## 2 The 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 vibrate under the flow-induced force. Then, the noise will be radiated from the model by the flow excitation. In the theory of vibration and sound radiation from the shell under the flow-induced force, some assumptions need to be upheld, as follows.  
1. The turbulent pressure field is spatially uniform and static relative to time. That is, the time-space correlation of wall pressure fluctuation only depends on the 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 elasticity.  
If \(p(x,y,t)\) denotes the turbulent fluctuation pressure, which excites the shell of the model, then \(p(x,y,t)\) can be decomposed by the wavenumber-frequency spectrum:  
\[p(x,y,t)=\int\int\int S\big{(}k_{x},k_{y},\omega\big{)}e^{i(k_{x}x+k_{y}y- \omega t)}dk_{x}dk_{y}d\omega. \tag{9}\]  
If \(H\big{(}k_{x},k_{y},z,\omega\big{)}\) is the function of the wavenumber-frequency transformation and is introduced to express the response of the excitation of the infinite plate by turbulent fluctuation pressure, then the pressure at any point in the system can be shown as:  
\[F(x,y,z,t)=\int\int\int S\big{(}k_{x},k_{y},\omega\big{)}H\big{(}k_{x},k_{y}, z,\omega\big{)}e^{i(k_{x}x+k_{y}y-\omega t)}dk_{x}dk_{y}d\omega. \tag{10}\]  
In the random fields, the time-space correlation function can be written in plural form:  
\[R(L,D,z_{1},z_{2},\tau)=F(x,y,z_{1},t)F*(x-L,y-D,z_{2},t-\tau) \tag{11}\]  
\[\left\langle S\big{(}k_{x},k_{y},\omega\big{)}S*\Big{(}k_{x}^{\prime},k_{y}^{ \prime},\omega\Big{)}\right\rangle=G_{s}\big{(}k_{x},k_{y},\omega\big{)} \delta(\omega-\omega t)\delta\big{(}k_{x}-k_{x}^{\prime}\big{)}\delta\Big{(}k _{y}-k_{y}^{\prime}\big{)}, \tag{12}\]  
where \(\left\langle\ \right\rangle\) denotes the average. If Equation (10) is substituted into Equations (11) and (12), then the function of the time-space correlation of the random field is obtained:  
\[R(L,D,z_{1},z_{2},\tau)=\int\int\int H\big{(}k_{x},k_{y},z_{1},\omega\big{)}G _{s}\big{(}k_{x},k_{y},\omega\big{)}H*\big{(}k_{x},k_{y},z_{2},\omega\big{)} \times e^{i(k_{x}L+k_{y}D-\omega\tau)}dk_{x}dk_{y}d\omega, \tag{13}\]  
where \(G_{s}\big{(}k_{x},k_{y},\omega\big{)}\) is the density function of the wavenumber-frequency spectrum. Then,  
\[G_{s}\big{(}k_{x},k_{y},\omega\big{)}=\frac{1}{\left(2\pi\right)^{2}}\int\int g _{s}(\xi,\zeta,\omega)e^{-(k_{x}\xi+k_{y}\xi)}d\xi d\xi, \tag{14}\]where \(g_{s}(\xi,\zeta,\omega)\) is the function of cross-spectrum density. Then,  
\[G(L,D,z_{1},z_{2},\omega)=\int\int H\big{(}k_{x},k_{y},z_{1},\omega)G_{s}\big{(} k_{x},k_{y},\omega\big{)}H\ast\big{(}k_{x},k_{y},z_{2},\omega\big{)}\times e^{i(k_{ x}L+k_{y}L)}dk_{x}dk_{y}.\]  
There was a minor difference in the sound pressure level in the frequency range between the two results. The reason was that microphones are the practical tools with some spacing, while the points in the numerical simulation were virtual, only picked up according to the receiver's position, as described in [36]. However, we see the trend of the sound pressure level changing with the frequency in the numerical simulation as being very similar to that in the experimental test. The total sound pressure level difference between them was only 0.6%, which can be neglected. Therefore, the established process of numerical calculation was feasible and could be used to calculate the flow-induced noise from the SUBOFF model.