## 1 Introduction  
Separated flow is a complex phenomenon in fluid mechanics. It is widely encountered in flow around various geometries such as bluff cylinders and hydrofoils and in turbo-machineries [1]. As the impeller runs under off-design conditions, due to the nonzero angle of attack of flow with respect to the blades, separated vortices may be generated in the impeller. The continuous generation and shedding of vortices often induce low-frequency pressure pulsation, which increases noise and intensifies vibrations, resulting in deteriorated performance or even accidents [2]. The vortices in the impeller are normally unstable because of the shear stress which finally viscously diffuses the vortices, and new vortices still form. From an energy point of view, these downstream moving vortices take away a majority part of energy, which is equivalent to an additional drag acting on the surrounding flow; this will lower the dynamic efficiency of the turbo-machineres. In order to understand the complex separation mechanisms for flow in a centrifugal impeller, a number of works have been conducted either by numerical simulations or experiments. [PERSON] and [PERSON] [3] performed a large-eddy simulation based on the second-order cumulative rate method combined with the eddy current viscosity sub-grid scalemodel and pointed out that flow instability such as a rotating stall can lead to severe vibrations in turbo-machines if the eigenfrequency of the impeller equals the stall frequency. [PERSON] et al. [4] conducted a detailed experimental measurement on flow in a shrouded centrifugal impeller and reported a two-passage phenomenon (stalled and unstalled passages) for flow rate \(Q=0.25Q_{d}\), which was not previously reported. [PERSON] et al. [5] studied the unsteady cavitation flow of centrifugal pumps operating under off-design conditions by using the improved cavitation mode. The authors found that, under partial discharge, the pressure fluctuation of the cavitation flow is higher than that of the non-cavitation flow. [PERSON] and [PERSON] [6] used OpenFOAM to carry out steady and unsteady numerical simulations of a centrifugal pump with vane diffusers to study the rotor-stator interaction. They found that the steady simulation produces results which are in reasonable agreement with the experimental data. [PERSON] et al. [7] used a third-order SGS model to simulate the stall phenomenon in the impeller of a centrifugal pump under quarter-load condition. Further study of the two components of the production and the rotation shows that they jointly transport certain energy from the Reynolds component \(R_{vv}\) to \(R_{uu}\). [PERSON] et al. [8] used the SST \(k\)-\(\omega\) turbulence model to study flow in a centrifugal pump under four different flow rates (\(1.0Q_{d}\), \(0.7Q_{d}\), \(0.5Q_{d}\), and \(0.3Q_{d}\), where \(Q_{d}\) is the designed flow rate). The results show that, as the flow rate decreases, the stall can be categorized as preliminary stall and stationary stall. As the flow rate decreases from \(1.0Q_{d}\) to \(0.3Q_{d}\), the inlet angle of the impeller increases from 35 degrees to 70 degrees, which is the direct cause of the stationary stall. [PERSON] et al. [9] analyzed the performance of the first stage of a multi-stage pump through experimental and numerical data and concluded that the reason for the positive slope characteristic in the low flow rate region is the increased loss in the diffuser. In addition, the vortex on the trailing edge of the diffuser blades will block airflow and cause a separate airflow on the leading edge, resulting in a rotation stall. [PERSON] et al. [10] used particle image velocimetry (PIV) and pressure fluctuation measurement to study the generation and development of stall in a centrifugal pump with a vaned diffuser. It is found that, as the flow velocity decreases, the size of the leakage and reversed flow in the stalled diffuser passage increases and the size of the stall unit diffuses from one diffuser passage to two diffuser passages.