# Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils
## 4 Results of the Parametric AFL Sensitivity Study
### Acoustic Radiation Frequency Structure  
[PERSON] and [PERSON] (2018) suggests that a transitional airfoil emitting tonal noise is the result of two mechanisms: the first is the scattering of vortices along the airfoil surface into the wake at the airfoil TE, and the second is the natural vortex shedding. The current section focuses on investigating the BL dynamics of the airfoil surface to examine the instability point of inception and the subsequent behavior of the flow. To achieve this task,  
Figure 21: Instability chordwise amplification for \(Re_{c}\) = 144,000 (**left**) and \(Re_{c}\) = 288,000 (right). the effect of mean flow velocity (Table 6) is further analyzed from the previous section to establish a connection between the LSB and its effect on the AFL. The selected cases correspond to an airfoil with the chord c = 0.12 m installed at the geometric AoA of 0\({}^{\circ}\), closely representing the experimental conditions of [PERSON] et al. [10; 31]. Figure 22a illustrates the Z-vorticity contours of an airfoil subjected to a 16 m/s (\(Re_{c}\) = 128,000) uniform upstream flow, revealing the expected vortex roll up and shedding pattern. As described by [PERSON] and [PERSON] [3], the onset of instabilities occur at 0.86c due to the T-S waves that grow and transition to discrete vortices, which convect along the airfoil surface towards the TE. Once the vortices reach the TE, their interaction with the natural shedding of the wake coalesce to amplify the noise generated by the airfoil. As the mean flow velocity is increased to 19 m/s (\(Re_{c}\) = 152,000) (Figure 22b), the overall behavior of the flow remains unchanged with the exception of the triggering of T-S to K-H instabilities. Unlike the 16 m/s (\(Re_{c}\) = 128,000) case, at 19 m/s (\(Re_{c}\) = 152,000), the transition from T-S to K-H instabilities appears further upstream as the increase in the flow velocity shifts the location of the separation region towards the LE. A similar pattern is exhibited as the mean flow velocity is increased to 21 m/s (\(Re_{c}\) = 168,000) (Figure 22c) and 25 m/s (\(Re_{c}\) = 200,000) (Figure 22d). It is worth mentioning that for all the flow regimes shown in Figure 22, the LSB is present and is the primary source to trigger instability amplification and the sustainment of the AFL. To further demonstrate the effect of \(Re_{c}\) on the tonal noise mechanism, spectral analysis is employed for various experimental probe locations along the airfoil surface (shown in Figure 1) to obtain the peak and neighboring tones. Details of the Fourier analysis performed to obtain the spectra are similar to those employed in the previous sections. However, for all subsequent near-field (2c above TE) spectra in this chapter, the results were obtained by sampling a signal that was 5 times longer, thus allowing for averaging of the FFT (using 5 segments) to smooth out the noise and accentuate the peaks of the signal. Figure 23 reveals the peak tonal noise (corresponding to Strouhal number \(St\)\(\sim\)0.2) predicted by the numerical simulations for the mean flow velocity of 16 m/s (\(Re_{c}\) = 128,000) to occur at a frequency of 636 Hz for every selected point along the airfoil, which correlates well with [PERSON]'s [1] empirical formula (14) for vortex shedding frequency,  
\[f_{s}=\frac{0.011U_{\infty}^{1.5}}{\left(Cv\right)^{1/2}} \tag{14}\]  
where \(c\) is the airfoil chord and \(\
u\) is the flow kinematic viscosity. Although the neighboring tones are more difficult to distinguish due to the low amplitudes, the equally distant spaced peaks, \(\Delta f\), are indicative of their existence. Furthermore, the peak amplitude of 120 db is revealed to occur at point A (0.945c) close to the TE, with the amplitude decreasing for upstream probes. This behavior confirms that the main noise source is, in fact, located at the TE where the scattering takes place.