Øye Dynamic Wake Model

The classical BEM model assumes that the wake and hence the induced velocities react instantaneously to changes of the pitching moment and thereby the rotor load. However, it is clear that the changes of the rotor load affect the vorticity that is trailed into the rotor wake and thereby change the induced flow field with a certain time lag. This effect has also been observed in full scale experiments as described by Øye in (Øye, 1991b). The aerodynamics associated with this phenomenon is referred to as dynamic wake or dynamic inflow that is the preferred term in this context. A simple model for this unsteady aerodynamics is described and proposed by Øye (Øye, 1995) where a simple first order time system response is used for describing the effect of a change of the pitch angle on the induced velocities. This model is based on filtering the induced velocities calculated by the fundamental quasi-steady momentum equilibrium that includes the tip (and root) loss correction. Hereby the dynamic wake calculation can be calculated over an entire annulus or over an annular segment of size \(\bscalar{2\pi \dfrac{r}{N_B}}\) where \(\bscalar{N_B}\) is the number of blades. In the following the average induced quasi-steady velocity components are denoted as \(\bscalar{\bar{v}_{qsn}}\) and \(\bscalar{\bar{v}_{qst}}\).

The applied filter is composed by two first order filters connected in series with time constants \(\bscalar{\tau_1}\) and \(\bscalar{\tau_2}\) , respectively. For the average induced velocity \(\bscalar{\bar{v}_{qsn}}\) in the normal direction, the combined response of the filters is described by the following two differential equations:

\[ \begin{flalign} \bscalar{\tau_1 \dfrac{d \bar{u}_n}{dt} + \bar{u}_n} &= \bscalar{\bar{v}_{qsn} + k_w \tau_1 \dfrac{d \bar{v}_{qsn}}{dt}} \\[1ex] \bscalar{\tau_2 \dfrac{d \bar{v}_n}{dt} + \bar{v}_n} &= \bscalar{\bar{u}_n} \end{flalign} \]

where \(\bscalar{k_w \approx 0.6}\), \(\bscalar{\bar{v}_n}\) is the final filtered value, while \(\bscalar{\bar{u}_n}\) is an intermediate variable. A similar scheme is applied for the tangential induced velocity. The time constants \(\bscalar{\tau_1}\) and \(\bscalar{\tau_2}\) are calculated according to

\[ \begin{flalign} \bscalar{\tau_1} &= \bscalar{F_w \dfrac{1.1}{1 - 1.3 \bar{a}_{eff}} \dfrac{R}{V_0}} \\[1ex] \bscalar{\tau_2} &= \bscalar{\left(0.39 - 0.26 \left( \dfrac{r}{R} \right)^2\right)\tau_1} \end{flalign} \]

where \(\bscalar{R}\) is rotor radius. The symbol \(\bscalar{V_0}\) represents a reference flow speed calculated as the averaged free flow speed at the blades at 70% radius, \(\bscalar{F_w \approx 1}\) is a predefined factor, while the symbol \(\bscalar{\bar{a}_{eff}}\) denotes the mean effective axial induction. The value of \(\bscalar{\bar{a}_{eff}}\) is limited to maximum 0.5.

Last updated 30-08-2024