## 1 Introductory  
To address the dual challenges of energy consumption and environmental pollution, improving the efficiency of energy production and utilization has become increasingly urgent on a global scale [1, 2]. Fans have widespread applications in various fields such as energy, construction, machinery, chemical engineering, and aerospace [3, 4, 5, 6, 7], with the impeller and blade components being the most critical parts of the fan, playing a primary role in energy transmission and significantly impacting fan efficiency [8, 9, 10]. In the design of centrifugal impellers, the assumption is often made that the impeller inlet has uniform inflow. However, in practical applications, factors such as limited installation space, structural design requirements, and the layout of inlet piping often lead to non-uniform inflow in centrifugal impellers [11; 12; 13; 14], as illustrated by the occurrence of the pre-rotation phenomenon in engineering applications shown in Figure 1. In actual operation, the non-uniform inflow caused by the inlet piping contradicts the assumption of uniform inflow made during design, which can negatively impact the performance and stable operation of the equipment [15]. Curved pipe inflow is one of the most common causes of non-uniform inflow in centrifugal impellers [16]. Some researchers [17; 18] have conducted experiments on the flow field in curved pipes and found that the axial velocity of the main flow begins to distort at the entrance of the bend, forming a recirculation zone, or vortex, as the flow develops through the curved pipe. [PERSON] [19] discovered that the additional losses and secondary flows caused by the curved inlet pipe lead to non-uniform velocity and pressure distributions at the centrifugal pump inlet. This non-uniform distribution can alter the inlet angle of attack of the impeller, thereby affecting the transient flow within the impeller passage. Many scholars have conducted extensive research on the performance of impellers under non-uniform inlet flow conditions. [PERSON] [20] found that non-uniform inflow reduces the efficiency of jet pumps and increases axial force fluctuations on the impeller, leading to performance instability. [PERSON] [21] indicated that the impact of non-uniform inflow on pump performance might stem from the non-uniform pressure field, which can generate low-pressure regions on the impeller, causing cavitation. [PERSON] [22] showed that reverse-rotating distorted vortices lead to greater efficiency losses than co-rotating ones. [PERSON] [23] revealed that inlet distortion causes blockage in the rotor tip passages, and the greater the distortion, the more significant the inlet blockage, resulting in greater flow losses. [PERSON] [24] also pointed out that flow separation due to asymmetric chamber structures and abrupt contraction sections are the main causes of non-uniform flow, which leads to a threefold increase in flow fluctuation in one impeller passage and a twofold increase in the other, ultimately reducing the stability of the pump's operation. High-pressure gas quenching heat treatment offers many advantages, including oxidation-free, decarbonization-free, pollution-free, and high surface finish [25; 26; 27]. In the gas quenching process, the rapid flow of gas is primarily relied upon to carry away  
Figure 1: Scenario of pre-rotation phenomenon occurrence. heat, achieving rapid cooling. Improving the flow conditions of high-pressure gas within the gas quenching furnace is crucial to enhancing its performance [28; 29]. Among the key components, the centrifugal fan impeller serves as the power source for air circulation in the high-pressure gas quenching cooling system [30]. It must overcome the system's resistance to deliver gas into the heat treatment furnace, and its performance directly affects the entire air-cooling system, with efficiency being a critical factor impacting overall system energy consumption.