# Near-Field Aeroacoustics of Spanwise Forcing on a Transonic Wing: A DNS Study
## 4 The Near-Field Acoustics
### Frequency Content  
The OASPL analysis can be enhanced by studying the changes in the frequency content of the radiated noise. We computed the square \(\widehat{p^{\prime}}^{2}\) of the Fourier coefficients of the pressure signals, obtained through [PERSON]'s method [26]. Figure 8 presents the pressure spectra for cases REF, RREF, C10, and R10. Specifically, the curves are computed starting from the pressure time histories measured by the probes corresponding to the largest overall sound pressure levels, whose angular position \(\theta\) is indicated by the colored diamonds in Figure 6. The signals have been subdivided into partially overlapping windows with a time span of \(\Delta T\approx 5c/U_{\infty}\) and 50% overlap; additionally, a Hanning window was applied to each segment. The Fourier transforms employed in the computation of the spectra were computed on each segment and then averaged together. The Nyquist frequency \(f_{N}\) is constrained by the temporal time step of the simulation (hence \(f_{N}=1/2\Delta t\approx 3.3\times 10^{3}U_{\infty}/c\)), while the frequency resolution is \(\Delta f=1/\Delta T\approx 0.2U_{\infty}/c\). Figure 7: Variation \(\Delta\) of the OASPL with respect to the REF case, as a function of the angular position of the observer \(\theta\): cases at \(\alpha=4^{\circ}\) (C10) and \(\alpha=3.45^{\circ}\) (R10). RR10 portraits changes in the OASPL between REF and R10. The horizontal solid lines mark the azimuthally averaged value, and the diamonds represent the local maxima. Figure 8, with the logarithmic scaling that provides a balanced view on the whole range of frequencies, shows clearly that StTWs do not significantly alter the frequency content of noise. Despite the difference in the position of the observers (\(\theta\approx 251^{\circ}\) for simulations REF and C10, and \(\theta\approx 224^{\circ}\) for simulations RREF and R10, see Figure 6), some analogies across the curves can be identified. In fact, all the spectra feature a rather large low-frequency content, while the Fourier coefficients for \(f>3.5U_{\infty}/c\) are drastically reduced. The angle of attack seems to be the most influential variable in terms of frequency content, too. In fact, when \(\alpha=4^{\circ}\), the frequency corresponding to the largest Fourier coefficient is fixed at \(f\approx 1.2U_{\infty}/c\), while at \(\alpha=3.45^{\circ}\) the peak shifts from \(f\approx 1U_{\infty}/c\) (case RREF) to \(f\approx 0.8U_{\infty}/c\) (case R10). In the intermediate frequency range \(fc/U_{\infty}\in[1\div 5]\), the spectra appear to collapse to a power-law dependence on the frequency of \(f^{-20/3}\). At the highest frequencies, instead, the decay of the spectra is slower, and proportional to \(f^{-2}\). Both power laws have been already observed, albeit in a different framework, by [PERSON] and [PERSON] (2017), who detected them in the spectrum of the pressure signals in the near-field around a subsonic jet flow. The theoretical background for these power laws was laid down by [PERSON] et al. (2018), who derived them starting from the quadrupole solution to the spherical wave equation, coupled with the unsteady Bernoulli equation of the velocity potential in the irrotational part of the flow. They highlighted that the decays of the spectra with frequency descends either from the assumption of intertial subrange of the turbulent energy spectrum of the flow or from energy conservation, and denoted the \(f^{-20/3}\) and \(f^{-2}\) dependencies as 'near-field' and 'far-field' behavior, respectively. The reason for such nomenclatures lies in that \(\widehat{p}^{2}\) actually depends on the product of the frequency \(f\) and the distance from the sound source \(r\): if the former is constant, the spectrum of the acoustic signals evolves either as \(r^{-20/3}\) or \(r^{-2}\), depending on whether the probe is located in the near-field or in the far-field. Finally, [PERSON] et al.