# Study on Actuator Line Modeling of Two NREL 5-MW Wind Turbine Wakes
## 2 Theoretical Model  
### Basic Theory of Actuator Line  
The Navier-Stokes (N-S) equations and the actuator line concept are combined as follows:  
\[\frac{\partial V}{\partial t}+V\cdot\
abla V=-\frac{1}{\rho}\
abla p+v\
abla^{ 2}V+f_{\epsilon} \tag{1}\]  
\[\
abla\cdot V=0 \tag{2}\]  
where \(V\) is the wind speed; \(t\) is the time; \(\rho\) is the air density, and \(\rho=1.225\) kg/m\({}^{3}\) in this paper; \(p\) is the pressure; \(v\) is the kinematic viscosity, and it is set as \(1.5\times 10^{-5}\) m\({}^{2}\)/s; and \(f_{\epsilon}\) is the loading on the blade as the body force that is added into the momentum equations.  
The body forces are located along the actuator lines, which represent the blades, as shown in Figure 1, and they are calculated by the blade-element theory with two-dimensional airfoil data. The lift and drag acting on the blade element are, respectively:  
\[L=\frac{1}{2}C_{l}(\alpha)\rho{U_{rel}}^{2}cdr \tag{3}\]  
\[D=\frac{1}{2}C_{d}(\alpha)\rho{U_{rel}}^{2}cdr \tag{4}\]  
where \(L\) and \(D\) represent the lift and drag, respectively; \(C_{l}(\alpha)\) and \(C_{d}(\alpha)\) are the lift and drag coefficients, correspondingly; \(\alpha\) is the local angle of attack; \(U_{rel}\) is the local velocity relative to the blade; \(c\) is the chord length; and \(dr\) is the thickness of the blade element. Figure 2 shows that local velocity relative to the blade can be calculated by:  
\[U_{rel}=\sqrt{U_{z}{}^{2}+(\Omega r-U_{\theta})^{2}} \tag{5}\]where \(U_{z}\) and \(U_{\theta}\) are the axial and tangential velocity, respectively; \(\Omega\) is the rotating angular velocity; and \(r\) is the rotating radius. Additionally, the angle of attack is:  
\[\alpha=\phi-\gamma \tag{6}\]  
where \(\phi\) is the angle of the inflow and \(\gamma\) is the local pitch angle. Therefore, the force per spanwise unit length is:  
\[f=(L,D)=\frac{1}{2}\rho U_{rel}{}^{2}c(C_{l}\overset{\rightarrow}{e}_{L}+C_{d }\overset{\rightarrow}{e}_{D}) \tag{7}\]  
where \(\overset{\rightarrow}{e}_{L}\) and \(\overset{\rightarrow}{e}_{D}\) are the unit direction vectors of \(L\) and \(D\), respectively.  
As the source term in Equation (1) is given in the form of the curl of the load, it acts as a singular vorticity source along the rotor blades. To avoid singularity, a constant \(\varepsilon\) is added to adjust the strength of the regularization kernel function, and one form can be shown as:  
\[\eta_{\varepsilon}(d)=\frac{1}{\varepsilon^{3}\pi^{3/2}}\exp[-(\frac{d_{i}}{ \varepsilon})^{2}] \tag{8}\]  
where \(d_{i}\) is the distance between the measured point (\(x_{i}\), \(y_{i}\), \(z_{i}\)) and initial force points (\(x\), \(y\), \(z\)) on the blade. \(\eta_{\varepsilon}\) is the kernel function between the measured and initial force points, which is quite similar to the kernel function used in Smoothed Particle Hydrodynamics (SPH) [22]. Finally, \(f_{\varepsilon}\) on the nearby mesh can be calculated by:  
\[f_{\varepsilon}(x,y,z,t)=f\otimes\eta_{\varepsilon}=\sum_{j=1}^{N}f(x_{i},y_{ i},z_{i},t)\frac{1}{\varepsilon^{3}\pi^{3/2}}\exp[-(\frac{d_{i}}{\varepsilon})^{2}] \tag{9}\]  
where \(N\) is the number of neighbouring blade sections. It means that the directed force on each blade section can be interpolated to the neigbouring mesh nodes smoothly. More details on the solution of Equation (9) can be referred to in [10]. To add the tower and hub effect into ALM, these will be achieved by following a similar technology.  
Figure 1: Blades represented by actuator lines.  
Figure 2: Cross-sectional airfoil element.