# Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils
## 3 Flow Configuration and Comparison of 2D vs. 3D Numerical Approaches  
### Test Cases for NACA-0012 Airfoil  
Following the experimental conditions of [PERSON] et al. [8] and [PERSON] et al. [10], the selected NACA-0012 was employed in a set of numerical simulations with a uniform upstream flow velocity of 16 m/s (\(M_{\infty}\) = 0.0465) and \(Re_{c}\) = 140,000 to closely mimic the experimental setup. The corresponding time-averaged U-velocity contours comparing the 2D and 3D ILES results are shown in Figure 2. Overall, the time-averaged U-velocity contours shown in Figure 2a,b are identical from the leading-edge (LE) to mid-chord due to the inherently 2D, and thus laminar, flow, as well as the absence of separation. Beyond the mid-chord, the flow similarities begin to diverge and the differences between the 2D and 3D ILES flow-fields become more apparent. Upon closer inspection, the 3D simulations reveal a noticeably smaller LSB and wake structures, which is a result of vortex breakdown captured in 3D ILES. To supplement the time-averaged U-velocity contours and characterize the thinner wake profile of the 3D simulations, near-wake U-velocity profiles are shown in Figure 3a. The results, which are measured 1% chord away from TE, confirms the size of the velocity deficit to be smaller in 3D simulations due to the energy redistribution along the span-wise direction. \begin{table}
\begin{tabular}{c c c c c} \hline
**Cases** & **Dimension** & \(\Delta\)**y/c** & \(\Delta\)**x/c** & \(\Delta\)**z/c** \\ \hline
2D FINE & 1281 \(\times\) 789 \(\times\) 3 & 2.5 \(\times\) 10\({}^{-5}\) & 0.5 \(\times\) 10\({}^{-3}\) & - \\
3D FINE & 1281 \(\times\) 789 \(\times\) 101 & 2.5 \(\times\) 10\({}^{-5}\) & 0.5 \(\times\) 10\({}^{-3}\) & 9.9 \(\times\) 10\({}^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 1: NACA-0012 grids employed in 2D and 3D studies. These findings are further validated in Figure 3b, which shows the surface pressure coefficient comparison between the numerical predictions and experimental measurements. The surface pressure coefficient correlates well with the time-averaged U-velocity (Figure 2) by revealing 2D and 3D results to be identical up to 70% chord. Beyond 70% chord, the results diverge due to the larger fluctuations inherent in the 2D results. The comparison between the numerical predictions and experimental measurements all indicate that the 3D simulations show a stronger correlation with the measurements than the 2D results; specifically, in the aft section of the airfoil where separation driven fluctuations exist. It should be noted, the slight parallel shift in \(Cp\) is a result of the freejet configuration of the wind tunnel, as pointed out by [PERSON] et al. (2018) and [PERSON] et al. (2019). Based on the findings of Figures 2 and 3, boundary layer (BL) velocity profiles (Figure 4) are generated between 70% and 95% chord, in increments of 5%, where most discrepancies are shown to occur. In agreement with Figure 3b, immediately after the separation point between 70% and 75% chord, numerical predictions and experimental measurements show similar profiles, with the exception of the latter revealing a thinner BL. At 80% and 85% chord, where the surface pressure coefficient shows the 2D and 3D results diverging, the 3D results show much better agreement with experiment as the 2D analysis overpredicts the height of the velocity profile. However, at 90%, the 3D simulations begin to underpredict the height of the BL while the 2D results continue with overprediction. Figure 3: (**a**) Near-wake U-velocity profile and (**b**) comparison of predicted vs. measured surface pressure coefficient. Baseline case for NACA-0012 at a 0\({}^{\circ}\) AoA and U = 16 m/s. Figure 2: Time-averaged U-velocity contours: (**a**) 2D; (**b**) 3D Implicit Large-Eddy Simulations (ILES). Baseline case for NACA-0012 at a 0\({}^{\circ}\) angle of attack (AoA) and U = 16 m/s.