## 3 Development of Basic Aerodynamic Theories  
The basic equations governing the aerodynamics of HAWTs are derived in this section. The ADT, the RAST and the VRT are used in deriving useful equations for the analysis of the flow past HAWT rotors. It is assumed in all the three theories that the wind flows at right angles to the WT rotor. The WT analyzed in this work is an unshrouded commercial-scale WT and not a shrouded WT that in most cases is a small WT. The ADT [1; 2; 3], first thought of by [PERSON], considers the rotor of a HAWT as a uniformly loaded permeable disk of area \(A\), or a rotor with an infinite number of blades. The basic equations underlying the theory have been derived in Reference [2]. In this work, it is sufficient to state the resulting equations without proof. The pressure drop across the WT rotor is given by  
\[p_{ad}=\frac{\rho(U_{0}^{2}-U_{3}^{2})}{2}. \tag{1}\]  
The axial thrust on the rotor is given by  
\[T_{ax}=\pi R^{2}P_{ad}=\pi R^{2}\frac{\rho({U_{0}}^{2}-U_{3}^{2})}{2}, \tag{2}\]  
and the speed of the air flowing through the rotor is given by  
\[U_{R}=\frac{U_{0}+U_{3}}{2}. \tag{3}\]  
Thus, the speed of the air flowing through the actuator disk is the average of the free-stream speed far upstream the disk and the speed of the air in the wake of the disk. If the induced speed at the rotor is \(\Delta U\), then \(\Delta U=U_{0}-U_{R}\), the axial induction coefficient \(a\) is given by  
\[a=\frac{\Delta U}{U_{0}}=\frac{U_{0}-U_{R}}{U_{0}}, \tag{4}\]  
from which one has  
\[U_{R}=U_{0}(1-a). \tag{5}\]  
The wind speed at the wake of the rotor is  
\[U_{3}=U_{0}(1-2a). \tag{6}\]  
For wind speeds that vary with the radial distance from the hub to the tip of the rotor, the actuator disk can be discretized as shown in Figure 1. If an element of the disk at radius \(r\) and thickness \(dr\) is considered, then the area of the element is given by \(A(r)=2\pi rdr\). The corresponding speeds for the elemental disk are \(U_{0}(r)\), \(U_{R}(r)\) and \(U_{3}(r)\), respectively. The pressure drop across the disk element is  
\[p_{ad}(r)=\frac{\rho(U_{0}^{2}(r)-U_{3}^{2}(r))}{2}. \tag{7}\]  
The axial induction factor at each disk element \(a(r)\) is given by  
\[a(r)=\frac{U_{0}(r)-U_{R}(r)}{U_{0}(r)}, \tag{8}\]  
from which one has  
\[U_{R}(r)=U_{0}(r)(1-a(r)). \tag{9}\]  
Additionally,  
\[T_{ax}=\pi R^{2}P_{ad}=\pi R^{2}\frac{\rho({U_{0}}^{2}-U_{3}^{2})}{2}, \tag{2}\]0The elemental or local axial thrust is given by  
\[T_{ax}=\pi R^{2}P_{ad}=\pi R^{2}\frac{\rho({U_{0}}^{2}-U_{3}^{2})}{2}, \tag{2}\]1  
and the local thrust coefficient per unit span is given by  
\[T_{ax}=\pi R^{2}P_{ad}=\pi R^{2}\frac{\rho({U_{0}}^{2}-U_{3}^{2})}{2}, \tag{2}\]2  
The local power captured is given by  
\[T_{ax}=\pi R^{2}P_{ad}=\pi R^{2}\frac{\rho({U_{0}}^{2}-U_{3}^{2})}{2}, \tag{2}\]3  
The local power coefficient is given by  
\[T_{ax}=\pi R^{2}P_{ad}=\pi R^{2}\frac{\rho({U_{0}}^{2}-U_{3}^{2})}{2}, \tag{2}\]4  
The elemental or local tip speed ratio is  
\[T_{ax}=\pi R^{2}P_{ad}=\pi R^{2}\frac{\rho({U_{0}}^{2}-U_{3}^{2})}{2}, \tag{2}\]5  
The RAST predicates on the generalized momentum theory of [PERSON] (1929) [2; 3; 24]. It is assumed that the pressure head at the far wake is equal to the pressure head in the free-stream, that pressure, wake rotation and the induction factors depend on the elemental or local annular radius and that the rotor is uniformly loaded. Additionally, it is assumed that the torque generated by the rotor is equal to the change in the angular momentum per unit time of the wake. All the assumptions made for the ADT are valid for the flow in the axial direction. If the angular speed of the rotor is \(\omega\) and the induced angular speed on the rotor by the wake is \(\omega^{\prime}\), then the angular speed on the rotor relative to the wake-induced angular speed is \(\omega+\omega^{\prime}\). It is worth noting that the wake rotates in the opposite direction to the rotor.