# Near-Field Aeroacoustics of Spanwise Forcing on a Transonic Wing: A DNS Study
## 5 Discussion  
In this section, we connect the application of flow control via StfWs to the variation in noise intensity, previously discussed in Section 4. In particular, we consider the relationship between the changes \(\Delta\) plotted in Figure 7 and the percentage increase in the lift and drag coefficients of the wing (\(\Delta C_{L,\%}\) and \(\Delta C_{D,\%}\), respectively) that derive from the activation of spanwise forcing. We have also observed that mean, azimuthally averaged changes in OASPL \(\overline{\Delta}\) and the lift coefficient do not share a similar dependency on the control parameters. A quantitative link between OASPL and the drag coefficient, instead, is provided in Figure 10. Here, we compare the averaged change \(\overline{\Delta}\) of OASPL with the percentage change in the drag coefficient \(\Delta C_{D,\%}\) for the simulations at constant forcing intensity \(A\) and angle of attach \(\alpha\), namely cases C1-C20. The figure shows that the larger is \(\Delta C_{D,\%}\), the larger is the mean increase in the overall sound pressure level, and vice versa. The mean position of the shock wave \(x_{s}/c\), which affects the lift and the drag coefficients [13], is color-coded in Figure 10. While the cases with the largest drag reduction also feature the most delayed shock, Figure 10 does not provide evidence for the existence of a clear relationship between \(x_{s}\) and \(\overline{\Delta}\). For instance, the data point corresponding to the largest \(\overline{\Delta}\) is the one featuring the largest \(x_{s}\), but the simulation with the lowest drag increase (\(\Delta C_{D,\%}=-1.3\%\)) does not yield the most upstream shock. Hence, we conclude that the observed noise increase should be arguably related to the increase in the aerodynamic drag, and not to the displacement of the shock wave. In Section 4.2, we have described the variations in the frequency content of the noise signal measured by a single specific observer. It is useful to verify whether or not those observations can be extended to probes in other azimuthal positions. Hence, we investigate the Fourier transforms for the acoustic signals of all observers in the reference uncontrolled case (REF), and in one controlled case at \(\alpha=4^{\circ}\) (C10). Figure 11 represents at once the absolute value of the Fourier coefficients obtained from the data of all the probes in the REF case (left panel) and in case C10 (right panel). In each panel, the distance of a point from the center of the picture represents a fixed value of frequency (for instance, the radius of the red circle corresponds to the temporal frequency of the forcing in case C10). The angle identified by the position vector of a point in the plot and the horizontal axis, instead, represents the angular position \(\theta\) of the probe. Since only 40 discrete probes are available, the values of this 'directional' Fourier transform between the probes have been azimuthally interpolated for the plot. Figure 10: Comparison between the azimuthally averaged change \(\overline{\Delta}\) in the OASPL and the drag coefficient increase \(\Delta C_{D,\%}\). Data points are colored according to the position of the shock wave \(x_{s}/c\). The dataset includes the simulations at constant forcing amplitude (cases C1–C20); note that two points overlap. In the left panel, the largest values of the Fourier transform are found, as expected by the previous analysis of the OASPL presented in Section 4, in the direction opposite to the shock wave, i.e., at \(\theta\approx 225^{\circ}\). The distinct peaks at \(\theta\approx 225^{\circ}\) and \(\theta\approx 270^{\circ}\) could be the trace of tonal noise, probably generated at the trailing edge of the wing. In fact, the presence of a separation bubble in this region (previously mentioned in Section 3) is the necessary condition for the generation of tonal noise, as explained by [PERSON] et al. [31].