## 3 Aerodynamic Noise Analysis of the Panel
### Incompressible Numerical Simulation
#### 3.1.3 Aerodynamic Noise Analysis
Therefore, the highest excitation occurs when the modal wavenumber (\(k_{x},k_{y}\)) close to \(k_{x}=\omega/U_{c}\), \(k_{y}=0\), namely the hydrodynamic coincidence condition. The side window is usually approximated as a thin panel with thickness \(h\) and density \(\rho\). The lateral displacement \(d\left(\overset{\rightarrow}{x},t\right)\) of the panel under a pressure loading \(p\left(\overset{\rightarrow}{x},t\right)\) described by Kirchhoff plate theory [24] is written as follows:  
\[ph\frac{\partial^{2}d}{\partial^{2}t}+D\
abla^{2}(d)=p \tag{13}\]  
where, \(\
abla^{2}\) is the Laplacian operator. \(D\) is the bending stiffness of the panel and is the product of Young's Modulus \(E\) and the moment of inertia \(I\). \[D=EI,\,I=\frac{h^{3}}{12(1-\
u^{2})} \tag{14}\]  
where, \(\
u\) is the [PERSON]'s ratio. The acoustic wavenumber, convective wavenumber and panel wavenumber are three important wavenumbers to analyze the hydrodynamic coincidence and acoustic coincidence effect. The panel wavenumber is calculated as follows:  
\[k_{b}=\sqrt[4]{\frac{\partial h^{2}\omega^{2}}{D}} \tag{15}\]  
The acoustic wavenumber is simply the frequency divided by the speed of sound. The convective wavenumber is the frequency divided by the convective speed:  
\[k_{a}=\frac{\omega}{C_{a}},\,k_{c}=\frac{\omega}{U_{c}} \tag{16}\]  
According to the physical parameters and Equation (15), the wavenumber curve of the panel with frequency can be calculated, and the wavenumber of the convective and acoustic components can also be calculated according to Equation (16). The results are shown in Figure 15. In this figure, we can see that there is an intersection point between the wavenumber of the panel and the wavenumber of the convective component and the wavenumber of the acoustic component at 400 Hz and 3150 Hz, corresponding to the aerodynamic coincidence effect and the vibro-acoustic coincidence effect. Under these two frequency points, the excitation of the convective component and the acoustic component reaches the maximum. \[J_{mn}^{2}(\omega)=\frac{\widetilde{\Phi}_{mn}(\omega)}{\Phi(\omega)S^{2}} \tag{10}\]0 Squared magnitude of the shape function with \(m=7\), \(k_{x}=0\). \[J_{mn}^{2}(\omega)=\frac{\widetilde{\Phi}_{mn}(\omega)}{\Phi(\omega)S^{2}} \tag{10}\]1 The wavenumbers. (Red curve, blue curve and green curve represent the wavenumbers of the acoustic component, the convective component and the panel, respectively). (b) Aerodynamic Noise Analysis Model  
For the incompressible numerical simulation of the FSP, the convective component is obtained by the Corcos model, random modal force and deterministic modal force, and the acoustic component is obtained by the BEM method. Therefore, for the incompressible numerical simulation of the FSP, the Corcos model coupling BEM method, random modal force coupling BEM method and the deterministic modal force coupling BEM method are used to solve the aerodynamic noise of the panel. Firstly, the Corcos model in TBL is used to calculate the convective component and DAF is used to represent the acoustic component. The model of Corcos model and DAF is shown in \[J_{mn}^{2}(\omega)=\frac{\widetilde{\Phi}_{mn}(\omega)}{\Phi(\omega)S^{2}} \tag{10}\]2 The panel is a SEA panel, and a semi-infinite fluid (SIF) is used to obtain the SPL at the SIF location. For the convection component characterized by the random modal force or deterministic modal force, the loading object is required to be a FEA structure. Therefore, the panel should be meshed. The boundary conditions by simulating the real side window and the material properties of the panel should be set up, and then the modal of the panel should be solved. The modal results of the panel solved in Nastran are shown in \[J_{mn}^{2}(\omega)=\frac{\widetilde{\Phi}_{mn}(\omega)}{\Phi(\omega)S^{2}} \tag{10}\]3 The first natural frequency is 153.2 Hz and the second natural frequency is 398.49 Hz.