Øye Model

The Øye model adopted in Bladed is a slightly modified version of the original implementation in Øye, 1991. This model was implemented in Bladed as a simplification of the incompressible Beddoes-Leishman model (Hansen, 2004), see also Incompressible Beddoes-Leishman model. In the Øye model, only the dynamics of the separation position are taken into account. The state-space equation then becomes:

\[ \begin{equation} \bscalar{ {\dot{x}}_{1} + \frac{1}{T_{f}T_{u}}x_{1} } = \bscalar{f(\alpha)} \end{equation} \]

The separation point is quantified by using:

\[ \begin{equation} \bscalar{ f(\alpha) } = \bscalar{ \left( 2\sqrt{\dfrac{C_N^{st}}{ \dfrac{d C_N^{st}}{d \alpha} (\alpha - \alpha_{0})}} - 1 \right)^{2} } \end{equation} \]

Note that the model uses \(\bscalar{\alpha}\) instead of \(\bscalar{\alpha_f}\) in contrast to the incompressible Beddoes-Leishman model. This implies that the attached flow hysteresis is not included in the Øye model.

The dynamic coefficients are then calculated as:

\[ \begin{flalign} \label{eq_ds_incomp_CNDyn} \bscalar{C_N^{dyn} &= \dfrac{d C_N^{st}}{d \alpha}\left( \alpha - \alpha_{0} \right)x_{1} + C_N^{fs}\left( \alpha \right)\left( 1 - x_{1} \right)} \\[1ex] \bscalar{C_C^{dyn} &= C_C^{st}(\alpha)} \\[1ex] \bscalar{C_M^{dyn} &= C_M^{st} (\alpha) - \dfrac{1}{2} \pi T_{u}\dot{\alpha} \phi_2^{impulsive} \phi^{damping}} \end{flalign} \]

It is to be noted that \(\bscalar{C_N^{fs}}\) is the normal force coefficient for a fully separated flow according to Hansen, 2004 defined as:

\[ \begin{equation} \bscalar{ C_N^{fs} = \dfrac{C_N^{st} - \dfrac{d C_N^{st}}{d \alpha} \left( \alpha - \alpha_{0} \right)f(\alpha)}{1 - f(\alpha)} } \end{equation} \]

Note that there is no impulsive contribution for the normal force in the Øye model. Meanwhile, the added mass (impulsive) contribution factor is applied for the pitching moment. This effect is always active (\(\bscalar{\phi_2^{impulsive}} = 1\)). However, similar with the Beddoes-Leishman model, a damping factor (\(\bscalar{\phi^{damping}}\)) is always applied which decreases the contribution for fully separated flow or when the rate of change of the angle of attack (\(\bscalar{\dot{\alpha}}\)) becomes large. The scaling is done through the following relationship:

\[ \begin{equation} \bscalar{ \phi^{damping} = \max \left[ 0,\left( 1 - 8 \left( \dfrac{\dot{\alpha} c }{2 U} \right)^3 \right) \right] x_1 } \end{equation} \]

which is helpful to avoid potential instability of the solutions.

Last updated 30-08-2024