## 2 CFD Simulation Method  
### Governing Equation and Solution Method  
In order to improve the accuracy of the calculation flow field, the N-S governing equations considering the gas viscosity and compressibility are used to solve the rotor flow field. Considering the continuity equations, momentum equations, and energy equations comprehensively, the conserved Reynolds-averaged Navier-Stokes equation of three-dimensional, viscous, unsteady, and compressible flow in the inertial coordinate system can be obtained:  
\[\frac{\partial}{\partial\mathbf{t}}\iint\limits_{V}\iint\limits_{W}\overset{ \rightarrow}{W}dV+\iint\limits_{S}\left(\overset{\rightarrow}{F}_{c}- \overset{\rightarrow}{F}_{v}\right)\cdot\overset{\rightarrow}{n}dS=0 \tag{1}\]  
where, \(\overset{\rightarrow}{W}=\left[\rho,\rho u,\rho v,\rho v,\rho e\right]^{T}\) is the conservation variable, \(\overset{\rightarrow}{F}_{c}=\left(f,g,h\right)\) is the convection flux, and \(\overset{\rightarrow}{F}_{v}=\left(a,b,c\right)\) is the viscous flux. Shear-Stress Transport (SST) K-Omega turbulence model [16] was used to estimate the eddy viscosity of the flow field. The problem of sensitivity to free-stream/inlet conditions was addressed by SST turbulence model. The SST model has seen fairly wide application in the aerospace industry, where viscous flows are typically resolved, and turbulence models are applied throughout the boundary layer. The turbulent eddy viscosity \(\mu_{t}\) is calculated as:  
\[\mu_{t}=\rho kT \tag{2}\]  
where, \(\rho\) is the density, \(T\) is the turbulent time scale. The transport equations for the kinetic energy \(k\) and the specific dissipation rate \(\omega\) are:  
\[\frac{\partial}{\partial\mathbf{t}}(\rho k)+\
abla\cdot(\rho k\overline{v})= \
abla\cdot[(\mu+\sigma_{k}\mu_{t})\
abla k]+P_{k}-\rho\beta^{*}f_{\beta^{*}}( \omega k-\omega_{0}k_{0})+S_{k} \tag{3}\]  
\[\frac{\partial}{\partial\mathbf{t}}(\rho\omega)+\
abla\cdot(\rho\omega \overline{v})=\
abla\cdot[(\mu+\sigma_{\omega}\mu_{t})\
abla\omega]+P_{\omega }-\rho\beta f_{\beta}\left(\omega^{2}-{\omega_{0}}^{2}\right)+S_{\omega} \tag{4}\]  
where, \(\overline{v}\) is the mean velocity, \(\mu\) is the dynamic viscosity, \(\sigma_{k}\) and \(\sigma_{\omega}\) are model coefficients, \(P_{k}\) and \(P_{\omega}\) are production terms, \(f_{\beta^{*}}\) is the free-shear modification factor, \(f_{\beta}\) is the vortex-stretching modification factor, \(S_{k}\) and \(S_{\omega}\) are the user-specified source terms, \(k_{0}\) and \(\omega_{0}\) are the ambient turbulence values that counteract turbulence decay. The advection upstream splitting method (AMSM) format in the upwind format is adopted. This format has both high computational efficiency and intermittent high resolution, low numerical dissipation and no entropy correction is required. Assuming that the unit volume of each control body \(V_{i,j,k}\) does not change with time,  
\[\frac{\partial}{\partial\mathbf{t}}\iint\limits_{V_{i,j,k}}W_{i,j,k}dV=V_{i,j, k}\frac{\partial W_{i,j,k}}{\partial\mathbf{t}} \tag{5}\]  
and the rotor is in the forward flight,  
\[\frac{\partial W_{i,j,k}}{\partial\mathbf{t}}=-\frac{1}{V_{i,j,k}}\iint \limits_{S}\left(F_{c}^{M}-F_{v}\right)\cdot ndS \tag{6}\]  
where, \(\iint\limits_{S}\left(F_{c}^{M}-F_{v}\right)\cdot\overset{\rightarrow}{n}dS\) is net flux on the boundary. In this paper, the dual-time advance method with higher computational efficiency is selected for time discretization, and the Runge-Kutta explicit solution method is used in the sub-iteration process. [PERSON] [38] explained in detail the dual-time method when using explicit advance. Figure 1 shows the flow chart of the dual-time advance format using the motion nested grid. This time-advance method is widely used in the unsteady CFD simulation of the rotor. When calculating the rotor flow field in the hovering flight, the initial conditions are as shown below:  
\[\left\{ \begin{aligned} &\rho=\rho_{\infty}\\ & p=p_{\infty}\\ & u=0\\ & v=0\\ & w=0\end{aligned} \right.