## 4 Numerical Simulation Results and Analysis
### Simulation Analysis  
Figure 4(a) shows the pressure distribution of the smooth, riblet, and placoid-scale skin in the flow fields.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Description** & **Grid 1** & **Grid 2** & **Grid 3** & **Grid 4** & **Grid 5** & **Grid 6** \\ \hline Grid number & \(6.8\times 10^{5}\) & \(1.15\times 10^{6}\) & \(1.62\times 10^{6}\) & \(1.90\times 10^{6}\) & \(2.07\times 10^{6}\) & \(2.27\times 10^{6}\) \\ \(C_{f}\) & 0.09967 & 0.09407 & 0.09041 & 0.08953 & 0.08952 & 0.08951 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of grid convergence analysis.  
Figure 4: Processed products for different skin types.  
In Figure 5, the three skins generate pressure vortices of varying sizes (highlighted by red circles in Figure 5a). The presence of these pressure vortices can lead to disparities in pressures across different regions, subsequently influencing structural vibration and other factors. Smooth skin produces a pressure vortex with substantial negative-pressure regions at its core, which may result in intense vibrational effects and potentially severe fluctuations in sound pressure. This may lead to an increase in low-frequency flow noise. In contrast, both riblet skin and placoid-scale skin transform larger single-pressure vortices into two separate negative-pressure regions (indicated by red circles in Figure 5a). The reduction in pressure vortices is more pronounced for placoid-scale skin as its negative-pressure region and numerical values are considerably diminished.  
The velocity distribution in the flow fields of smooth, riblet, and placoid-scale skin is shown in Figure 5b.  
The velocity distribution enables the observation of speed changes between the different skin types, providing a more direct representation of shedding vortices. The three skins exhibit certain degrees of shedding vortex. Smooth skin has the strongest intensity, with highly concentrated regions of shedding vortices. Placoid-scale skin demonstrates larger shedding vortex regions, but with an evident decrease in intensity compared to smooth skin. Riblet skin exhibits the weakest shedding vortex, characterized by smaller regions and significantly lower intensity. The riblet skin reduces the shedding vortices level by level, resulting in a weaker vortex.  
Upon examining the TBL of the three skin types, it becomes apparent that thickness follows specific pattern: placoid-scale skin has the lowest thickness, followed by smooth skin, while riblet skin possesses the highest thickness. The varying TBL thickness also highlights the effect of passive fluid control on the three skins, which is demonstrated by the strength of nonlinear changes when a fluid flows through structural surfaces. The numerical simulation results suggest that placoid-scale skin should have a more pronounced effect on reducing flow noise.  
To illustrate the vortex alterations as fluid flows through various skin surfaces, Figure 6 presents the vortex diagrams for the flow fields of smooth, riblet, and placoid-scale skin.  
Figure 5: Pressure distribution and velocity distribution. (**a**) Pressure distribution, (**b**) velocity distribution.  
The vortex distribution provides comprehensive and intuitive visualization of the vortex generation and dissipation process of various structural surfaces. The vortex distribution for both smooth skin and riblet skin exhibits a decreasing trend from the inflow to the outflow direction, accompanied by a gradual reduction in vortex intensity along the surface. In the vortex diagram of the placoid-scale skin, the vortex weakens through two distinct stages (labeled 1 and 2 in Figure 6c).