Nacelle Windage Loads

The nacelle windage loads can be calculated using a simple drag model.

The drag model

This model utilizes the drag equation \(\eqref{eq:dragForce}\), by only using the horizontal wind component and the horizontally projected area of the nacelle. The resulting load is then applied at the centre of volume defined using the hub height.

Figure 1: A upwind turbine illustrating the centre of volume, \(\bvector{CoV}\), nacelle angle, \(\bscalar{\gamma}\), and the wind direction, \(\bscalar{\delta}\).

The combined yaw angle of both the wind and the nacelle is defined as

\[ \begin{equation} \bscalar{\beta}=\bscalar{\delta}+\bscalar{\gamma} \end{equation} \]

where \(\bscalar{\delta}\) is the horizontal wind direction and \(\bscalar{\gamma}\) is the yaw angle of the nacelle. The combined yaw, \(\bscalar{\beta}\), is then used to calculate the projected area for the incoming wind as follows:

\[ \begin{equation} \bscalar{A}_{p}=\bscalar{N}_\text{height}\left(\bscalar{N}_\text{width}\cos{\beta}+\bscalar{N}_\text{length}\sin{\beta}\right) \end{equation} \]

where the \(\bscalar{N}_\text{height}\), \(\bscalar{N}_\text{width}\) and \(\bscalar{N}_\text{length}\) are the dimensions of the nacelle illustrated in Figure 1.

The total drag force in the direction of the incoming wind:

\[ \begin{equation} \bscalar{F}=\frac{1}{2}\bscalar{\rho}_\text{air}\bscalar{V_h}^2\bscalar{c_d}\bscalar{A}_{p} \label{eq:dragForce} \end{equation} \]

where \(\bscalar{\rho}_\text{air}\) is the density of the incoming air, \(\bscalar{V_h}\) is the horizontal wind component and \(\bscalar{c_d}\) is the drag coefficient, which is assumed constant for all angles and conditions.

The force is then transformed into the nacelle coordinate system denoted \(x^\prime,y^\prime,z^\prime\) in Figure 1.

\[ \begin{flalign} \bscalar{F}_{Nx}=\bscalar{F}\cos{\bscalar{(-\beta)}} \\[1ex] \bscalar{F}_{Ny}=\bscalar{F}\sin{\bscalar{(-\beta)}} \end{flalign} \]

Finally, this is transformed into the global \(x,y,z\) coordinate system, using a \(2D\) rotation and the yaw of the nacelle.

\[ \begin{equation} \bvector{F_G}=\left[\begin{matrix}\cos{\bscalar{\gamma}}&-\sin{\bscalar{\gamma}}\\\sin{\bscalar{\gamma}}&\cos{\bscalar{\gamma}}\\\end{matrix}\right]\left[\begin{matrix}\bscalar{F_{Nx}}\\\bscalar{F_{Ny}}\\\end{matrix}\right] \end{equation} \]

The drag force in global coordinates is then applied at the centre of volume, shown in Figure 1 and in Figure 2.

Figure 2: Illustration of the distance \(\bscalar{h_\text{hub}}\) in a deflected tower which is used to formulate the centre of volume, \(\bvector{CoV}\).

The centre of volume vector is defined in the global coordinate system as

\[ \bvector{CoV}=\left[\begin{matrix}0\\0\\\bscalar{h_{hub}}\\\end{matrix}\right] \]

where the \(\bscalar{h_\text{hub}}\) is defined using the global tower height and the global hub height:

\[ \begin{equation} h_\text{hub}=H_\text{hub}-H_\text{tower} \end{equation} \]

The resulting moment is calculated using the centre of volume vector and the global drag force by taking the 3D cross product as follows:

\[ \begin{equation} \bvector{M_G}=\bvector{CoV}\times \bvector{F_G} \label{eq:momentCalc} \end{equation} \]

Note that for downwind turbines the yaw-bearing coordinate system is rotated 180 degrees, hence there is no need to adjust the centre of volume when changing between upwind and downwind configurations.

Last updated 15-11-2024