# Investigation on Aerodynamic Noise Characteristics of Coaxial Rotor in Hover
## 2 Methodology  
### CFD Method and Trim Model  
The CFD solver used in this paper was developed in Refs. [18; 19]. Navier-Stokes equations [21] in an inertial Cartesian coordinate system are employed to simulate the rotor flow field, which can be written as  
\[\frac{\partial}{\partial t}\iiint_{\partial V}WdV+\oiint_{\partial S}(F_{c}-F _{v})\cdot ndS=0 \tag{1}\]  
where \(W\) represents the vector of conservative variables, \(F_{c}\) is the vector of the convective flux, and \(F_{v}\) is viscous flux.  
The governing equations are spatially discretized by the finite volume method. The Roe scheme is employed to compute the inviscid flux terms. A three-order monotone upwind-centered scheme for conservation laws (MUSCL) [22] is applied to reconstruct variables in a control volume. The viscous flux terms are discretized by adopting a second-order central difference scheme. The one-equation Spalart-Allmaras turbulence model [23] is used to simulate the turbulent viscosity. The dual time-stepping approach is employed for temporal discretization. The implicit lower-upper symmetric Gauss-Seidel (LUSGS) [24] scheme is used for the calculation of the pseudo-timestep. In the following computations, the sub-iteration of the pseudo-timestep is set as 8 for the single rotor, and for the coaxial rotor, it is 20 to recover higher time accuracy. The physical timestep is set as 1440 in one resolution, meaning that each time interval corresponds to 0.25\({}^{\circ}\) azimuth.  
A moving overset mesh system is employed to simulate the contra-rotating of rotors. The mesh system consists of the Cartesian background mesh and the body-fitted blade mesh with C-O type. The mesh of the blade tip is refined spanwise. The background mesh in the region where the rotors lie is also refined to capture the details of rotor wake. The mesh system of blade and background is shown in Figure 1.  
The research is conducted with a two-bladed model rotor. The blades use NACA0012 airfoil sections with no twist. The parameters are given in Table 1. Each blade mesh has \(221\times 78\times 102\) points in the streamwise, normal, and spanwise directions, respectively. The background mesh has 235 \(\times\) 196 \(\times\) 235 points in the x, y, and z directions. The background mesh is refined in the region of rotor wake. The refined mesh spacing of the blade tip and background near the rotor tip is approximately 0.05 c for both, which is set according to the previous calculation experience of coaxial rotors. The total number of mesh points is approximately 17.85 million.  
In order to obtain a specific value of thrust and achieve torque balance, an efficient CFD trim model [18] for a coaxial rotor is adopted. This trim model was developed from the \"delta method\", which has been applied in the single rotor [25; 26]. In the model, the blade element moment theory (BEMT) and CFD solver of the coaxial rotor are coupled with each other, which can significantly improve the efficiency and ensure the accuracy. A detailed introduction to the trim model can be found in Refs. [18; 19]. The trimmed collective pitches for various rotor thrusts in hover are given in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Values** \\ \hline Rotor radius & 2.0 m \\ Rotor cutout & 0.212 R \\ Chord & 0.22 m \\ Number of blades & 2 + 2 \\ Twist & None \\ Blade tip Mach number & 0.5878 \\ Rotor vertical distance & 0.15 R \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the model coaxial rotor in this paper.  
Figure 1: Moving overset mesh system for the coaxial rotor.