# Aerodynamics and Motion Performance of the H-Type Floating Vertical Axis Wind Turbine
## 2 The Aerodynamic Loads of H-Type Floating VAWT  
Based on the double multiple stream tube (DMST) theory, the aerodynamic loads acting on the blades were calculated considering dynamic stall, tip losses, aspect ratio, tower shadow effect and the influence of the surge-heave-pitch motions of the floater. The whole floating VAWT was dealt with as a rigid body. The stream tubes were divided into upwind and downwind sectors, respectively. Coordinate system is shown in Figure 1, where the coordinate origin \(o\) is in the center of gravity of the floating VAWT, and \(\xi_{1}\), \(\xi_{3}\), and \(\xi_{5}\) are the displacements of surge, heave and pitch of the floating VAWT, respectively. The positive directions of the surge and heave motions are along the positive directions of the \(x\) axis and \(z\) axis, respectively; the positive direction of the pitch motion is clockwise. The vector relations of the wind turbine are shown in Figure 2.  
Figure 1: Coordinate system. The steady wind was considered here, it was a function of the height above the water level, and for a local height \(z_{i}\), the wind speed can be written as follow,  
\[V_{\rm ooi}(z_{i})=V_{\infty}(z_{i}/z_{ref})^{a}, \tag{1}\]  
where \(V_{\infty}\) is the average wind velocity at a reference height \(z_{ref}\), the vertical distance from the center of the blades to the water level is taken as the reference height in this paper, and with power \(a=0.14\)[19]. Assuming the velocity parallel to the direction of the steam tube affects the induced velocities and the velocity perpendicular to the direction of the steam tube only affects the local relative velocities, the coordinate and the velocity of each stream tube were deduced considering the motions of the floating foundation in the following parts. Induced velocities at the upwind and downwind sectors \(V\) and \(V^{\prime}\), and the induced equilibrium velocity \(V_{e}\) are given by [20],  
\[V=uV_{\rm ooi}\cos\xi_{5}, \tag{2}\]  
\[V_{\rm e}=(2u-1)V_{\rm ooi}\cos\xi_{5}, \tag{3}\]  
\[V^{\prime}=u^{\prime}V_{e}=u^{\prime}(2u-1)V_{\rm ooi}\cos\xi_{5}. \tag{4}\]  
where \(u\) is the interference factor, and \(u^{\prime}\) is the second interference factor. For the upwind sector, the distance from local blades to the gravity center of floating wind turbine system is expressed as,  
\[L_{2}=\sqrt{\left(R\cos\theta\right)^{2}+\left(z+L_{1}+H_{g}\right)^{2}}, \tag{5}\]  
where \(R\) is the radius of rotor, \(\theta\) is the azimuth of rotor, \(z\) is the distance from each stream tube to equator (below equator is negative direction), \(L_{1}\) is the distance from equator to still water level, and \(H_{g}\) is the distance from still water level to the gravity center of floating wind system when the system is in a state of stillness. The angle between the local position of the blade and the tower is written as  
\[\gamma=\arctan(\frac{R\cos\theta}{z+L_{1}+H_{g}}), \tag{6}\]  
Then, the local wind velocity of the blade is given by,  
\[V_{\rm ooi}=V_{\infty}\cdot\left(\frac{L_{2}\cos(\xi_{5}-\gamma)-\left(H_{g}- \xi_{3}\right)}{H+L_{1}}\right)^{\rm a}, \tag{7}\]  
Figure 2: Equatorial velocity vector. where \(H\) is the half-height of rotor. The velocities of local blades can be derived from the surge, heave and pitch velocities of floater by using transformation matrix. Then, the resultant velocities, which are caused by the motion of the floating foundation, parallel to the direction of the steam tube for the local position can be written as,  
Figure 1:0  
Using a similar deducing progress, we get the resultant velocities parallel to and vertical to the direction of the steam tube for the downwind sector, as follows,  
Figure 1:1  
For the rotor of upwind sector, namely, the range of azimuth \(\theta\) is \([-\pi/2,\,\pi/2]\), and the local relative velocity of the rotor can be written as,  
Figure 1:2  
where \(\omega\) is the rotation velocity of the blade.