## 3 Unsteady Flow Analysis and Decomposition of Surface Pressure
### Decomposition of Surface Pressure  
The wavenumber-frequency transform was applied to the unsteady surface pressure fields on the side windows and the front windshield of the HSM shown in Figure 2 to separate the compressible part from the incompressible one. The wavenumber-frequency analysis was conducted by applying Equation (14) to the surface pressure fields. The sampling rate \(f_{s}\) and the frequency interval \(\Delta f\) were 80,000 Hz and 20.0 Hz, respectively. Wavenumber-frequency analysis is available only for a rectangular surface. However, the side window and the front windshield glass of the HSM were not precisely a rectangle. Figure 8: Snapshot of iso-contours of surface wall pressure on left window with reattachment line (at \(t=2.8\) s). Figure 7: Instantaneous iso-surfaces of Q-criterion (at \(t=2.8\) s). Figure 9: Comparison of power spectral density levels of surface pressure fluctuations monitored at probe locations between numerical and experimental results: (**a**) probe locations; (**b**) probe 1; (**c**) probe 2; (**d**) probe 3; (**e**) probe 4; (**f**) probe 5. To resolve this issue, the targeted surface for the wavenumber-frequency analysis was extended to become a rectangle. Figure 10 illustrates the extended rectangular surface pressure field, including the side window surface. Furthermore, signal processing information is shown in Table 3.  
The power spectral density obtained from the wavenumber-frequency transform is plotted in Figure 11a. The slanted Dirac cone (\(-c_{0}+U_{0}\leq\omega/k\leq c_{0}+U_{0}\)) could be clearly identified with the convection hydrodynamic components. To highlight the characteristics of the power spectral density diagram more clearly, the two-dimensional spectrum at the cross-sectional plane of \(k_{z}=0\) is shown in Figure 11b. Significant compressible components were identified between the characteristic lines for which the slopes corresponded to the phase speeds of \(|-c_{0}+U_{0}|=310.56\) m/s and \(|-c_{0}+U_{0}|=370.56\) m/s, while most of the incompressible components were located on the characteristic wave line with slope \(U_{c}\cong 0.9U_{0}=27.5\) m/s. It is known that the turbulent vorticity within the boundary layer is convected at the speed of \(0.9U_{0}\) or less. These distinctly separated regions manifest that the LES simulation effectively captured the acoustic wave propagation, as well as the aerodynamic noise generation, in the external flow of the HSM. The corresponding power spectral density levels were obtained by integrating the power spectral density over the entire wavenumber ranges in the following form:  
\[S(\omega)=\int\int S(\mathbf{k},\ \omega)dk_{x_{1}}k_{x_{2}}. \tag{45}\]  
Figure 11c shows the computed PSD of the fluctuating pressure field. Although the predicted PSD of surface pressure showed good agreement with the measured one in the frequency range below 3.5 kHz, the decomposed PSDs were compared in the frequency range up to 5 kHz to highlight the general characteristics of the relative contribution between the compressible and incompressible pressure fields. It can be seen that the incompressible pressure parts dominated in most of the frequency range, but the relative contribution of compressible ones became comparable in the frequency range higher than 5 kHz. Figure 10: Virtual rectangular monitoring area for left side window. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Spatial–Temporal Domain**} & \multicolumn{3}{c}{**Wavenumber–Frequency Domain**} \\ \hline \(\Delta x\) (mm) & \(\Delta z\) (mm) & \(\Delta t\) (s) & \(\Delta k_{x}\) (m\({}^{-1}\)) & \(\Delta k_{z}\) (m\({}^{-1}\)) & \(\Delta f\) (Hz) \\
2 & 2 & 0.0000125 & 1.1003 & 1.6523 & 20.0 \\ \(N_{x}\) & \(N_{x}\) & \(N\) & \(k_{xmax}\) (m\({}^{-1}\)) & \(k_{xmax}\) (m\({}^{-1}\)) & \(f_{max}\) (Hz) \\
455 & 303 & 4000 & 500.6 & 500.6 & 80,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of three-dimensional DFT for wall pressure fluctuation signal of side windows.