Tip and Root Loss Models

This extension of the classical BEM model corrects the assumption of an infinite number of blades, and it describes the effect of the tip vortex on the induced velocities and consequently the lift coefficient \(\bscalar{C_L}\). The wake of the rotor is made up of helical sheets of vorticity trailed from each rotor blade, and consequently the induced velocity at a fixed point in the rotor plane is not constant, but fluctuates as a result of the passage of the blades. The effect on the induced velocity is most pronounced near the blades tips, where the energy loss is largest resulting in a reduction of the power production. A similar loss takes place at the blade root, where the bound circulation must decrease to zero and therefore a vortex must be trailed into the wake like the tip vortex.

In Bladed, the effect of the tip loss is modelled using the Glauert theory (Glauert, 1935) that is used widely for aeroelastic simulations of turbines. This is a simplification of the Prandtl tip loss model (Prandtl & Betz, 1927). The main result of the theory is summarized in the tip loss factor \(\bscalar{F}\) by the equation

\[ \begin{equation} \bscalar{F} = \bscalar{\dfrac{2}{\pi} \cos^{-1} \left( e^{-f} \right) } \end{equation} \]

where

\[ \begin{equation} \bscalar{f} = \bscalar{\dfrac{N_B}{2} \dfrac{R - r}{r \left| \sin \phi \right|}} \end{equation} \]

Here \(\bscalar{N_B}\) is the number of blades, \(\bscalar{R}\) is the rotor radius, \(\bscalar{r}\) is the local radius of the blade element, while \(\bscalar{\phi}\) is the inflow angle shown in the modelling of flow speed components at an aerofoil section. The momentum equations are now modified as:

\[ \begin{flalign} \bscalar{v_{n}} &= \bscalar{- \dfrac{p_{n}}{2JF\dot{m'}}} \\[1ex] \bscalar{v_{t}} &= \bscalar{- \dfrac{p_{t}}{2JF\dot{m'}}} \end{flalign} \]

The correction of the root loss effect is also done in a similar manner.

Last updated 30-08-2024