# Single and Multivariate Statistics of Jet-Induced Pressure Fluctuations over an Infinite Plate
## 3 Results  
### Single-Point Statistics  
The wall pressure statistics were characterized in the frequency domain through the power spectral density (PSD) evaluated with the Welch method [19] and presented as a function of the Strouhal number, defined as follows:  
\[St=\frac{fD}{U_{j}}, \tag{1}\]  
where \(f\) is the frequency and U\({}_{\text{j}}\) is the nozzle exhaust velocity. The axial evolution of the pressure spectra for two different plate positions (H/D = 0.75 and H/D = 2) is reported in Figure 2.  
Figure 1: Experimental setup. Different shapes are detected for H/D = 0.75 (Figure 2a) with respect to H/D = 2 (Figure 2b). This difference is ascribed to the different level of jet-surface interaction, which is related to the jet impact position over the flat plate. According to [PERSON] and [PERSON] [18], the jet spreading angle is about 10\({}^{\circ}\); therefore, in the case of H/D = 0.75, the jet is expected to impact the plate surface at about x/D \(\approx\) 1.5. In agreement with [15; 16], the jet impact point seems to be neither influenced by the different jet Mach numbers nor by the nozzle exhaust Reynolds numbers. Figure 2a shows peaks at x/D = 5 in the frequency range associated with the jet Kelvin-Helmholtz mode (i.e., between St = 0.2 and St = 0.5). At larger x/D, far downstream of the impact point, the shapes of the spectra becomes similar and the energy content decreases moving downstream. As reported in [15], the spectral shape follows the classical behavior expected for a fully developed TBL. Concerning the jet-plate configuration H/D = 2, the jet impacts the plate surface at about x/D \(\approx\) 8.5 [15], thus inducing the development of a TBL-like zone further downstream with respect to the H/D = 0.75 configuration. The spectral behavior and the energy content vary accordingly with the more limited jet-surface interactions; three different interaction zones are presented in Table 1[15; 16; 20]. Figure 2b shows that a significant energy bump is present at low frequencies for small axial distances (x/D \(\leq\) 10) (i.e., free-jet zone), and this feature is the signature of the hydrodynamic pressure field, whereas the change in the spectral energy decay observed at high frequencies can be ascribed to the dominance of the acoustic pressure in the near region of the jet [21]. Furthermore, comparing x/D = 5 at H/D = 0.75, Figure 2a, with x/D = 10 at H/D = 2, Figure 2b, we point out that both positions were close to the jet impact point, although they showed a different spectral behavior influenced by the jet flow conditions. Indeed, we observe, at x/D = 5, a peak related to the Kelvin-Helmholtz roll-up, whereas at x/D = 10, a bump can be observed at lower St numbers due to the larger-scale turbulent structures generated by the jet development. The effect of the jet Mach number at H/D = 0.75 is reported in Figure 3a,b. It is shown that, for a fixed x/D, the spectral shape is weakly affected by M\({}_{\rm j}\), in agreement with the results presented in our previous studies [15] where this behavior was observed for larger H/D vales. \begin{table}
\begin{tabular}{c c c c} \hline \hline H/D & **Free Jet Zone** & **Impact Zone** & **TBL-Like Zone** \\ \hline
0.75 & Not appreciable & 1D-7D & 7D-25D \\
2 & 1D-10D & 10D-20D & 20D-25D \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Jet-plate interaction zones. TBL: turbulent boundary layer. Figure 2: Axial evolution of the wall pressure autospectra at M\({}_{\rm j}\) = 0.5: (**a**) H/D = 0.75; (**b**) H/D = 2. The wall pressure fluctuations were investigated in the span-wise direction as well, and results for two different stream-wise positions--i.e., x/D = 5 and x/D = 25--are reported in Figure 4. We note that both spectral energy and shape are affected by the different transverse position considered at x/D = 5 (see Figure 4a). Specifically, the energy content significantly decreases as the span-wise distance increases, and this behavior could be likely ascribed to the lower flow-surface interaction far away from the core of the jet.