## 2 Theory of Numerical Simulation
### Theory of Vibration and Sound Radiation by the Flow-Induced Force  
When the fluid flows around the model, the shell will be vibrated under the flow-induced force. Then, the noise will be radiated from the model by the excitation of the flow. In the theory of the vibration and sound radiation from the shell under the flow-induced force, some assumptions need to be held as follows. (1) The turbulent pressure field is spatially uniform, and is static relative to time. That is, the time-spatial correlation of wall pressure fluctuation only depends on spatial distance and time interval. (2) The sound radiation from the vibration of the model is under the excitation of turbulent fluctuation pressure, while that from the turbulent fluctuation pressure itself is ignored. (3) The properties of the model are isotropic and obey the theory of the elasticity. If \(p(x,y,t)\) denotes turbulent fluctuation pressure, which excites the shell of the model, \(p(x,y,t)\) can be decomposed by the wavenumber-frequency spectrum:  
\[p(x,y,t)=S\big{(}k_{x},k_{y},\omega\big{)}e^{i(k_{x}x+k_{y}y-\omega t)}dk_{x} dk_{y}d\omega, \tag{9}\]  
If \(H\big{(}k_{x},k_{y},z,\omega\big{)}\) is the function of wavenumber-frequency transform and introduced to express the response of the excitation of the infinite plate by turbulent fluctuation pressure, the pressure at any point in the system can be shown as:  
\[F(x,y,z,t)=S\big{(}k_{x},k_{y},\omega\big{)}H\big{(}k_{x},k_{y},z,\omega\big{)} e^{i(k_{x}x+k_{y}y-\omega t)}dk_{x}dk_{y}d\omega, \tag{10}\]  
In the random fields, the time-space correlation function can be written in the plural form:  
\[R(L,D,z_{1},z_{2},\tau)=\left\langle F(x,y,z_{1},t)F*(x-L,y-D,z_{2},t-\tau) \right\rangle, \tag{11}\]  
\[\left\langle S\big{(}k_{x},k_{y},\omega\big{)}S*\Big{(}k_{x}^{\prime},k_{y}^{ \prime},\omega\big{)}\right\rangle=G_{s}\big{(}k_{x},k_{y},\omega\big{)} \delta(\omega-\omega t)\delta\big{(}k_{x}-k_{x}^{\prime}\big{)}\delta\Big{(}k _{y}-k_{y}^{\prime}\big{)} \tag{12}\]where \(\langle\ \rangle\) denotes the average. If Equation (10) is substituted into Equation (11) and Equation (12), then the function of the time-space correlation of the random field is obtained:  
\[R(L,D,z_{1},z_{2},\tau)=H\big{(}k_{x},k_{y},z_{1},\omega\big{)}G_{s}\big{(}k_{x },k_{y},\omega\big{)}H\ast\big{(}k_{x},k_{y},z_{2},\omega\big{)}\times e^{i(k_ {x}L+k_{y}D-\omega\tau)}dk_{x}dk_{y}d\omega, \tag{13}\]  
where \(G_{s}\big{(}k_{x},k_{y},\omega\big{)}\) is the density function of wavenumber-frequency spectrum. Then,  
\[G_{s}\big{(}k_{x},k_{y},\omega\big{)}=\frac{1}{\left(2\pi\right)^{2}}g_{s}( \xi,\zeta,\omega)e^{-\left(k_{x}\xi+k_{y}\zeta\right)}d\xi d\zeta, \tag{14}\]  
where \(g_{s}(\xi,\zeta,\omega)\) is the function of cross-spectrum density. Then,  
\[G(L,D,z_{1},z_{2},\omega)=H\big{(}k_{x},k_{y},z_{1},\omega\big{)}G_{s}\big{(} k_{x},k_{y},\omega\big{)}H\ast\big{(}k_{x},k_{y},z_{2},\omega\big{)}\times e^{i(k_ {x}L+k_{y}L)}dk_{x}dk_{y}, \tag{15}\]  
To describe the pressure fluctuation of the turbulent boundary layer, the Corcos model is adopted. The function of cross-spectrum density can be acquired:  
\[g_{s}(\xi,\zeta,\omega)=g_{s}(\omega)e^{-C_{1}K_{\rm C}|\xi|+C_{2}K_{\rm C}| \zeta|}e^{-iK_{\rm C}\xi}, \tag{16}\]  
where \(C_{1}\) and \(C_{2}\) are two constants, which are related to the surface roughness, \(K_{\rm C}=\omega_{C}/U_{\rm C}\) is the migration wave number, and \(U_{\rm C}\) is the migration turbulence velocity.