The forces and the velocities acting in a Darrieus turbine are depicted in Figure 1. The resultant velocity vector, \vec{W}, is the vectorial sum of the undisturbed upstream air velocity, \vec{U}, and the velocity vector of the advancing blade, -\vec{\omega }\times\vec{R}. : \vec{W}=\vec{U}+\left( -\vec{\omega }\times\vec{R} \right) Thus the oncoming fluid velocity varies during each cycle. Maximum velocity is found for \theta =0{}^\circ and the minimum is found for \theta =180{}^\circ , where \theta is the azimuthal or orbital blade position. The
angle of attack, \alpha , is the angle between the oncoming air speed, W, and the blade's chord. The resultant airflow creates a varying, positive angle of attack to the blade in the upstream zone of the machine, switching sign in the downstream zone of the machine. It follows from geometric considerations of angular velocity as seen in the accompanying figure that: : V_t=R \omega + U\cos(\theta) and: : V_n=U \sin(\theta) Solving for the relative velocity as the resultant of the tangential and normal components yields: : W= \sqrt{V_t^2+V_n^2} Thus, combining the above with the definitions for the
tip speed ratio \lambda =(\omega R) /U yields the following expression for the resultant velocity: : W=U\sqrt{1+2\lambda \cos \theta +\lambda ^{2}} Angle of attack is solved as: : \alpha = \tan^{-1} \left( \frac{V_n}{V_t} \right) Which when substituting the above yields: : \alpha =\tan ^{-1}\left( \frac{\sin \theta }{\cos \theta +\lambda } \right) The resultant aerodynamic force is resolved either into
lift (L) -
drag (D) components or normal (N) - tangential (T) components. The forces are considered to be acting at the quarter-chord point, and the
pitching moment is determined to resolve the aerodynamic forces. The aeronautical terms
lift and
drag refer to the forces across (lift) and along (drag) the approaching net relative airflow. The tangential force acts along the blade's velocity, pulling the blade around, and the normal force acts radially, pushing against the shaft bearings. The lift and the drag force are useful when dealing with the aerodynamic forces around the blade such as
dynamic stall, boundary layer etc.; while when dealing with global performance, fatigue loads, etc., it is more convenient to have a normal-tangential frame. The lift and the drag coefficients are usually normalised by the dynamic pressure of the relative airflow, while the dynamic pressure of the undisturbed upstream fluid velocity usually normalises the normal and tangential coefficients. : C_{L}=\frac{F_L}{{1}/{2}\;\rho AW^{2}}\text{ };\text{ }C_{D}=\frac{D}{{1}/{2}\;\rho AW^{2}}\text{ };\text{ }C_{T}=\frac{T}{{1}/{2}\;\rho AU^{2}R}\text{ };\text{ }C_{N}=\frac{N}{{1}/{2}\;\rho AU^{2}} A = Blade Area (not to be confused with the Swept Area, which is equal to the height of the blade/rotor times the rotor diameter), R = Radius of turbine The amount of power, P, that can be absorbed by a wind turbine: : P=\frac{1}{2}C_{p}\rho A\nu^{3} Where C_{p} is the power coefficient, \rho is air density, A is the swept area of the turbine, and \nu is the wind speed. == Types ==