## 3 Aerodynamic Noise Analysis of the Panel
### Incompressible Numerical Simulation
#### 3.1.3 Aerodynamic Noise Analysis  
(a) Mechanism of Radiated Sound under Aerodynamic Load  
Compared with the aircraft TBL, the flow speed past a car is relatively small and--particularly--the vortex happens when it meets irregularly shaped bodies. The flow past a car is an aero-vibro-acoustic problem. [PERSON] [22] had proposed the sound power \(\prod_{\Delta\omega}^{rad}\) radiated from the panel vibration as follows:  
\[\prod_{\Delta\omega}^{md}=\rho_{0}c_{0}A\sigma_{\Delta\omega}^{rad}\int\Phi_{v }(\omega)d\omega \tag{7}\]  
Figure 12: The RMS of the space average pressure on the side mirror surface. Figure 13: The pressure RMS distributions of the panel for 315 Hz, 1000 Hz, 3150 Hz and the overall pressure under the FSP of the side mirror: (**a**) For 315 Hz; (**b**) For 1000 Hz; (**c**) For 3150 Hz; (**d**) For the overall pressure. where, \(\rho_{0}\) is air density. \(c_{0}\) is air speed. \(A\) is area of the panel. \(\sigma_{\Delta\omega}^{nd}\) is radiation efficiency. \(\Phi_{v}(\omega)\) is velocity spectral density. The radiation efficiency in terms of the wavenumber and the velocity spectral density is given by [23]:  
\[ \begin{array}{c}\sigma_{mn}=\frac{1}{4\pi^{2}A}\iint\limits_{ \omega}\left|S_{mn}(k)\right|^{2}\left[1-\left(\frac{k}{k_{0}}\right)^{2} \right]^{-0.5}d^{2}k\\ \Phi_{v}(\omega)=\sum\limits_{mn}\frac{\[PERSON]_{mn}(\omega)}{\left|Z_{mn}(\omega )\right|^{2}}=\sum\limits_{mn}\frac{\omega^{2}\widetilde{\[PERSON]}_{mn}(\omega)}{ m_{mn}^{2}|Y_{mn}(\omega)|^{2}}\\ \widetilde{\[PERSON]}_{mn}(\omega)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \Phi_{p}\left(k_{x},k_{y},\omega\right)\left|S_{mn}\left(k_{x},k_{y}\right) \right|^{2}dk_{x}dk_{y}\end{array}  \tag{8}\]  
where, \(k_{0}=\omega/c_{0}\), \(S_{mn}(k)\) is the shape function. \(\widetilde{\[PERSON]}_{mn}\) is the excitation source. \(Z_{mn}(\omega)\) is the structural modal impedance. \(m_{mn}\) is the modal mass and \(Y_{mn}(\omega)\) is the modal admittance. \(\Phi_{p}\left(k_{x},k_{y},\omega\right)\) is the WFS.  
Later, [PERSON] [14] separated the formula of the sound power \(\prod_{\Delta\omega}^{rad}\) into resonant and non-resonant contributions without considering stiffness-controlled modes. Then Equation (7) can be written as:  
\[\prod_{\Delta\omega}^{nd}\propto\rho_{0}c_{0}A\int d\omega\left\{\sum\
olimits _{\omega_{nm}\in\Delta\omega}\frac{\omega^{2}\Phi(\omega)J_{mn}^{2}(\omega) \sigma_{nm}(\omega)}{m_{mn}^{2}|Y_{mn}|^{2}}+\sum\
olimits_{\omega_{nm}\in \Delta\omega}\frac{\Phi(\omega)J_{mn}^{2}(\omega)\sigma_{mn}(\omega)}{m_{mn}^ {2}}\right\} \tag{9}\]  
where, the modal joint acceptance \(J^{2}(\omega)\) is as follows:  
\[J_{mn}^{2}(\omega)=\frac{\widetilde{\Phi}_{mn}(\omega)}{\Phi(\omega)S^{2}} \tag{10}\]  
where, \(S\) is the surface area of the panel. The (\(m\), \(n\))th simply supported mode shape is as follows:  
\[S_{mn}\left(k_{x},k_{y}\right)=\frac{2}{\sqrt{ab}}\int_{0}^{a} \int_{0}^{b}sin(\frac{m\pi y}{a})sin(\frac{n\pi x}{b})e^{-ik_{x}x}e^{-ik_{y}y }dxdy \tag{11}\]  
where, \(a\) and \(b\) are the length and width of the panel, respectively. The amplitude of the shape function is given by:  
\[\left|S_{mn}\left(k_{x},k_{y}\right)\right|^{2}=\frac{2(m\pi)^{2} \left[1-\left(-1\right)^{m}\!\!\cos(k_{x}a)\right]}{\left[\left(k_{x}a\right) ^{2}-\left(m\pi\right)^{2}\right]^{2}}\times\frac{2(n\pi)^{2}\left[1-\left(- 1\right)^{m}\!\!\cos\!\left(k_{y}b\right)\right]}{\left[\left(k_{y}b\right)^ {2}-\left(n\pi\right)^{2}\right]^{2}} \tag{12}\]  
The amplitude of the shape function reaches peak around the modal wavenumbers \(k_{x}(=m\pi/a)\) and \(k_{y}(=n\pi/b)\). There are an infinite number of side lobes at each side of the peak, as shown in Figure 14. We can see that \(\left|S_{mn}\left(k_{x},k_{y}\right)\right|^{2}\) has peaks around the modal wavenumber \(k_{x}(=m\pi/a)\). Meanwhile, the WFS of the boundary layer pressure achieves the maximum value when \(k_{x}=\omega/U_{c}\), \(k_{y}=0\) (\(U_{c}\) is convective velocity).