Incompressible Beddoes-Leishman Model

An overview of the state-space implementation of the incompressible Beddoes-Leishman model is given in Hansen, 2004. To a large extent the model in Bladed is based on this theory. The main formulations and modifications to the model are summarized in this article. The unsteady normal force (\(\bscalar{F_N}\)) is defined as:

\[ \begin{equation} \bscalar{F_N} = \bscalar{\dfrac{1}{4} \pi\rho c^{2} \left( U\dot{\alpha} + \dot{U}\alpha + \ddot{h}_{\frac{1}{2}} \right) + \pi\rho cU \left( w(0)\phi(s) + \int_{0}^{s}{\frac{\partial w}{\partial\sigma}\phi(s - \sigma)d\sigma} \right)} \end{equation} \]

The added mass terms \(\dot{U}\alpha + \ddot{h}_{\frac{1}{2}}\) are neglected in this implementation.

The model is expressed in state-space form through a closed set of equations which is defined as:

\[ \begin{flalign} \bscalar{& \dot{x}_{1} + \dfrac{1}{T_{u}} \left( b_{1} + \dfrac{c\dot{U}}{2U^{2}} \right)x_{1} } = \bscalar{\dfrac{1}{T_{u}} b_{1}A_{1}\alpha_{3/4} } \\[1ex] \bscalar{& {\dot{x}}_{2} + \dfrac{1}{T_{u}} \left( b_{2} + \frac{c\dot{U}}{2U^{2}} \right)x_{2} } = \bscalar{\dfrac{1}{T_{u}} b_{2}A_{2}\alpha_{3/4} } \\[1ex] \bscalar{& {\dot{x}}_{3} + \dfrac{1}{T_{p}T_{u}} x_{3} } = \bscalar{ \dfrac{1}{T_{p}T_{u}} \left( \dfrac{d C_N^{st}}{d \alpha} \left( \alpha_{e} - \alpha_{0} \right) + \pi T_{u}\dot{\alpha} \right) } \\[1ex] \bscalar{& {\dot{x}}_{4} + \frac{1}{T_{f}T_{u}}x_{4} } = \bscalar{f(\alpha_f)} \\[1ex] \label{eq_ds_incomp_state5} \bscalar{& {\dot{x}}_{5} + \dfrac{1}{T_{v}T_{u}} x_{5} } = \bscalar{ {\dot{C}}_{v} } \end{flalign} \]

with \(\bscalar{A_1}\), \(\bscalar{A_2}\), \(\bscalar{b_1}\) and \(\bscalar{b_2}\) representing the attached flow constants which are given in Leishman, 1988, \(\bscalar{c}\) being the local chord length of the blade section and \(\bscalar{U}\) denoting the local relative wind speed. Variable \(\bscalar{T_{u}}\) is defined as \(\bscalar{T_{u} = c / (2U)}\) and the constants \(\bscalar{T_f}\), \(\bscalar{T_p}\) and \(\bscalar{T_v}\) describe the separated flow, pressure lag and vortex lift decay effects, respectively. Note that \(\bscalar{\alpha_{3/4}}\) represents the angle of attack sampled at the 3/4 chord location of the local blade section. As can be seen, this model uses the normal force coefficient as the basis instead of the lift coefficient originally proposed by Hansen, 2004 in conjunction with the formulation from Beddoes-Leishman in the following references (Leishman, 1988; Leishman, 1989a; Leishman, 1989b). On that note, the inclusion of the fifth state (\(\bscalar{x_{5}}\)) which describes the vortex lift effect also marks the difference with the model according to Hansen, 2004.

The separation point is quantified by using:

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

which depends on the delayed angle of attack (including the pressure lag effect) \(\bscalar{ \alpha_f }\). This parameter can be computed as:

\[ \begin{equation} \label{eq_ds_incomp_aoa_delayed} \bscalar{ \alpha_f } = \bscalar{ \alpha_0 + \dfrac{x_3}{ \left( d C_N^{st} / d \alpha \right) } } \end{equation} \]

The instantaneous value of vortex lift \(\bscalar{ C_v }\) is found from Leishman, 1989b as:

\[ \begin{equation} \bscalar{ C_{v} } = \begin{cases} C_{N}^{C}\left( 1 - \left( 1 + \sqrt{x_{4}} \right)^{2}/4 \right) & 0 \leq \ \tau_{vortex} \leq T_{vl} \\[1ex] 0 & \text{otherwise} \end{cases} \end{equation} \]

