Pre-4.8 Beddoes-Leishman Model
The dynamic stall models described in Incompressible Beddoes-Leishman model and in Compressible Beddoes-Leishman model are based
directly on the original work by Leishman (Leishman, 1988; Leishman, 1989a; Leishman, 1989b) and on the
method provided by Hansen, 2004. In Bladed version 4.7 and earlier, a version of the Beddoes-Leishman model was implemented with some modifications. This legacy model was re-implemented in Bladed 4.8 and later by adopting similar algorithms as closely as possible, denoted as the "Pre 4.8 Beddoes-Leishman model". This model is not recommended for usage in design load cases, and instead should only be used for verification studies or comparison against old load case results.
Attached flow states:
\[
\begin{flalign}
\bscalar{& \dot{x}_{1} + \dfrac{1}{T_{u}} b_{1} \beta^2 x_{1} } = \bscalar{\dfrac{1}{T_{u}} b_{1} \beta^2 A_{1}\alpha } \\[1ex]
\bscalar{& {\dot{x}}_{2} + \dfrac{1}{T_{u}} b_{2} \beta^2 x_{2} } = \bscalar{\dfrac{1}{T_{u}} b_{2} \beta^2 A_{2}\alpha }
\end{flalign}
\]
Pressure lag & trailing edge separation states:
\[
\begin{flalign}
\bscalar{& {\dot{x}}_{3} + \dfrac{1}{T_{p}T_{u}} x_{3} } = \bscalar{ \dfrac{1}{T_{p}T_{u}} \left( \dfrac{d C_N^{st}}{d \alpha} \left( \alpha_{e} - \alpha_{0} \right) + \dfrac{4}{M}\dot{x_6} + \dfrac{2 T_u}{M}\dot{x_9} \right) } \\[1ex]
\bscalar{& {\dot{x}}_{4} + \frac{x_{4}}{T_{f}T_{u}} } = \bscalar{f(\alpha_f)}
\end{flalign}
\]
Vortex lift state:
\[
\begin{equation}
\bscalar{ {\dot{x}}_{5} + \dfrac{1}{T_{v}T_{u}} x_{5} } = \bscalar{ {\dot{C}}_{v} }
\end{equation}
\]
Impulsive lift/moment states due to angle of attack:
\[
\begin{flalign}
\bscalar{& {\dot{x}}_{6} + \dfrac{1}{K_{\alpha}T_{1}} x_{6} } = \bscalar{ \alpha } \\[1ex]
\bscalar{& {\dot{x}}_{7} + \dfrac{1}{K_{\alpha,m}T_{1} b_3} x_{7} } = \bscalar{ \alpha } \\[1ex]
\bscalar{& {\dot{x}}_{8} + \dfrac{1}{K_{\alpha,m}T_{1} b_4} x_{8} } = \bscalar{ \alpha }
\end{flalign}
\]
The dynamic normal force coefficient is found according to:
\[
\begin{equation}
\bscalar{ C_N^{dyn} = C_N^{circ} + x_{5} + \dfrac{4}{M} \dot{x}_{6} \phi_1^{impulsive} \phi^{damping} \phi^{comp} }
\end{equation}
\]
The circulatory term is calculated using:
\[
\begin{equation}
\bscalar{C_N^{circ}} = \begin{cases}
\dfrac{d C_N^{st}}{d \alpha}\left( \dfrac{1 + \sqrt{x_4}}{2} \right) \left( \alpha_{E} - \alpha_0 \right) & \text{for} \ f(\alpha_f) \gt 1e-5 \\[1ex]
C_L^{st} (\alpha) \cos \alpha_f + C_D^{st} (\alpha) \sin \alpha_f & \text{for} \ f(\alpha_f) \leq 1e-5
\end{cases}
\end{equation}
\]
The chordwise force component is calculated using:
\[
\begin{equation}
\bscalar{C_C^{dyn}} = \begin{cases}
C_L^{st}\left( \alpha_{E} \right)\sin \alpha - C_D^{st}\left( \alpha_{E} \right)\cos \alpha \sqrt{\dfrac{x_{4}}{f(\alpha_f)} } - x_{5}\left( 1 + \max\left( - \dfrac{t_{vortex}}{T_{vl} T_u }, - 1 \right) \right) & \text{for} \ f(\alpha_f) \gt 0.01 \\[1ex]
