequency will be 1/5-1/2 impeller rotating frequency in different cases [19]. Based on the mechanism studies, [PERSON] [22], [PERSON] [23], [PERSON] et al. [24], and [PERSON] et al. [25] investigated the controlling method of rotating stall, using different methods in the past. The sound generated by the rotating stall was also studied by [PERSON] et al. [26]. Low-frequency noise from the impeller was found to be relative to the rotating stall. Currently, researchers still focus on the rotating stall phenomenon in turbomachinery, which has complicated flow characteristics [27; 28; 29]. In pumps which usually transport incompressible fluids, researchers have also studied the rotating stall characteristics. [PERSON] et al.