## 3 Discussion
### Noise Reduction Effect of Free-Slip Fraction  
This section mainly studies the influence of GF on the drag and the noise reduction effect of superhydrophobic surfaces. The simulation of the free-slip surface with different proportions is realized by controlling the wall shear stress at the bottom of the channel in a period.  
Figure 16 shows the velocity distribution near the wall at different \(GF=0.5,0.6,0.7,0.8\) with same numbers of spanwise strips. The results show that slip velocity is greater when  
Figure 15: Comparison of frequency spectra for different strips number at \(GF=0.5,0.6,0.7,0.8\). (**a**) \(GF=0.5\). (**b**) \(GF=0.6\). (**c**) \(GF=0.7\) (**d**) \(GF=0.8\).  
the strip number is smaller, and the GF is larger. Under CPG conditions, we define the drag reduction rate as Equation (11)  
\[DR=\frac{U_{0}-U_{i}}{U_{0}} \tag{11}\]  
where \(U_{0}\) is the average velocity of ordinary no-slip channel, and \(U_{i}\) is the average velocity of superhydrophobic channels with different parameters. The results are shown in Figure 17.  
At \(GF=0.8\), DR reaches a maximum value of 32%. When the strip number in a spanwise period is 8, the drag reduction effect is approximately 16%, as shown in Figure 17. When \(n\) is 1, the velocity RMSE and Reynolds stress of different GF are shown in Figure 18. As the velocity RMSE and the Reynolds stress increase, the linear sublayer becomes thinner. Otherwise, the increase in strip number near the wall accelerates the exchange of fluid momentum and forms more small vortices, therefore the shear stress between the wall and the fluid are increased. Then, the drag reduction effect is decreased.  
Figure 16: Comparison of mean velocity for same strips number at different GF. (**a**) \(Strip\ number=1\), \(GF=0.5,0.6,0.7,0.8\). (**b**) \(Strip\ number=2\), \(GF=0.5,0.6,0.7,0.8\). (**c**) \(Strip\ number=4\), \(GF=0.5,0.6,0.7,0.8\). (**d**) \(Strip\ number=8\), \(GF=0.5,0.6,0.7,0.8\).  
Figure 19 is the noise level under different GF when the number of strips is fixed. As shown in Figure 19, the superhydrophobic surfaces with different GF have a certain noise reduction effect. Under the same strip number \(n\), the noise reduction effect increases firstly as the GF increases and then decreases. When \(GF=0.8\), the increase of the overall flow velocity in the channel increases the overall noise level and makes the noise reduction effect of superhydrophobicity no longer obvious. In summary, both GF and strip number should have a suitable range. If the strip number is too small, the gas layer must be unstable. If the GF is too large, the noise level increases. Therefore, comprehensive consideration should be given to noise reduction and drag reduction effect when the superhydrophobic surface is used. The overall sound pressure level is exhibited in Table 4. When strips number is 4 and the GF is 0.5, the best noise reduction effect are obtained as 10.7 dB.  
Figure 17: Drag reduction effect of superhydrophobic surfaces at different scenarios. (\(GF=0.5,0.6,0.7,0.8\), \(Strip\)\(number=1,2,4,8\)).  
Figure 18: Comparison of statistics for \(Strip\)\(number=1\) at different GF. (**a**) Velocity RMSE. (**b**) Reynolds number.