# Accurate RANS Simulation of Wind Turbine Stall by Turbulence Coefficient Calibration
## 1 Introduction  
Aerodynamics is one of the most important topics in wind turbine technology. Accurate calculation of the aerodynamic loads is essential for the power prediction and the structural design. A large number of computational tools has been developed based on the blade element momentum (BEM) theory [1; 2; 3; 4], the prescribed/free vortex wake models (VWM) [5; 6], or the computational fluid dynamics (CFD) method [7; 8]. BEM and VWM are specially developed for rotary machinery, such as propellers, helicopter rotors, and wind turbine rotors. CFD is a general method that can be used to solve almost all common flow problems based on detailed numerical solutions of the Navier-Stokes equations and gives physical quantities of the whole computational domain. This brings many benefits to the computational research of wind turbines. For example, the rotational effect and the three-dimensional (3-D) effect are naturally included in CFD, and therefore the additional models are not required as in BEM. On the other hand, CFD still faces some problems, among which turbulence is the most intractable one, since it is too computationally expensive to be directly simulated. Turbulence plays an important role in the aerodynamic performance of wind turbines. It exists in the wind, in the blade boundary-layer, and in the wake behind the rotor. There are mainly two ways to deal with turbulence in modern CFD applications. One is the Reynolds Averaged Navier-Stokes equations (RANS) simulation, the other is the Large Eddy Simulation (LES). Although LES provides a much better description of turbulence compared to RANS, it is much more computationally expensive. In order to make the computational cost acceptable, LES is usually used together with the actuator disk (AD) or the actuator line (AL) for wind turbine CFD. The methods of AD-LES and AL-LES have made great progress in the study of rotor wake [9; 10]. However, they do not describe the geometry of rotors, and therefore the detailed flow around blades is not simulated. For simulations with blade geometry, RANS simulation is the most practical option. The simplification of turbulence in RANS increases the inaccuracy of CFD, especially for phenomena directly related to turbulence. The prediction of stall, which is determined by the simulation of turbulence in the boundary-layer, becomes one of the most challenging problems for RANS. Stall is related to flow separation that occurs on the suction side of an airfoil (or blade) when the angle of attack (AoA) exceeds a certain value. It can be divided into two stages of light stall and deep stall, corresponding to trailing edge flow separation and complete flow separation, respectively. Light stall decreases the slope of lift curve, while deep stall causes the lift to drop sharply from the peak value. A typical example of the inaccuracy of stall prediction is the RANS simulations of the S809 airfoil [11]. A large number of simulations using various turbulence models have been carried out since the end of the twentieth century. As early as 1994, [PERSON] [12] made RANS simulation of the S809 airfoil using a 2-D incompressible solver and the Baldwin-Lomax algebraic turbulence model. The predicted lift coefficient significantly exceeds the experimental data when light stall occurs. Researchers later attempted to improve the simulation by using various turbulence models such as \(k\)-\(\omega\) used by [PERSON] [13], S-A, SST and SA-DES used by [PERSON] [14], and SST and RNG \(k\)-\(\epsilon\) used by [PERSON] [15]. Their results repeatedly showed the same inaccuracy: predicting a late onset of stall and therefore overestimating the maximum lift coefficient. Instead of fully turbulent simulations, laminar/turbulent transitional simulations were carried out by other researchers. [PERSON] [16] specified a transition point, while [PERSON] [17] and [PERSON] [18] employed the [PERSON] model and the \(\gamma\)-\(Re\theta\) model [19], respectively.