## 3 Performance-Based Analysis of the Target Axial-Flow Fan Unit
### Numerical Analysis of the Target Axial-Flow Fan System
#### 3.2.2 Validation of Numerical Methods  
In this study, a virtual fan performance tester is conceptualized as the computational region, aiming to enhance the precision of the numerical simulations. The geometry of this computational region mimics that of a rectangular duct, proportionally scaled according to the diameter of the fan, as depicted in Figure 3. The dimensions of the computational domain are judiciously chosen to ensure ample space. The boundary conditions are strategically defined to emulate the actual experimental setup of the fan performance test. At the inlet and outlets of the domain, pressure boundary conditions are imposed, replicating the conditions encountered during experiments. This involves manipulating the pressure levels to establish a pressure differential between the inlet and outlet across the entire domain. Additionally, the remaining surfaces within the field are designed as non-slip walls to encompass a comprehensive simulation environment. Prior to the validation of the numerical methods used in the current study, a grid refinement study is carried out at the operational pressure condition of the target fan system, specifically at a rotational speed of 800 RPM. The outcomes of this grid refinement analysis are presented in Figure 4, showing the variation in the relative error of the predicted volume flow rate with respect to the actual measured value, versus the total count of grids employed. It is observed that as the grid number is augmented, the relative error progressively diminishes and ultimately converges, particularly in cases characterized by more than 16 million grids. Guided by these results, the subsequent numerical simulations are executed by employing approximately 16 million grids. This selection of the grid count is motivated by the intention to strike a balance between simulation accuracy and computational expenditure, thus circumventing the need for excessively high computational costs. To validate the accuracy of the numerical techniques and the appropriateness of the computational setup, a comparative analysis is conducted against the experimental outcomes. The comparison of the predicted and experimental P-Q curves, illustrated in Figure 4a, demonstrates a notable concordance between the numerical predictions and experimental observations. This alignment is substantiated by relative errors spanning from 0.6% to 6.6%, implying a robust agreement between the two sets of results. Figure 3: Dimensions of the virtual fan tester with details of the fan blades and shroud used as the integral impermeable surface for the FW–H equations. The prediction of the aeroacoustics noise emitted from the target fan system is conducted in this study using the FW-H integral equation, and the results are presented in the form of a time history of the acoustic pressure. The predicted acoustic pressure is then transformed into the frequency domain using the Fast Fourier Transform (FFT). The time step size is determined in accordance with a sampling frequency of 4800 Hz and a frequency resolution of 3.3 Hz. After the data set has sufficiently converged, a total of four full revolution of data are processed, with a frequency resolution of 3.3 Hz. Figure 4b displays the predicted spectral density spectrum of the sound pressure level alongside the experimental results, with a relatively low target frequency range due to the Blade Passage Frequency (BPF) of 41.67 Hz. However, for the noise experiment, the outer casing  
Figure 4: Comparison to confirm the validity of the numerical methods: (**a**) comparison of the predicted volume flow rate (blue line) under two given pressure conditions with the measured P–Q curve (dotted-black line) and (**b**) comparison of the predicted sound pressure spectral density (blue line) with the measured sound pressure spectral density (black line) at a location 1 m away from the rotating center of the fan.