## 2 Problem Formulation and Simulation Methods
The filter function represented by the above-mentioned formula is applied into the Navier-Stokes and the continuity equations under transient conditions. The equation can be obtained as follows:  
\[\frac{\partial}{\partial t}(\rho\overline{u_{i}})+\frac{\partial}{\partial x_ {j}}\big{(}\rho\overline{u_{i}u_{j}}\big{)}=-\frac{\partial p}{\partial x_{i} }+\frac{\partial}{\partial x_{j}}\Bigg{(}\mu\frac{\partial\overline{u_{i}}}{ \partial x_{j}}\Bigg{)}-\frac{\partial\tau_{ij}}{\partial x_{j}}, \tag{5}\]  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho\overline {u_{i}})=0, \tag{6}\]  
Equations (5) and (6) constitute the governing equations in the LES. The oversored variable in Equation (6) denotes the filtered field variable. \(\tau_{ij}\) is defined as follows,  
\[\tau_{ij}=\rho\overline{u_{i}u_{i}}-\rho\overline{u_{i}u_{j}}, \tag{7}\]  
where \(\tau_{ij}\) refers to the subgrid-scale stress, which embodies the effect of the small-scale vortex motion on the governing equations. The simulation method is LES and the sub-grid stress model is Wall-Adapting Local Eddy-viscosity (WAIE) model. The space discretization method is central difference. The grid is Cartesian with uniform spacing in the streamwise and spanwise directions, and stretching in the wall-normal direction. The grids number of no-slip boundary channel flow is 150 \(\times\) 60 \(\times\) 80. Other scenarios are shown in \[\rho\frac{Du}{Dt}=-\
abla p+\mu\
abla^{2}u, \tag{2}\]0  
According to the LES, the \(y^{+}\) is preferably less than 1 and the time step size need to satisfy the Courant-Friedrichs-Lewy (CFL) condition. Our time step size is \(\Delta t=1\times 10^{-5}\) and \(y^{+}\approx 0.3\). The near-wall velocity profile is verified with the logarithmic law as shown in \[\rho\frac{Du}{Dt}=-\
abla p+\mu\
abla^{2}u, \tag{2}\]1 The grid and Wale constant independence verification are shown in \[\rho\frac{Du}{Dt}=-\
abla p+\mu\
abla^{2}u, \tag{2}\]2 We have calculated the drag coefficients of four sets of grid. The results show that under the condition of \(y^{+}\approx 0.3\), the change of grid number have a slight influence on the accuracy of flow field. Two experiential Wale constants have been chosen for verification and the results show that the Wale constant is appropriate. \begin{table}
\begin{tabular}{l c c c c c c c} \hline
**Domain Size** & **Gas Fraction** & **Strip Number** & **Grid Number (\(N_{x}\times N_{y}\times N_{z}\))** & \(Re_{\tau}\) & \(x^{+}\approx\) & \(y^{+}\approx\) & \(z^{+}\approx\) \\ \hline \multirow{4}{*}{\(7\delta\times 2\delta\times 3.5\delta\)} & GF = 0.5 & 1,2, 4, 8 & 150 \(\times\) 60 \(\times\) 80 & 180 & 3 & 0.3 & 3 \\  & GF = 0.6 & 1, 2, 4, 8 & 150 \(\times\) 60 \(\times\) 80 & 180 & 3 & 0.3 & 3 \\  & GF = 0.7 & 1, 2, 4, 8 & 150 \(\times\) 60 \(\times\) 96 The Lighthill analogy theory and hybrid numerical method [42; 43] are used to calculate the flow noise of channel. The velocity variable in the flow simulation is extracted and used as source term in Equation (8). \[\rho\frac{Du}{Dt}=-\
abla p+\mu\
abla^{2}u, \tag{2}\]3  
\[\rho\frac{Du}{Dt}=-\
abla p+\mu\
abla^{2}u, \tag{2}\]4  
where \(c_{0}\) is sound speed. \(Q\) and \(F\) are mass and momentum changes in unit time, respectively. \(T_{ij}\) is Lighthill stress tensor. In particular, at the calculation of flow noise, the Lighthill stress tensor can simplified to \(T_{ij}\approx\rho U_{i}U_{j}\). According to Equation (8), the flow field data can be used and interpolated to the sound calculation grids. Then, the Discrete Fourier Transform(DFT) is used to convert sound source from time domain to frequency domain. The finite element method is used to calculate the generation and propagation of flow noise. At the boundary, the infinite element method is used to simulate the no-reflection boundary. The accuracy of this method is proven by comparing with the flow noise experiment results of [PERSON] [44]. The flow noise of cavity in wind tunnel is tested, as shown in Figure 4 and \[\rho\frac{Du}{Dt}=-\
abla p+\mu\
abla^{2}u, \tag{2}\]5