Compressible Beddoes-Leishman Model
An overview of a state-space implementation of the incompressible Beddoes-Leishman model is described in Hansen, 2004. However, this paper
excludes compressibility corrections and impulsive lift contributions as
described in Leishman, 1989b. The incompressible model is therefore
expanded to include these extra terms, which in turn increases the number of state equations to be solved. A closed set of 11 aerodynamic states is defined as follows:
Attached flow states:
\[
\begin{flalign}
\bscalar{& \dot{x}_{1} + \dfrac{1}{T_{u}} \left( b_{1} \beta^2 + \dfrac{c\dot{U}}{2U^{2}} \right)x_{1} } = \bscalar{\dfrac{1}{T_{u}} b_{1} \beta^2 A_{1}\alpha_{3/4} } \\[1ex]
\bscalar{& {\dot{x}}_{2} + \dfrac{1}{T_{u}} \left( b_{2} \beta^2 + \frac{c\dot{U}}{2U^{2}} \right)x_{2} } = \bscalar{\dfrac{1}{T_{u}} b_{2} \beta^2 A_{2}\alpha_{3/4} }
\end{flalign}
\]
Pressure lag & trailing edge separation states:
\[
\begin{flalign}
\bscalar{& {\dot{x}}_{3} + \dfrac{1}{T_{p}T_{u}} x_{3} } = \bscalar{ \dfrac{1}{T_{p}T_{u}} \left( \dfrac{d C_N^{st}}{d \alpha} \left( \alpha_{e} - \alpha_{0} \right) + \dfrac{4}{M}\dot{x_6} + \dfrac{2 T_u}{M}\dot{x_9} \right) } \\[1ex]
\bscalar{& {\dot{x}}_{4} + \frac{1}{T_{f}T_{u}}x_{4} } = \bscalar{f(\alpha_f)}
\end{flalign}
\]
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:
\[
\begin{flalign}
\bscalar{& {\dot{x}}_{6} + \dfrac{1}{K_{\alpha}T_{1}} x_{6} } = \bscalar{ \alpha } \\[1ex]
\bscalar{& {\dot{x}}_{7} + \dfrac{1}{K_{\alpha,m}T_{1} b_3} x_{7} } = \bscalar{ \alpha } \\[1ex]
\bscalar{& {\dot{x}}_{8} + \dfrac{1}{K_{\alpha,m}T_{1} b_4} x_{8} } = \bscalar{ \alpha }
\end{flalign}
\]
Impulsive lift/moment states due to pitch rate:
\[
\begin{flalign}
\bscalar{& {\dot{x}}_{9} + \dfrac{1}{K_{q}T_{1}} x_{9} } = \bscalar{ \dot{\alpha} } \\[1ex]
\bscalar{& {\dot{x}}_{10} + \dfrac{\beta^2 b_5}{T_{u}} x_{10} } = \bscalar{ \dot{\alpha} } \\[1ex]
\bscalar{& {\dot{x}}_{11} + \dfrac{1}{K_{q,m}T_{1}} x_{11} } = \bscalar{ \dot{\alpha} }
\end{flalign}
\]
Variable \(\bscalar{\beta}\) represents the compressibility correction factor and is defined as a function of Mach number (\(\bscalar{M}\)) as:
\[
\begin{equation}
\bscalar{ \beta^{2} = (1 - M^{2}) }
\end{equation}
\]
The time constant \(\bscalar{T_{1}}\) represents the time required for the pressure disturbance to convect over the chord (\(\bscalar{c}\)) as a function of the speed of sound (\(\bscalar{S_s}\))
\[
\begin{equation}
\bscalar{ T_{1} = \dfrac{c}{S_s} }
\end{equation}
\]
The non-circulatory time constants \(\bscalar{K_{\alpha}}\) and \(\bscalar{K_{\alpha,m}}\) are defined as:
\[
\begin{flalign}
\bscalar{& K_{\alpha} = \dfrac{1}{(1 - M) + \pi\beta M^{2}(A_{1}b_{1} + A_{2}b_{2}) } } \\[1ex]
