## 2 Problem Formulation and Simulation Methods  
We studied a fully developed turbulent channel flow. The top-wall and bottom-wall are no-slip boundary and superhydrophobic surfaces, respectively. Superhydrophobic surfaces are streamwise strip composed of alternating no-slip and free-slip boundary conditions. The space of strip is W. The length of microstructure which contains a no-slip strip and a free-slip is L. The details are shown in Figure 1a. The channel flow is governed by incompressible Navier-Stokes equation. \[\
abla\cdot u=0, \tag{1}\]  
\[\rho\frac{Du}{Dt}=-\
abla p+\mu\
abla^{2}u, \tag{2}\]  
where \(\mu\) is dynamic viscous and \(\rho\) is density of water. We have simplified the boundary condition of superhydrophobic surfaces. Therefore, the liquid-solid interface and the liquid-air interface are equal to no-slip and free-slip boundary condition. The velocity distribution profiles of slip boundary are shown in Figure 1b, which satisfies the Navier slip equation. \[u_{s}=L_{S}\frac{\partial u}{\partial y}|_{wall}, \tag{3}\]  
where \(u_{s}\) is slip velocity and \(L_{S}\) is slip length. This simplified method is proven to be reliable. When the liquid-air interface deformation is negligible compared with the width of microstructures, the gas-liquid viscosity ratio is sufficiently small, and the depth is equal to the width of the microstructure. The reasonability of the hypothesis has been verified [25; 28; 29; 41]. Although this assumption improves the drag reduction effect, the overall trend of Reynolds stress is still consistent in the prediction of turbulence characteristics. Therefore, the alternating boundary conditions of no-slip and free-slip have been used in this work to simulate superhydrophobic surfaces with streamwise strips. Through the change of the percentage of free-slip boundary conditions and the number of strips, the flow noise of superhydrophobic surfaces is calculated. The size of flow field is set to \(7\delta\times 2\delta\times 3.5\delta\) at streamwise, wall-normal, and spanwise direction, respectively. Where \(\delta\) is the half height of channel. The flow Reynolds number in this work is \(\text{Re}=U_{c}\delta/\
u=4200\) and the friction Reynolds number is \(\text{Re}_{\tau_{0}}=u_{\tau}\delta/\
u\approx 180\), where \(U_{c}\) is the velocity in channel centerline and \(u_{\tau}\) is wall friction velocity. The flow region size is consistent with that of [PERSON] and [PERSON] [24] to ensure the accuracy of simulation. The top boundary is no-slip wall and the bottom boundary is superhydrophobic surfaces with alternating strips. Other sides are periodic boundary. The flow is driven by CPG. At the initial time, we produced some random turbulence to help the flow to fully develop. The details are shown in Figure 2.  
The large eddy simulation (LES) is a mathematical solved model for turbulence in computational fluid dynamics. This simulation method can achieve flow regimes more  
Figure 1: Schematic of streamwise strip on superhydrophobic surface. (**a**) Alternating free-slip and no-slip boundary condition. (**b**) Diagram of slip velocity. Figure 2: Diagram of flow domain size and boundary condition. precisely than the Reynolds-averaged Navier-Stokes equations method. The vortices are the main cause of flow noise according to the vortex sound theory. Each variable in LES is divided into two parts by the filter function. For example, the transient variables \(\phi\) is divided into \(\overline{\phi}\) and \(\phi^{\prime}\), which are referred to as the large- and the small-scale component, respectively. \[\overline{\phi}\int\limits_{D}\phi G\big{(}x,x^{\prime}\big{)}dx^{\prime}, \tag{4}\]  
where \(D\) refers to the flow field, \(x^{\prime}\) is the spatial coordinate in the original flow area, \(x\) is the filtered spatial coordinate on the large-scale space, and \(G(x,x^{\prime})\) is the filter function that determines the vortices scale.