## 3 Discussion
### Noise Reduction Effect of Strip Numbers  
Under the same GF, namely free-slip boundary fraction on superhydrophobic surfaces, different numbers of microstructures in a spanwise period can generate different slip velocities and further influences the noise reduction effect. In this study, the velocity is united by the friction velocity \(u_{\tau}\), \(u^{\prime}v^{\prime}\), and other second-order variables are united by \(u_{\tau}^{2}\). Then, our results are compared with the logarithmic law. The obtained velocity distribution near the wall has a certain deviation due to the accuracy of LES, and the parameters in the logarithmic law need to be changed. \[u^{+}=\frac{1}{\kappa}\ln\bigl{(}y^{+}\bigr{)}+C \tag{10}\]  
In this equation, \(\kappa=0.41\) is Karmen constant and \(C\) in our study is 0.65. After time average in a flow time period and space average in the streamwise and spanwise direction, the following calculation results are obtained, as shown in Figure 13. The near-wall velocity profile corresponds to the logarithmic law. At the range of \(y^{+}<10\), the linear law of \(y^{+}=u^{+}\) is satisfied ; at the range of \(y^{+}>10\), the logarithmic law of \(u^{+}=2.44\ln(y^{+})+6.5\) is satisfied. Figure 12: Comparison of vorticity for different GF at \(Strip\ number=8\). (**a**) \(GF=0.5\). (**b**) \(GF=0.6\). (**c**) \(GF=0.7\) (**d**) \(GF=0.8\). Under the same \(GF\), we calculated scenarios of different numbers \(n\) of free-slip strips in a spanwise period. Notably, \(n=1,2,4,8\). Figures 13 and 14 are the near-wall velocity profiles on superhydrophobic surfaces at \(GF=0.5\) and no-slip wall, respectively. The slip velocity can be observed on superhydrophobic surfaces at \(GF=0.5\). Furthermore, it decreases with the increase in the number of free-slip stripes. Figure 13b indicates that the root mean square error (RMSE) of velocity on superhydrophobic surface has contact with the number of free-slip stripes. The RMSE of the near-wall velocity is greater and the fluctuation of the fluid is stronger when more free-slip strips are generated in a period. This condition results in a larger Reynolds stress value in the logarithmic region. Figure 13c shows that the Reynolds stress increases significantly at \(y^{+}=30\) when \(n=1\). However, when the \(n=2,4,8\), the Reynolds stress increases significantly at \(y^{+}=10\). These changes show that the enhancement in velocity fluctuation is affected by Reynolds stress. As the strip number increases, the influence range of Reynolds stress increases. At the same time, the thickness of viscous sublayer reduces, and the range of log law region increases. As a result, the near-wall velocity profiles are shown in Figure 13a. Similarly, velocity distribution on the top wall has same changes. We calculate the flow noise of channel flow at CPG condition and the noise level comparation of different strip numbers at the same GF is shown in Figure 15. The embedded graph in the upper right corner of Figure 15 shows the curve of overall sound pressure level in different scenarios. Figure 14: Comparison of statistics for \(GF=0.5\) and \(Strip\ number=1,2,4,8\) at the top wall of the channel (no-slip surface). (**a**) Mean streamwise velocity profile. (**b**) Velocity RMSE fluctuations. (**c**) Reynolds stress. Figure 13: Comparison of statistics for \(GF=0.5\) and \(Strip\ number=1,2,4,8\) at the bottom wall of channel (superhydrophobic surface). (**a**) Mean streamwise velocity profile. (**b**) Velocity RMSE fluctuations. (**c**) Reynolds stress. When \(GF=0.5\), the overall sound pressure level(OSPL) has a significant reduction compared with that in the no-slip channel at different strip numbers \(n\). The reason is that the free-slip boundary cannot produce a viscous sublayer, the near-wall flow does not have a velocity gradient. Therefore, the energy exchange decreases at different fluid layers and the flow is more stable. As a result, the noise level has a significant reduction. When \(GF=0.5\), the strip numbers \(n=1,2,4,8\) have apparent noise reduction effect.