# Near-Field Aeroacoustics of Spanwise Forcing on a Transonic Wing: A DNS Study
## 4 The Near-Field Acoustics
### Frequency Content
(2019) explained that the former power law is related to pressure fluctuations induced by large turbulent structures, while the latter is linked to the propagation of the weaker acoustic waves. Two characteristics emerge in the comparison among the cases considered. In particular, significant differences in the energetic content of the signals are detected at \(fc/U_{\infty}\in[2\div 3]\) and \(fc/U_{\infty}\in[4\div 10]\). We have considered whether these could be directly related to StTWs by comparing them with characteristic frequencies of this forcing. The temporal frequency of StTWs of cases C10 and R10 (\(f=\omega/2\pi\approx 5.3U_{\infty}/c\)) is denoted in Figure 8 by the vertical black solid line. Additionally, it is explained in Ref. [5] how  
Figure 8: Energy spectrum of the acoustic signals measured at the azimuthal location of the largest OASPL (visible in Figure 6), for simulations REF, RREF, C10, and R10. The colored diamonds mark the peak value of the spectrum. The vertical solid line denotes the temporal frequency of StTWs in cases C10 and R10, while the (nearly overlapping) vertical dashed line denotes the ‘equivalent frequency’ (see text) of the forcing. an 'equivalent oscillation period' \(\mathcal{T}\) (and therefore frequency) can be defined for StTWs, derived from the relative velocity between the convecting coherent, near-wall turbulent structures, and the traveling wave of the forcing. In particular,  
\[\mathcal{T}=\frac{\lambda_{x}}{U_{c}-U_{\phi}} \tag{5}\]  
where \(\lambda_{x}=2\pi/\kappa_{x}\) is the forcing wavelength, \(U_{c}\) is the convective velocity of the turbulent structures in the near-wall region, which can be considered constant in viscous units [30] at 10 times the friction velocity \(u_{\tau}\), and \(U_{\phi}=\omega/\kappa_{x}\) is the phase velocity of StTWs. In Figure 8, the 'equivalent frequency' \(1/\mathcal{T}\) nearly overlaps with the temporal frequency of StTWs, consistently with the small phase speed of the forward-traveling waves. The figure makes it quite clear that the presence of noise caused by, or directly related to StTWs, can be ruled out, as no peak is discernible at the two aforementioned forcing frequencies. The increase in \(\widehat{p}^{2}\) in the two ranges described above for cases REF and C10 cannot be attributed to the forcing, as the former simulation is uncontrolled. Therefore, differences in the spectra are arguably due to the difference in the angle of the attack. Figure 9 plots the spectrum of the acoustic signal detected at the probe of the largest OASPL for the REF case (the same presented in Figure 8), and compares it with the spectrum computed for the time history of the signal expressing the instantaneous position of the shock wave, namely for the quantity \(x_{s}^{\prime}(t)\). The latter expresses the excursion of the instantaneous position \(x_{s}(t)\) of the shock from its time-averaged position. Both quantities are computed with a fast Fourier transform algorithm using the time histories truncated at \(120c/U_{\infty}\) and modified with a Hanning window. The signal \(x_{s}(t)\) was recorded during the simulation at the somewhat lower temporal resolution of \(370\Delta t\); however, this only hampered the very high-frequency content of the transform. The important observation deduced from this figure is that the movement of the shock wave contains much smaller frequencies than those important for pressure fluctuations, suggesting that buffeting takes place at time scales separated enough from those significant for noise production. The frequency corresponding to the peak of the acoustic signal (\(f\approx 1.2U_{\infty}/c\), which identifies a tonal noise) is decoupled from the peak corresponding to the frequency of the movement of the shock wave (\(f\approx 0.01U_{\infty}/c\)). Hence, the most intense tonal noise observed is not caused by an oscillation of the shock wave.