# Evaluation of Tip Loss Corrections to AD/NS Simulations of Wind Turbine Aerodynamic Performance
## 2 Axisymmetric AD/NS Method
### Force on Actuator Disc  
The conceptual idea of the AD/NS method is to solve the aerodynamic force of rotor blades by using the blade-element theory and then applying its counterforce as an external body force into the momentum equations. At each time step of solving the NS equations, the velocity passing through the actuator disc is detected and used to determine the axial induced velocity \(W_{a}\) and the tangential induced velocity \(W_{t}\),  
\[W_{a}=V_{0}-u_{z}, \tag{5}\]  
\[W_{t}=-u_{\theta}. \tag{6}\]  
The axial interference factor \(a\) and tangential interference factor \(a^{\prime}\) are then determined by,  
\[a=\frac{W_{a}}{V_{0}}, \tag{7}\]  
\[a^{\prime}=\frac{W_{t}}{\Omega r}, \tag{8}\]  
where \(V_{0}\) is the wind speed, and \(\Omega\) is the rotating speed of the wind turbine rotor. The above interference factors are azimuthally unchanged according to the axisymmetric condition, which implies an infinite number of blades and zero tip loss. In order to estimate the tip loss of the realistic rotor with a finite number of blades, the interference factors need to be corrected as  
\[\bar{a}=f_{corr}(a), \tag{9}\]  
\[\bar{a}^{\prime}=f^{\prime}_{corr}(a^{\prime}), \tag{10}\]  
where \(\bar{a}\) and \(\bar{a}^{\prime}\) denote the corrected axial and tangential interference factors, \(f_{corr}\) and \(f^{\prime}_{corr}\) represent correction functions. The axial and tangential induced velocities after the correction are written as  
\[\tilde{W}_{a}=V_{0}\bar{a}, \tag{11}\]  
\[\tilde{W}_{t}=\Omega r\bar{a}^{\prime}. \tag{12}\]  
According to the velocity triangle shown in Figure 2, the flow angle \(\phi\), angle of attack \(\alpha\), and relative velocity \(V_{rel}\) are then determined by  
\[\phi=\tan^{-1}\left(\frac{V_{0}-\tilde{W}_{a}}{\Omega r+\tilde{W}_{t}}\right), \tag{13}\]  
\[\alpha=\phi-\beta, \tag{14}\]  
\[W_{t}=-u_{\theta}. \tag{6}\]0  
where \(\beta\) is the local pitch angle of a blade cross-section. With the determined angle of attack and relative velocity, the lift coefficient \(C_{l}\) and drag coefficient \(C_{d}\) of each cross-section of the blade can be obtained from tabulated airfoil data. According to the relationship of force projection, the normal and tangential force coefficients are determined by  
\[W_{t}=-u_{\theta}. \tag{6}\]1  
\[W_{t}=-u_{\theta}. \tag{6}\]2  
The axial force \(F_{2}\) and tangential force \(F_{\theta}\) per radial length of the rotor are then given by  
\[W_{t}=-u_{\theta}. \tag{6}\]3  
\[W_{t}=-u_{\theta}. \tag{6}\]4  
where \(c\) is the local chord length, and \(B\) is the number of blades. Ignoring the effect of the radial flow, the vector form of the counterforce acting on the air is  
\[W_{t}=-u_{\theta}. \tag{6}\]5  
Rather than distributing the force only on the disc, the above force is regularized by the following Gaussian distribution along the axial direction in order to avoid a numerical singularity. The Gaussian distribution has been widely used and proven to be proper for AD/NS and AL/NS simulations [10; 14; 15; 16; 17]. \[W_{t}=-u_{\theta}. \tag{6}\]6  
where \(z_{0}\) is the axial position of the disc. The parameter \(\varepsilon\) serves to adjust the concentration of the regularized force, which is in the present study set to \(\varepsilon=0.02R\) where \(R\) is the rotor radius. The source terms of the external body force in the momentum Equations (2)-(4) need to be replaced with the above force per unit volume. In the axisymmetric coordinate system, a 2D grid cell with an axial length of \(\Delta z\) and a radial height of \(\Delta r\) represents a volume of \(2\pi r\Delta r\Delta z\), and thus, the force per unit volume is  
\[W_{t}=-u_{\theta}. \tag{6}\]7  
Using Equations (20) and (21), it reads  
\[W_{t}=-u_{\theta}. \tag{6}\]8  
\[W_{t}=-u_{\theta}. \tag{6}\]9 Illustration of the velocity and the aerodynamic force for a blade cross-section. i.e.,  
\[\left\{ \begin{array}{c}f_{z}=-\frac{\eta_{e}F_{z}}{2\pi r}\\ f_{r}=0\\ f_{\theta}=-\frac{\eta_{e}F_{\theta}}{2\pi r}\end{array} \right.. \tag{24}\]