Skew Wake Correction

The classical BEM model does not include any effect of yaw misalignments, which have proved to be important in order to describe the yawing moments of a turbine physically correctly at high yaw angles. The model of yaw misalignments implemented in Bladed is based on an early work of Glauert and described in details in Refs. (Glauert, 1926a; Glauert, 1926b; Glauert, 1935). It is assumed that the variation of the induced velocity in the normal direction due to yaw misalignment can be described by the algebraic equation

\[ \begin{equation} \bscalar{v_n} = \bscalar{\bar{v}_n} \left( 1 + F \dfrac{r}{R} \tan \left(\dfrac{\chi}{2}\right) \cos (\psi_{rel} - \psi_{w,rel}) \right) \end{equation} \]

where \(\bscalar{F}\) is the tip (or root) loss factor, \(\bscalar{\psi_{rel}}\) is the relative azimuth angle of the blade, and \(\bscalar{\psi_{w,rel}}\) is the relative azimuth angle where a blade points downstream.

The calculation of the wake characteristics \(\bscalar{\chi}\) and \(\bscalar{\psi_{w,rel}}\) are based on the quasi-steady instantaneous wake velocity given in terms of its components in the inertial frame of reference by the expression

\[ \begin{equation} \label{eq_quasisteady_wake_vel} \bscalar{V_{qsw}} = \bscalar{\bar{V}_{0} + A_{w,red}\bar{v}_{qsn}} \end{equation} \]

where \(\bscalar{\bar{V}_{0}}\) is the reference velocity calculated as the mean undisturbed flow velocity at a characteristic radius, i.e. at 70% radius and \(\bscalar{\bar{v}_{qsn}}\) is the normal component of the quasi-steady induced velocity at 70% radius. Note that the reference flow velocity is also used for expressing the dynamic inflow described in the modelling of dynamic wake using the Øye model. The correction factor \(\bscalar{A_{w,red} \approx 0.75}\) is introduced in Equation \(\ref{eq_quasisteady_wake_vel}\) in order to model the direction of the first part of the wake instead of the far wake. The dynamics of the wake direction are taken into account by applying a simple first order time lag filter to the wake velocity, that is

\[ \begin{equation} \bscalar{\tau_{w, dir} \dfrac{d V_w}{dt} + V_w} = \bscalar{V_{qsw}} \end{equation} \]

Here \(\bscalar{V_w}\) is the filtered value, while \(\bscalar{\tau_{w, dir}}\) is the time constant defined as

\[ \begin{equation} \bscalar{\tau_{w, dir}} = \bscalar{\dfrac{A_{w,dir} R}{\bar{V}_{0}}} \end{equation} \]

where the factor \(\bscalar{A_{w,dir} = 0.5}\) is a constant.

The direction of this vector is used for calculating the skewed angle as:

\[ \begin{equation} \bscalar{\chi} = \cos^{-1} \dfrac{{\bvector{n}^T} {\bvector{V}_w}}{\left| {\bvector{V}_w} \right|} \end{equation} \]

where \(\bvector{n}\) denotes normal vector. The downstream relative azimuth angle, which can be computed as:

\[ \begin{equation} \bscalar{\cos (\psi_{w, rel})} = \dfrac{\bvector{r}^T {\bvector{V}_w}}{ \left| {\bvector{V}_{w,tr}} \right| } \end{equation} \]

where \(\bvector{r}\) and \(\bscalar{T}\) denote radial vector and matrix transpose, respectively. Note that \({\bvector{V}_{w,tr}}\) represents the projection of the wake velocity on the rotor plane, defined as:

\[ \begin{equation} {\bvector{V}_{w,tr}} = (\bscalar{I} - \bvector{n} {\bvector{n}^T}) {\bvector{V}_w} \end{equation} \]

with \(\bscalar{I}\) being the identity matrix.

Last updated 30-08-2024