In this formulation, \(\bscalar{C_{N}^{C}}\) describes the circulatory part of the normal force coefficient for attached flow. The constant \(\bscalar{T_{vl}}\) represents the non-dimensional time required for the vortex to convect from the leading edge to the trailing edge. Lastly, \(\tau_{vortex}\) represents the time from detachment from the leading edge. The circulatory part of the attached flow component can be computed as:

\[ \begin{equation} \label{eq_ds_incomp_CNCirc_att} \bscalar{ C_{N}^{C} } = \bscalar{ \dfrac{d C_N^{st}}{d \alpha} \left( \alpha_{E} - \alpha_{0} \right) } \end{equation} \]

with the effective angle of attack defined as:

\[ \begin{equation} \label{eq_ds_incomp_aoa_eff} \bscalar{ \alpha_{E} } = \bscalar{ \alpha_{3/4}\left( 1 - A_{1} - A_{2} \right) + x_{1} + x_{2} } \end{equation} \]

In Equation \(\eqref{eq_ds_incomp_state5}\), \(\bscalar{\dot{C_{v}}}\) is the rate of change of vortex lift which can be conveniently expressed in terms of the known state values and state rates according to:

\[ \begin{flalign} \bscalar{& \dot{C_{v} } = \bscalar{ \dot{C}}_{N}^{C}\left( 1 - \frac{\left( 1 + \sqrt{x_{4}} \right)^{2}}{4} \right) - C_{N}^{C}\frac{\left( 1 + \sqrt{x_{4}} \right)}{4\sqrt{x_{4}}}{\dot{x}}_{4}} \\[1ex] \bscalar{& \dot{C}_{N}^{C} } = \bscalar{ \dfrac{d C_N^{st}}{d \alpha} \left( \dot{\alpha}\left( 1 - A_{1} - A_{2} \right) + {\dot{x}}_{1} + \ \ \dot{x_{2}} \right) } \end{flalign} \]

The vortex lift term will only become active when a critical leading edge pressure condition is met. This condition is:

\[ \begin{equation} \bscalar{x_{3} } > \bscalar{C_N^{MAX}} \end{equation} \]

with \(\bscalar{C_N^{MAX}}\) being the maximum value of the normal force coefficient obtained from the static polar data. If this condition is met, the vortex lift will start to accumulate while the vortex is convecting over the aerofoil. The vortex lift will cease to accumulate if:

• The vortex reaches the trailing edge

• The angle of attack rate of change direction reverses relative to the direction when the vortex initially formed at the leading-edge

Furthermore, a new vortex can only form after the initial vortex has detached from the trailing edge and/or when the pressure lift state has reduced back below \(\bscalar{ C_N^{MAX}}\).

The dynamic coefficients are then found according to:

\[ \begin{flalign} \label{eq_ds_incomp_CNDyn} \bscalar{C_N^{dyn} &= \dfrac{d C_N^{st}}{d \alpha}\left( \alpha_{E} - \alpha_{0} \right)x_{4} + C_N^{fs}\left( \alpha_{E} \right)\left( 1 - x_{4} \right) + \pi T_{u}\dot{\alpha\ } \phi_1^{impulsive} \phi^{damping} + x_{5}} \\[1ex] \bscalar{C_C^{dyn} &= C_C^{st}(\alpha_{E})} \\[1ex] \label{eq_ds_incomp_drag} \bscalar{C_D^{dyn} & = C_D^{st} (\alpha_{E}) + C_N^{dyn} (\alpha - \alpha_{E}) \\[1ex] \nonumber & + (C_D^{st} (\alpha_{E}) - C_{D,0})\left( \frac { \sqrt{f \left( \alpha_{E} \right)} - \sqrt{x_{4}}}{2} - \frac{f\left( \alpha_{E} \right) - x_{4}}{4} \right)} \\[1ex] \bscalar{C_M^{dyn} & = C_M^{st} (\alpha_{E}) + C_N^{circ} \left( x_{cp} (\alpha_{f}) - x_{cp} ( \alpha_{E} ) \right) - \dfrac{1}{2} \pi T_{u}\dot{\alpha} \phi_2^{impulsive} \phi^{damping} \\[1ex] \nonumber & - 0.2\left( 1 - \cos\left( \pi\dfrac{\tau_{vl}}{T_{vl}T_u} \right) \right)x_{5} } \end{flalign} \]

