# Accurate RANS Simulation of Wind Turbine Stall by Turbulence Coefficient Calibration
## 2 Simulation Methods  
In the present study, 2-D airfoil simulation and 3-D rotor simulation are carried out using ANSYS FLUENT on the S809 airfoil and the NREL (National Renewable Energy Laboratory) Phase VI rotor, respectively. The involved governing equations are the incompressible RANS equations. A comparison between an incompressible solver and a compressible solver was made by [PERSON] (2008) for the NREL Phase VI rotor. The results of the two solvers were almost identical. Therefore, it is reasonable to ignore the compressibility of air in the present study. Two kinds of 2-D computational domains are generated for the airfoil simulation. One contains the walls of the wind tunnel's experimental section, the other does not. The outer boundaries of the domain without tunnel walls are set to more than 20 times the chord length away from the airfoil. In each domain, 430 grid nodes are properly distributed on the airfoil surface, and about 100,000 grid cells are employed. The initial height of the first grid layer measured from the airfoil surface is about \(5\times 10^{-6}\) of the chord length. The values of \(y^{+}\) (see Figure 1) are found less than 1.0 on most areas of the airfoil surface, indicating that the initial height of the first grid layer is small enough to capture the boundary layer. The computational settings of the airfoil simulation are summarized in Table 1.  
The computational domain for the rotor simulation is shown in Figure 2. A pair of rotational periodic surfaces is defined to reduce the computational domain to a half, taking advantage of the axial symmetry of the rotor geometry. The grid nodes on the two periodic surfaces would overlap one to one if one surface were rotated \(180^{\circ}\) around the axis. The far boundaries are set to more than 15 times the rotor diameter away from the rotor, with no consideration of wind tunnel walls. The computational settings of the rotor simulation are summarized in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Items** & **Options** & **Settings** \\ \hline \multirow{3}{*}{Models} & Space & 2D \\  & Time & Steady \\  & Viscous & SST k-omega \\ \hline \multirow{2}{*}{Pressure-Velocity Coupling} & Type & Coupled \\  & Courant Number & 50 \\ \hline \multirow{3}{*}{Discretization Scheme} & Pressure & Second Order \\  & Momentum & Second Order Upwind \\ \cline{1-1}  & Turbulent Kinetic Energy & Second Order Upwind \\ \cline{1-1}  & Specific Dissipation Rate & Second Order Upwind \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational settings of the airfoil simulation. Figure 1: The wall \(y^{+}\) distribution around the airfoil at angles of attack of \(10^{\circ}\), \(20^{\circ}\) and \(30^{\circ}\) (Re = \(1\times 10^{6}\)). A high-quality mesh with enough cells and properly distributed nodes is essential for a reliable numerical simulation. In the present study, the mesh independency has been tested for both the airfoil and the rotor simulations. The five meshes involved in the rotor test are shown in Table 3, where they are denoted by M01, M02, M03, M04, and M05, respectively. The initial height of the grid cells adjacent to the blade is set to 5 \(\times\) 10\({}^{-6}\) meter equally for all of the five meshes, according to the criterion of y\({}^{+}\) < 1.0. The simulation results of M03, M04, and M05 are found very close to each other, and the M03 mesh is preferred, since it has the smallest number of cells among the three. For a further verification of sufficient mesh resolution, the Richardson extrapolation [35; 36] is applied to the predicted rotor torque from the simulations using M01, M02, and M03. Table 4 shows that the order of accuracy of the Richardson extrapolation is greater than 2 in conditions of various wind speeds. That means the M03 mesh (demonstrated in Figure 3) is fine enough for the present simulation based on the second order upwind scheme, and therefore it is finally adopted.