## 1 Introduction  
Low-speed axial cooling fans are frequently used to manage engine temperature by ensuring that adequate quantities of air pass through heat exchangers (radiators), even at low vehicle speeds or in the idle condition (Figure 1) [1, 2]. In some cases, especially with a plug-in hybrid electric vehicle, the cooling fan could be operational while a small high-output turbocharged internal combustion engine is not running [3]. An important constraint on fan usage is the generation of a flow-induced noise, called aerodynamic noise, which is emitted to the motors' cabin and causes much annoyance for passengers. Therefore, this problem poses a significant challenge for noise enhancement of axial fans. Compared to other sources, such as the engine and transmission, mechanical, and exhaust contributions, the cooling fan is a major contributor to overall noise. The reputation and customer satisfaction of a car brand can be severely affected if this quality issue is not properly counteracted prior to mass production [2, 4]. Therefore, reducing the axial fan noise in a cost-effective manner necessitatesa thorough investigation of the unsteady flow and aerodynamic noise characteristics of a baseline model and incorporation of the findings into the early design stage. The noise generated by fans is composed of structural noise and aerodynamic noise. The structural noise is caused by the duct emission and vibration of the fan motor, whereas the aerodynamic noise originates from the aerodynamics of the fan. According to the aerodynamic sound generation mechanism for Fans by Neise [5], aerodynamic noise sources can be classified into three types: monopole, dipole and quadrupole sources. For subsonic fans, the monopole source due to the blade thickness and the quadrupole source due to the unsteady momentum transport in turbulent flow can be disregarded [6; 7; 8], whereas the dipole source due to the unsteady rotating forces developed through the interaction of the fan blade with the inlet flow is considered as a dominant noise source. In the regular performance of an axial fan, the dipole source generates the harmonic characteristics of the fan noise at the blade passing frequency (BPF), called tonal noise, which is superimposed on the broadband noise, where energy is present over a wide range of frequencies. In other words, the tonal noise is characterized by the presence of higher intensity peaks at the BPF and its harmonics [9]. In 1952, [PERSON] [10; 11; 12] presented [PERSON]'s general theory for predicting jet noise by rearranging the Navier-Stokes (N-S) equations in the form of a linear wave equation for a medium not moving with a quadrupole source term that included density contribution, which [PERSON] [13] demonstrated to be more suitably described by a dipole source. In 1955, [PERSON] [14] extended [PERSON]'s general theory to incorporate the impact of solid boundaries in stationary motion, and in 1968, [PERSON] (FW-H) [15; 16; 17] further extended the [PERSON] theory of aerodynamic sound to include solid boundaries in arbitrary motion. The FW-H equation is essentially an inhomogeneous wave equation that can be induced by addressing the continuity equation and the N-S equations. To compute sound using the FW-H method, a well-resolved transient computational fluid dynamics (CFD) simulation is required to be performed only in the source region. With the rapid development of aerodynamics, aerodynamic noise, and computer technology, many investigations into aeroacoustic mechanisms and prediction have been conducted [2; 18; 19; 20; 21; 22; 23]. [PERSON] et al. [18; 19] proposed a new analytical treatment to estimate the sound pressure level (SPL) of turbulent noise emitted from low pressure axial flow fans. [PERSON] et al.