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:

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

If the effective angle of attack,\(\bscalar{\alpha_E}\), described in Equation \(\eqref{eq_ds_incomp_aoa_eff}\) is used in Equation \(\eqref{eq_ds_incomp_sep_point}\) instead of the delayed angle of attack (see Equation \(\eqref{eq_ds_incomp_drag}\) for calculating the drag force) then the separation point will slightly differ as this does not incorporate the pressure lag effect. Similarly, if the actual angle of attack is used instead without considering any delay at all (as employed in Equation \(\eqref{eq_ds_incomp_CNfs}\)), this will yield the static separation position.

Variable \(\bscalar{x_{cp}}\) describes the quasi-steady position of the aerodynamic centre. This parameter is calculated as:

\[ \begin{equation} \bscalar{x_{cp} = \dfrac{C_{M}^{st}(\alpha) - C_{M}^{st}(\alpha_{0})}{\ C_{N}(\alpha)} } \end{equation} \]

where \(\bscalar{C_M}\) denotes pitch moment coefficient. Parameter \(\bscalar{C_N^{circ}}\) represents the viscous circulatory contribution of the normal force coefficient (compare with the attached flow part in Equation \(\eqref{eq_ds_incomp_CNCirc_att}\)) represented as:

\[ \begin{equation} \label{eq_ds_incomp_CNCirc_visc} \bscalar{ C_N^{circ} = \dfrac{d C_N^{st}}{d \alpha} \left( \alpha_{E} - \alpha_{0} \right)x_{4} + C_N^{fs}\left( \alpha_{E} \right)\left( 1 - x_{4} \right) + x_{5} } \end{equation} \]

An additional option is introduced allowing Bladed user to use a different equation for the normal force coefficient instead of using Equation \(\eqref{eq_ds_incomp_CNDyn_Kirch}\). In this sense, the normal force coefficient is computed as:

\[ \begin{equation} \label{eq_ds_incomp_CNDyn_Kirch} \bscalar{ C_N^{dyn} = \dfrac{d C_N^{st}}{d \alpha} \left( \dfrac{1 + \sqrt{x_{4}}}{2} \right)^{2} \left( \alpha_{E} - \alpha_{0} \right) + \pi T_{u}\dot{\alpha\ } + x_{5} } \end{equation} \]

The usage of Equation \(\eqref{eq_ds_incomp_CNDyn_Kirch}\) does not result in significant calculation result differences in normal operating conditions. In parked/idling simulations where the blade experiences large angles of attack it is found that above equation results in improved aerodynamic damping. The validation document in DNV, 2018 discusses in detail what the implications on aerodynamic damping are.

The lift coefficient is subsequently calculated from the normal and chord force coefficient using the geometric angle of attack.

\[ \begin{equation} \bscalar{ C_L^{dyn} = \ C_N^{dyn}\cos(\alpha) + \ C_C^{dyn}\sin(\alpha) } \end{equation} \]

For the drag coefficient it is opted to not use decomposition from \(\bscalar{C_N}\) and \(\bscalar{C_C}\) but use the dynamic equations for \(\bscalar{C_D}\) from Hansen, 2004 directly.

Lastly, Bladed user has a control to deactivate the impulsive contribution factor \(\bscalar{\phi_1^{impulsive}}\) for the normal force. Its value is unity if the impulsive effect is activated and zero otherwise. Meanwhile, the added mass (impulsive) contribution factor (\(\bscalar{\phi_2^{impulsive}}\)) for the pitching moment is always set to unity. Regardless of the choice of setting the impulsive effect, 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:

\[ \begin{equation} \bscalar{ \phi^{damping} = \max \left[ 0,\left( 1 - 8 \left( \dfrac{\dot{\alpha} c }{2 U} \right)^3 \right) \right] x_4 } \end{equation} \]

which is helpful to avoid potential instability of the solutions.

Last updated 30-08-2024