# Pressure Fluctuation and Flow-Induced Noise of the Fin and Rudder in a Water Tunnel
## 3 Method
### Mesh Independence  
In order to obtain a computational model with mesh independence, models with different CFD mesh types, namely, coarse mesh, medium mesh, and fine mesh, were used to predict the flow field. The mesh number of these models increased gradually. By comparing the numerical results, a CFD mesh that meets the requirements for flow field prediction can be obtained. As an example, a midsection view of the medium CFD mesh with a mesh number of 30 million is shown in Figure 8. To better capture the flow characteristics and details of the flow field, mesh refinement is performed in the vicinity of the wall and areas with significant velocity gradient changes, such as the fin and rudder and the gap regions. Considering noise generation in the wake region, mesh refinement was applied to the region 2L downstream of the trailing edge of the fin and rudder. In the region far from the fin and rudder, a larger mesh size was employed to reduce the computational resources required. A sectional view of the local mesh and details of the gap between the fin and the rudder are shown in Figure 9. This figure shows that the gap between the fin and the rudder is filled with mesh, and the transition of mesh size is relatively smooth. The first layer of mesh on most of the fin and rudder satisfies a normal thickness with \(y+<1\), and 35 mesh layers are generated outward using a growth rate of 1.2 for height. After the prediction of the flow field, the drag coefficient and the pressure coefficient of the fin and rudder were calculated. The simulation results with different meshes are shown in Table 5. Considering the negligible differences in the drag coefficient and pressure coefficient compared to those with 37 million meshes, the medium CFD mesh was chosen for the flow field computation. The time-domain pressure at monitoring point #1 with a flow velocity of 5 m/s is shown in Figure 10a. To analyze the frequency characteristics of the pressure fluctuation, the temporal data of the pressure were processed using the Fast Fourier Transform (FFT) algorithm. To minimize spectral leakage and enhance the spectral resolution, the Hanning window was applied. Figure 8: Sectional view of medium CFD mesh. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**CFD Mesh** & **Number of Meshes (Million)** & **Drag Coefficient** & **Pressure Coefficient** \\ \hline Coarse mesh & 15 & 0.01376 & 0.20386 \\ Medium mesh & 30 & 0.01293 & 0.20147 \\ Fine mesh & 37 & 0.01261 & 0.20142 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Drag coefficient and pressure coefficient results with different CFD meshes. Figure 9: Sectional view of the local mesh and details of the gap between the fin and the rudder. The third-octave spectrum divides the audible frequency range into bands and presents the results of spectral analysis in terms of band energy. Firstly, the center frequency of the third-octave bands is predefined, allowing for the upper and lower band limits of each octave band and the number of spectral lines within each band to be determined. By using Equation (8), the pressure root-mean-square values for different octave bands can be calculated. Applying Equation (9) enables the acquisition of the corresponding sound pressure levels within each octave band. These values are then assigned to the center frequencies of the respective octave bands. Using the obtained frequency domain data, we can construct a spectrum, with the vertical axis representing decibels (dB), resulting in the pressure fluctuation spectrum in the third-octave shown in Figure 10b. \[\overline{p}^{2}=\sum_{i=1}^{n}p_{i}^{2} \tag{8}\]  
\[SPL=10 log\Bigg{(}\frac{\overline{p}^{2}}{p_{ref}^{2}}\Bigg{)} \tag{9}\]  
where \(p_{i}\) represents the pressure fluctuation data or sound pressure of the spectral line; \(\overline{p}^{2}\) represents the pressure fluctuation or the sound pressure root-mean-square value within the octave band; and \(p_{ref}\) denotes the reference sound pressure, which is \(1\times 10^{-6}\) Pa for underwater conditions.