## 2 Methodology
### Beddoes-Leishman Semi-Empirical Model
#### 2.3.3 Vortex Force  
The induced vortex force and the associated pitching moment are represented empirically in a time-dependent manner during dynamic stall. The total accumulated vortex normal force coefficient under unsteady conditions is given by the following:  
\[C_{n}^{p}=C_{n}^{p}(s-\Delta s)\exp(-\Delta s/T_{v})+[C_{v}(s)-C_{v}(s-\Delta s )]\exp(-\Delta s/2T_{v}) \tag{15}\]  
where  
\[C_{v}(s)=C_{n}^{c}(s)\left[1+\left(\frac{1+\sqrt{f^{\prime\prime}}}{2}\right) ^{2}\right]. \tag{16}\]  
\(T_{v}\) is the vortex time decay constant. The center of pressure on the airfoil also varies with the chordwise position of the shedding vortex and will achieve a maximum value when the vortex reaches the trailing edge after a non-dimensional time period \(T_{vl}\). A general representation of the center of pressure behavior (aft of the quarter-chord) is empirically formulated as follows:  
\[\left(CP\right)_{v}=0.25\bigg{(}1-\cos\frac{\pi T_{v}}{T_{vl}}\bigg{)} \tag{17}\]  
where \(T_{v}\) denotes the non-dimensional vortex time, \(0\leq T_{v}\leq 2T_{vl}\) (i.e., \(T_{v}=0\) at the onset of dynamic stall and \(T_{v}=T_{vl}\) when the vortex reaches the trailing edge). Thus, the increment in the pitching moment about the quarter-chord due to dynamic stall is given by the following:  
\[C_{m}^{p}=-\left(CP\right)_{v}\cdot C_{n}^{p}. \tag{18}\]  
The total unsteady air loads on the airfoil is then obtained by superposition as follows:  
\[ \begin{array}{l}C_{n}=C_{n}^{f}+C_{n}^{\mathrm{nc}}+C_{n}^{p}\\ C_{t}=C_{t}^{f}+\left(C_{n}^{\mathrm{nc}}+C_{n}^{p}\right)\sin a_{e}^{\mathrm{ c}}\\ C_{m}=C_{m}^{\mathrm{c}}+C_{m}^{\mathrm{nc}}+C_{m}^{f}+C_{m}^{p}\end{array}  \tag{19}\]  
Apart from dynamic stall, rotational augmentation is of crucial importance for wind turbine aerodynamic performance prediction. The rotational stall delay model of [PERSON] et al. (2018) is implemented. This model was developed on the difference in chordwise pressure distribution between the rotating blade section and the static airfoil. Although the obtained lift and drag coefficients were generally good, the modeled pitching moment coefficient indicated a poor agreement with the measurements, because the center of pressure was often underpredicted at large AOAs. Thus, a simple modification that the integrated location is changed from 0.25 to 0.1 was made to the pitching moment coefficient when fully separated flow occurs. Figure 4 implies an acceptable performance of this modification. It is generally acknowledged that the rotational augmentation is relatively apparent in separated flow; therefore, rotational augmentation mainly affects the nonlinear air loads. To incorporate the rotational stall delay model into the dynamic stall model, the static airfoil data are corrected by [PERSON]'s model firstly. Then, the static separation point is obtained from the Kirchhoff formulation (Equation (12)). The dynamic separation point is calculated by a first-order lag of the static separation point (Equation (10)), while the nonlinear normal force is reconstructed by Equation (12). Afterwards, the chordal force coefficient is obtained by Equation (13), with an increment added to consider the rotational augmentation. Finally, the nonlinear pitching moment coefficient is obtained by Equation (14), and a rotational increment is added as well. According to the recommendations of [PERSON] et al. (2018) and [PERSON] et al. (2018), the set of corresponding empirical time constants, \(T_{p}\), \(T_{f}\), \(T_{v}\), \(T_{vl}\), is as follows: 1.7, 8.0, 1.0, and 7.0, where \(T_{p}\) and \(T_{f}\) are related to the first-order lag of the leading-edgeseparation and trailing-edge separation, respectively, and both \(T_{v}\) and \(T_{vl}\) are the non-dimensional time constants related to the movement of the dynamic vortex. The difference between PO and OF is addressed in the circulatory part of unsteady linear air loads, based on the thin-airfoil theory.