## 1 Introduction  
The performance of an axial-flow fan can be evaluated using the flow rate, efficiency, and noise. Axial-flow fans are widely used in various industries, such as ventilation, air conditioning, and cooling systems, due to their high flow rate, high efficiency, and low noise level. However, the axial-flow fan blades experience a more complex flow feature than the other types of fans, so designing an axial-flow fan is challenging because it involves many design parameters that can significantly affect its performance. Therefore, a thorough understanding of the basic design principles of axial-flow fans is necessary for developing efficient and optimized fans. One of the most common design methods is based on the single airfoil theory, which originates from the wing theory of aircraft [1]. The basic design procedure based on the airfoil theory can be summarized as follows. First, the overall dimensions of the fan, such as the diameter of the fan and hub, are determined. The number of blades and solidity are subsequently determined. Then, the sectional airfoil shapes are designed using design parameters, such as the pitch angle, camber angles, thickness, etc., in two dimensions. Next, the designed airfoils are stacked along a spanwise baseline called a stacking line. In this step, three-dimensional design parameters are considered, like the sweep and lean angles [2]. At last, a fully three-dimensional design is adopted for a particular intention, such as a winglet shape [3; 4] at the tip or bionic curvature on the blade surface [5]. However, the complex flow features of axial-flow fans make it challenging to apply the single airfoil theory as a reliable design tool. An alternative approach is based on the cascade of airfoils [6; 7; 8; 9; 10], which is based on the strip theory. The blade strip theory is a theoretical model used to analyze the aerodynamic behavior of an axial-flow fan blade by dividing it into a series of blade strips along the spanwise direction. Each blade strip is assumed to be independent of the others, and the flow through each strip is treated as two dimensional. [PERSON] et al. [6] studied the impact of airfoil probes on a compressor cascade's aerodynamics using RANS and DDES simulations. The probes influence the load capacity and shock waves, reducing the separation bubbles but also causing extra flow losses. The DDES revealed detailed vortex structures, including shedding streamwise vortices, impacting wake behavior. [PERSON] et al. [7] introduced a deep reinforcement learning (DRL) framework for rotor airfoil optimization to address issues with conventional methods lacking interpretability and generalizability. Using neural networks and reinforcement learning, it enhances the rotor airfoil dynamic stall characteristics. The results of optimizing the OA212 rotor and 20 additional airfoils demonstrate significant improvements in the lift coefficients, up to 46.3% drag reduction. Numerical simulations confirm the superior aerodynamic performance of the optimized rotor compared to the baseline, showcasing the effectiveness of this DRL-based optimization approach. [PERSON] et al. [8] investigated a proper orthogonal decomposition (POD)-assisted inverse design optimization method for cascade airfoils, addressing a key limitation of existing inverse design techniques. By leveraging POD, this method automatically obtains an optimal pressure distribution via direct design optimization, showcasing its effectiveness in enhancing the aerodynamic performance of compressor cascade airfoils. [PERSON] et al. [9] studied the impact of periodic turbulence and pressure fluctuations on the heat transfer over the suction surfaces of a specific row of rotor blades. The study compared numerical results with experimental data, revealing significant impacts on the heat transfer performance, particularly near shock waves on the fore suction surface and due to turbulence on the rear suction surface.