Wagner + Section
Type Definition
Aeroelasticity.WagnerSection — TypeWagnerSectionCoupling model for coupling an unsteady aerodynamic model based on Wagner's function (see Wagner) and a two-degree of freedom typical section model (see Section). This coupling introduces the freestream velocity $U_\infty$, air density $\rho_\infty$, and the Prandtl-Glauert compressibility factor $\beta$ as additional parameters.
The parameters for the resulting coupled model (as defined by the parameter function) defaults to the parameters for each model concatenated into a single vector.
Example Usage
Theory
This model is defined by coupling Wagner's function unsteady aerodynamics
with the typical section model.
To facilitate this coupling, the freestream velocity components $u$ and $v$ are assumed to be aligned with the undeflected chordwise and normal directions, respectively, so that
\[u \approx U_\infty \\ v \approx \dot{h} \\ \omega \approx \dot{\theta}\]
where $U_\infty$ is the freestream velocity magnitude, $\theta$ is pitch, and $h$ is plunge. To capture the effect of twist on the circulatory lift (since it is no longer implicitly modeled by the $\frac{v}{u}$ quantity) twist is added to the effective angle of attack from the Wagner's function model so that the effective angle of attack is now given by
\[\alpha_\text{eff} = \left(\theta - \frac{v}{u} + \frac{b}{u} \left( \frac{1}{2} - a \right) \omega - \alpha_0 \right) \phi(0) + \frac{\lambda_1}{u} + \frac{\lambda_2}{u}\]
The original expression for the effective angle of attack may be used by defining the new variable $\bar{v} = u \theta + v$ such that
\[\alpha_\text{eff} = \left(-\frac{\bar{v}}{u} + \frac{b}{u} \left( \frac{1}{2} - a \right) \omega - \alpha_0 \right) \phi(0) + \frac{\lambda_1}{u} + \frac{\lambda_2}{u}\]
A small angle assumption is also used to define the lift about the reference location as
\[\mathcal{L} \approx \mathcal{N}\]
where $\mathcal{N}$ is the normal force per unit span at the reference location.