Wagner's Function

Type Definition

Wagner(; C1=0.165, C2=0.335, eps1 = 0.0455, eps2 = 0.3)

Wagner(; C1=0.165, C2=0.335, eps1 = 0.0455, eps2 = 0.3)

Example Usage

• Aeroelastic Analysis of a Typical Section
• Time Domain Simulation of a Typical Section
• Aeroelastic Analysis of the Goland Wing
• Steady State Aeroelastic Analysis of a Highly Flexible Wing
• Aeroelastic Stability Analysis of a Highly Flexible Wing

Theory

This model is a two-dimensional aerodynamic model with unsteady aerodynamics which are derived from Wagner's function.

Wagner's function models the indicial response of aerodynamic loads under a sudden change in downwash $w$ at the three-quarter's chord. The exact expression for Wagner's function is

\[\phi(t) = \frac{2}{\pi} \int_0^\infty \frac{Re(C) \sin ( \omega (u/b) t )}{\omega} d\omega\]

where $u$ is the freestream velocity in the chordwise direction, $\omega$ is the freestream angular velocity, $b$ is the semi-chord, and $C(\omega)$ is Theodorsen's function. In many cases, approximate versions of Wagner's function are used rather than the exact expression, of which one of the most common is the approximation of Wagner's function provided by R. T. Jones

\[\phi(t) = 1 - C_1 e^{-\varepsilon_1 (u/b) t} - C_2 e^{-\varepsilon_2 (u/b) t}\]

where $C_1 = 0.165$, $C_2 = 0.335$, $\varepsilon_1 = 0.0455$, and $\varepsilon_2 = 0.3$.

Normal Force, Axial Force, and Pitching Moment

Wagner's function may be used to model arbitrary airfoil motion using Duhamel's integral. We start by modeling the increment in the circulatory normal force $d \mathcal{N}_c(t)$ at time $t$ due to an increment in downwash $d w(t)$ at earlier time $\tau$ as

\[\frac{d \mathcal{N}_c(t)}{a_0 \rho_\infty u b} = \phi(t - \tau) d w(\tau)\]

where $\phi(t)$ is the impulse response function, which in this case is R. T. Jones' approximation of Wagner's function. Superimposing all previous impulse responses using Duhamel's integral yields the following expression for the instantaneous circulatory normal force.

\[\frac{\mathcal{N}_c}{a_0 \rho_\infty u b} = \int_{-\infty}^t d w(\tau) \phi(t - \tau) d\tau = w(0) \phi(t) + \int_{0}^t d w(\tau) \phi(t - \tau) d \tau\]

We can transform this equation using integration by parts, yielding

\[\frac{\mathcal{N}_c}{a_0 \rho_\infty u b} = w(t) \phi(0) - \int_{0}^t w(\tau) d\phi(t - \tau) d\tau\]

The integral in this expression may be expressed as a function of the aerodynamic states $\lambda_1$ and $\lambda_2$.

\[\lambda_1 = C_1 \varepsilon_1 \frac{u}{b} \int_0^t w(\tau) e^{-\varepsilon_1 (u/b) (t - \tau)} d \tau\]

\[\lambda_2 = C_2 \varepsilon_2 \frac{u}{b} \int_0^t w(\tau) e^{-\varepsilon_2 (u/b) (t - \tau)} d \tau\]

\[\lambda_1 + \lambda_2 = - \int_0^t w(\tau) d\phi(t-\tau) d\tau\]

The expression for the circulatory normal force then reduces to

\[\frac{\mathcal{N}_c}{a_0 \rho_\infty u b} = w(t) \phi(0) + \lambda_1 + \lambda_2\]

where the downwash at the three quarter's chord is given by

\[w(t) = v + b \left( \frac{1}{2} - a \right) \omega - u\alpha_0\]

and the aerodynamic states variables $\lambda_1$ and $\lambda_2$ are described by the ordinary differential equations

\[\dot{\lambda_1} = -\varepsilon_1 \frac{u}{b} \lambda_1 + C_1 \varepsilon_1 \frac{u}{b} w(t) \\ \dot{\lambda_2} = -\varepsilon_2 \frac{u}{b} \lambda_2 + C_2 \varepsilon_2 \frac{u}{b} w(t)\]

The same normal force, axial force, and pitching moment are used as in the quasisteady model, but with the following effective angle of attack

\[\alpha_\text{eff} = \left(\frac{v}{u} + \frac{b}{u} \left( \frac{1}{2} - a \right) \omega - \alpha_0 \right) \phi(0) + \frac{\lambda_1}{u} + \frac{\lambda_2}{u}\]

Compressibility Correction

A compressibility correction may be applied to the results of this model in order to extend their applicability. Applying a Prandtl-Glauert compressibility correction, the normal force, axial force, and pitching moment become

\[\mathcal{N}_\text{compressible} = \frac{\mathcal{N}}{\beta} \\ \mathcal{A}_\text{compressible} = \frac{\mathcal{A}}{\beta} \\ \mathcal{M}_\text{compressible} = \frac{\mathcal{M}}{\beta}\]

where $\beta=\sqrt{1 - M^2}$ is the Prandtl-Glauert factor, which is a function of the local section Mach number $M$.

Viscous Forces

After the Prandtl-Glauert compressibility correction has been applied, an extra force in the axial direction $\mathcal{F}_v$ may be added to account for viscous forces. The magnitude of this force is scaled using the $c_{d_0}$ coefficient.

\[\mathcal{F}_v = ρ b u^2 c_{d_0}\]