# Research on Characteristics of Flow Noise and Flow-Induced Noise
## 2 Computational Theory
### Flow-Induced Noise Calculation Theory  
As a result of the flow-induced vibration of the hull surface, the velocity boundary condition needs to be satisfied:  
\[v_{n}=\frac{j}{\rho_{0}\omega}\frac{\partial p}{\partial n}=\overline{v}_{n} \tag{12}\]  
Equation (2) is an inhomogeneous Helmholtz equation. According to the superposition principle, its solution can be considered as the superposition of the solution \(p_{a}\) to the homogeneous Helmholtz equation and the solution \(p_{b}\) to the inhomogeneous Helmholtz equation, i.e.,  
\[p^{\prime}=p_{a}+p_{b} \tag{13}\]  
where \(p_{b}\) is the flow noise without considering fluid\(-\)structure coupling, which was solved in the previous section. \(p_{a}\) is the solution to the homogeneous Helmholtz equation satisfying: the following  
\[\
abla^{2}p_{a}+k^{2}p_{a}=0 \tag{14}\]  
The velocity boundary condition to be satisfied for Equation (14) is as follows:  
\[\frac{j}{\rho_{0}\omega}\frac{\partial p_{a}}{\partial n}=\overline{v_{n}}- \frac{j}{\rho_{0}\omega}\frac{\partial p_{b}}{\partial n} \tag{15}\]  
Now, we just need to solve the homogeneous Helmholtz equation. Discretization is first performed. For any point on mesh cell \(\Omega_{ae}\), assuming the acoustic pressure and velocity have the following discrete functions, respectively:  
\[p(\overrightarrow{r}_{a})=\sum_{i=1}^{n_{e}}N_{i}^{e}(\overrightarrow{r}_{a })p_{i} \tag{16}\]  
\[v_{n}(\overrightarrow{r}_{a})=\sum_{i=1}^{n_{e}}N_{i}^{e}(\overrightarrow{r}_ {a})v_{ni} \tag{17}\]  
where the shape function \(N_{i}^{e}\) takes the value 1 at node \(i\) and 0 at other positions; \(n_{e}\) is the number of nodes on cell \(\Omega_{ae}\). The shape functions \(N_{i}^{e}\) in Equations (16) and (17) are only the local shape functions defined on cell \(\Omega_{ae}\). Expanding them to the entire mesh surface \(\Omega_{a}\) yields the global shape functions \(N_{i}\), which takes the value \(N_{i}^{e}\) on cell \(\Omega_{ae}\) and 0 at other positions, i.e.,  
\[p(\overrightarrow{r}_{a})=[N_{i}]\{p_{i}\} \tag{18}\]  
\[\
u_{n}(\overrightarrow{r}_{a})=[N_{i}]\{\
u_{ni}\} \tag{19}\]  
According to the theory of boundary elements, the solution satisfying Equation (14), velocity boundary condition (15), and the Sommerfeld radiation condition is as follows:  
\[Cp_{a}(\overrightarrow{r})=\int_{\Omega}(p(\overrightarrow{r}_{a})\frac{ \partial G(\overrightarrow{r}_{r},\overrightarrow{r}_{a})}{\partial n}-G( \overrightarrow{r},\overrightarrow{r}_{a})\frac{\partial p(\overrightarrow{ r}_{a})}{\partial n})\text{d}\Omega \tag{20}\]  
where the coefficients \(C\) take the following values:  
\[C=\left\{ \begin{array}{ll}0.5&\overrightarrow{r}\in\Omega\\ 1&\overrightarrow{r}\in V\mid\Omega\\ 0&\overrightarrow{r}\
otin V\end{array} \right. \tag{21}\]  
In Equation (20), letting an arbitrary point \(\overrightarrow{r}\) in the acoustic field be at node \(\overrightarrow{r}_{b}\), then  
\[[A_{b}]\{p_{ai}\}=[B_{b}]\{\frac{\partial p_{ai}}{\partial n}\} \tag{22}\]  
where the elements of column \(i\) of the coefficient matrices \(A_{b}\), \(B_{b}\) are, respectively,  
\[A_{bi}=\delta_{bi}\Bigg{(}1+\frac{1}{4\pi}\int_{\Omega}\frac{\partial}{ \partial\eta}(\frac{1}{|\overrightarrow{r}_{b}-\overrightarrow{r}_{a}|}) \text{d}\Omega\Bigg{)}-\Bigg{(}\int_{\Omega}N_{i}(\overrightarrow{r}_{a}) \frac{\partial G(\overrightarrow{r}_{b},\overrightarrow{r}_{a})}{\partial \eta}\text{d}\Omega\Bigg{)} \tag{23}\]  
\[B_{bi}=-\int_{\Omega}N_{i}(\overrightarrow{r}_{a})G(\overrightarrow{r}_{b}, \overrightarrow{r}_{a})\text{d}\Omega \tag{24}\]  
Applying Equation (22) to all the \(n_{a}\) nodes on the boundary \(\Omega\) yields the boundary solution equation:  
\[[A]\{p_{ai}\}=[B]\{\frac{\partial p_{ai}}{\partial n}\} \tag{25}\]  
where \(A\), \(B\) are \(n_{a}\times n_{a}\) matrices whose \(i\) row vectors are \(A_{bi}\) and \(B_{bi}\), respectively. Upon solving for \(p_{a}\) using Equation (25), their summation with \(p_{b}\) constitutes the total fluctuating pressure.