## 2 Numerical Methods and Target Model
### Wavenumber-Frequency Transform  
The wavenumber-frequency diagram of fluctuating surface pressure can be obtained by taking the three-dimensional Fourier transform of the surface pressure in the space-time domains, thereby revealing the distribution of the magnitude of the specific pressure wave component in terms of the wavenumber \(k\) and the frequency \(\omega\). One of the advantages of using the wavenumber-frequency spectrum over the space-time diagram is that the phase speed of the specific wave component constituting the wave-like surface pressure field in the space-time domain can be determined. The phase speed \(v_{p}=\omega/k\) can be used as a criterion to decompose the surface pressure fluctuation into the incompressible and compressible components. The former is convected at the velocity proportional to the mean flow velocity \(U_{0}\), while the latter propagates at the velocity equal to the vector sum of mean flow velocity and the sound speed \(c_{0}\). The detailed algorithm for the wavenumber-frequency analysis is described below. The spatial-temporal pressure field on a land vehicle can be characterized using the following correlation function:  
\[R(\mathbf{x};\mathbf{\upxi},\tau)=\big{\langle}p^{\prime}(\mathbf{x},t)\cdot p^{\prime}( \mathbf{x}+\mathbf{\upxi},t+\tau)\big{\rangle}, \tag{14}\]  
where \(\mathbf{\upxi}\) is the distance vector, \(\tau\) is the time delay, and \(p^{\prime}\) is the perturbed pressure. The correlation function of the spatial-temporal pressure field can be converted to the wavenumber-frequency domain using a Fourier transform. The Fourier transform from the two-dimensional space-time domain to the wavenumber-frequency domain is written as  
\[S(\mathbf{k},\omega)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty} ^{\infty}R(\mathbf{\upxi},\tau)e^{-i(-\mathbf{k}\mathbf{\upxi},+\omega\tau)}d\xi_{1}d\xi_ {2}d\tau. \tag{15}\]  
The power spectral density (PSD) of the wall pressure fluctuations can be estimated using the periodogram method in the following form:  
\[S(\mathbf{k},\omega)=\lim_{T,L_{1},L_{2}\rightarrow\infty}\frac{1}{L_{1}}\frac{1 }{L_{2}}\frac{1}{T}\ \big{\langle}\rho_{L,T}(\mathbf{k},\omega)\cdot\hat{p}_{L,T}^{*}(\mathbf{k}, \omega)\big{\rangle}, \tag{16}\]  
where  
\[\hat{p}_{L,T}(\mathbf{k},\omega)=\int_{-T/2}^{T/2}\int_{-L_{1}/2}^{L_{1}/2}\int_{- L_{2}/2}^{L_{2}/2}p^{\prime}(\mathbf{x},t)e^{-i(-\mathbf{k}\mathbf{x}+\omega t)}dx_{2}dx_{1}dt, \tag{17}\] \begin{table}
\begin{tabular}{c c} \hline
**Equations** & **Discretization Method** \\ \hline Pressure & Second-order \\ Momentum & Bounded central differencing \\ Energy & Second-order upwind \\ Transient formulation & Second-order implicit \\ \hline \end{tabular}
\end{table} 
Table 1: Discretization schemes of governing equations. and \(\hat{p}_{L,T}^{*}\) is a complex conjugate of \(\hat{p}_{L,T}\). In Equation (16), \(L_{1}\), and \(L_{2}\) are the length of the space domain in each direction, and \(T\) is the period. Lastly, the three-dimensional discrete Fourier transform equation used to obtain the three-dimensional averaged modified periodogram can be written in the following form [3; 4; 5]:  
\[ \begin{split} S_{m,n,o}&=S(m\Delta f,n\Delta k_{x_{1} },o\Delta k_{x_{2}})\\ &=\frac{\Delta x_{1}\Delta x_{2}}{N_{x_{1}}N_{x_{2}}N_{x_{2}}} \bigg{|}\sum\limits_{k=0}^{N_{1}-1}\sum\limits_{l=0}^{N_{3}-1}\sum\limits_{j= 0}^{N_{2}-1}w_{klj}P_{klj}e^{-2i\pi(\frac{k\pi}{N_{l}}-\frac{ul}{N_{x_{1}}}- \frac{ul}{N_{x_{2}}})}\bigg{|}^{2}\bigg{/}\frac{1}{N_{x_{1}}N_{x_{2}}}\sum \limits_{k=0}^{N_{x_{1}}-1}\sum\limits_{l=0}^{N_{x_{2}}-1}\sum\limits_{j=0}^{ N_{x_{2}}-1}\bigg{|}w_{klj}\bigg{|}^{2}\end{split}  \tag{18}\]  
where \(w_{klj}\) is a discretized Hanning window. In the three-dimensional periodogram, the incompressible and compressible parts can be separated by using the slanted Dirac cone, defined as  
\[\omega=c_{0}\sqrt{k_{x_{1}}^{2}+k_{x_{2}}^{2}}+k_{x_{1}}L_{0}, \tag{19}\]  
where \(U_{0}\) is the free stream velocity.