\bscalar{& K_{\alpha,m} = \dfrac{A_{3}b_{4} + A_{4}b_{3}}{(b_{3}b_{4}(1 - M) }}
\end{flalign}
\]
The dynamic coefficients are then found according to:
\[
\begin{flalign}
\bscalar{C_N^{dyn} & = \dfrac{d C_N^{st}}{d \alpha}\left( \alpha_{E} - \alpha_{0} \right)x_{4} + {C_N}^{fs}\left( \alpha_{E} \right)\left( 1 - x_{4} \right) + x_{5} + \left[ \dfrac{4}{M} \dot{x}_{6} + \dfrac{2T_{u}}{M} {\dot{x}}_{9} \right] \phi_1^{impulsive} \phi^{damping} \phi^{comp} } \\[1ex]
\bscalar{C_C^{dyn} & = C_C^{st}(\alpha_{E})} \\[1ex]
\label{eq_ds_incomp_drag}
\bscalar{C_D^{dyn} & = C_D^{st} (\alpha_{E}) + C_N^{dyn} (\alpha - \alpha_{E}) \\[1ex] \nonumber
& + (C_D^{st} (\alpha_{E}) - C_{D,0})\left( \frac { \sqrt{f \left( \alpha_{E} \right)} - \sqrt{x_{4}}}{2} - \frac{f\left( \alpha_{E} \right) - x_{4}}{4} \right)} \\[1ex]
\bscalar{C_M^{dyn} & = C_M^{st} (\alpha_{E}) + C_N^{circ} \left( x_{cp} (\alpha_{f}) - x_{cp} ( \alpha_{E} ) \right) - 0.2\left( 1 - \cos\left( \pi\frac{\tau_{vl}}{T_{vl}T_u} \right) \right)x_{5} \\[1ex] \nonumber
& + \left[ - \dfrac{1}{M} \left( - \frac{A_{3}}{b_{3}K_{\alpha,m}T_{1}}x_{7} - \dfrac{A_{4}}{b_{4}K_{\alpha,m}T_{1}}x_{8} \right) - \dfrac{1}{M}\alpha \right] \phi_2^{impulsive} \phi^{damping} \phi^{comp} \\[1ex] \nonumber
& + \left[ \dfrac{2 \pi T_{u}}{8\beta}{\dot{x}}_{10} - \dfrac{14T_{u}}{12M}{\dot{x}}_{11} \phi^{comp} \right] \phi_2^{impulsive} \phi^{damping} \\[1ex] \nonumber
& - \dfrac{\pi T_{u}}{4\beta}\dot{\alpha} \phi_2^{impulsive} \phi^{damping} }
\end{flalign}
\]
In this context, \(C_N^{circ}\) represents the circulatory contribution of the normal force coefficient represented as:
\[
\begin{equation}
\bscalar{ C_N^{circ} = \dfrac{d C_N^{st}}{d \alpha}\left( \alpha_{E} - \alpha_{0} \right)x_{4} + C_N^{fs}\left( \alpha_{E} \right)\left( 1 - x_{4} \right) + x_{5} }
\end{equation}
\]
Note that the contributions due to pitch rate on the moment coefficient are modified from Appendix C of Leishman, 1988. The equation is mathematically equivalent, but the constant circulatory part \(\bscalar{((\pi T_{u}\dot{\alpha})/(4\beta)}\) is
added explicitly rather than being part of the state-space equation. This latter contribution can be interpreted as the compressible version of the pitch-rate damping as in the incompressible model. However, for incompressible flow (\(\bscalar{\beta = 1}\)) the terms differ by a factor of 2. The high frequency states in this model are likely to slow down the aeroelastic simulations. It is therefore an option to switch on/off impulsive lift terms in the simulation.
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. A damping is also applied to control the Mach number dependent characteristics using \(\bscalar{\phi^{comp}}\) for small Mach values.
Last updated 30-08-2024