C_L^{st}\left( \alpha_{E} \right)\sin \alpha - C_D^{st}\left( \alpha_{E} \right)\cos \alpha & \text{for} \ f(\alpha_f) \leq 0.01
\end{cases}
\end{equation}
\]
The separation point is quantified by using:
\[
\begin{equation}
\bscalar{ f(\alpha_f) } = \bscalar{ \left( 2\sqrt{\dfrac{C_N^{st}}{ \dfrac{d C_N^{st}}{d \alpha} (\alpha_f - \alpha_{0})}} - 1 \right)^{2} }
\end{equation}
\]
which depends on the delayed angle of attack (including the pressure lag effect) \(\bscalar{ \alpha_f }\). This parameter can be computed as:
\[
\begin{equation}
\bscalar{ \alpha_f } = \bscalar{ \alpha_0 + \dfrac{x_3}{ \left( d C_N^{st} / d \alpha \right) } }
\end{equation}
\]
The drag coefficient is calculated by:
\[
\begin{equation}
\bscalar{C_D^{dyn}} = \begin{cases}
C_N^{dyn} \sin \alpha - C_C^{dyn} \cos \alpha & \text{for} \ f(\alpha_f) \gt 0.01 \\[1ex]
C_D^{st} (\alpha_{E}) & \text{for} \ f(\alpha_f) \leq 0.01
\end{cases}
\end{equation}
\]
and the pitching moment coefficient by:
\[
\begin{flalign}
\bscalar{C_M^{dyn} & = \dfrac{1}{1 - f(\alpha_f)} \left[ \left( \dfrac{d C_M^{st}}{d \alpha} (\alpha_{E} - \alpha_0) + C_{M0} \right) \left( x_4 - f(\alpha_f) \right) + C_M^{st}(\alpha) \left( 1 - x_4 \right) \right] \\[1ex] \nonumber
& - 0.2\left( 1 - \cos\left( \pi\frac{\tau_{vl}}{T_{vl}T_u} \right) \right)x_{5} \\[1ex] \nonumber
& + \left[ - \dfrac{1}{M} \left( - \frac{A_{3}}{b_{3}K_{\alpha,m}T_{1}}x_{7} - \dfrac{A_{4}}{b_{4}K_{\alpha,m}T_{1}}x_{8} \right) - \dfrac{1}{M}\alpha \right] \phi_2^{impulsive} \phi^{damping} \\[1ex] \nonumber
& - \dfrac{14}{24} C_D^{st} (\alpha) \dot{\alpha} T_u }
\end{flalign}
\]
The following key differences can be observed with the implementations in Incompressible Beddoes-Leishman model and in Compressible Beddoes-Leishman model:
- The effective angle of attack is not computed using the
three-quarter chord angle of attack
- The normal force coefficient is not an interpolation between fully
attached and fully separated flow.
- The vortex lift is added to the chordwise force with a fading factor
depending on the vortex travel time
- An own-developed method is used for the moment coefficient. It is
assumed the separation position can be used as an interpolant
between fully attached moment coefficient and the steady state
measured curve.
- The impulsive lift/moment contributions due to a step change in
pitch rate are excluded.
- An additional term is added which provides extra damping due to drag
at high angles of attack.
- The vortex lift will keep accumulating when the angle of attack
remains consistently high. This has an impact on lift/drag
particularly in high angles of attack.
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. For this model, the moment impulsive contribution factor (\(\bscalar{\phi_2^{impulsive}}\)) behaves equally the same as \(\bscalar{\phi_1^{impulsive}}\). 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