# Wing-tail configuration

Example simulation of a fixed-wing configuration

The wing-tail configuration has a wing aspect ratio $AR=10$, wing area $S_{ref}=0.3$m$^2$, wing chord $c_{ref}=0.1732$m. Both the wing and the tail have a taper ratio of 1. The distance between the quarter-chord points of the wing and the tail is $l=0.56$m. A body-fixed coordinate system is placed at the wing quarter-chord point. The reference speed is $|\mathbf{V}_{ref}|=25$m/s and the reference density, pressure, and viscosity are evaluated at an altitude of $h=0$m (ISA). An unstructured mesh with 6354 panels is used. The image below shows the pressure coefficient of the wing-tail configuration at an angle of attack $\alpha=5$deg.

The lift, drag and pitching moment coefficients of the wing-tail configuration are assumed to be given by:

$C_{X}=C_{X0}+C_{X,\alpha}\alpha+C_{X,\bar{q}}\bar{q}+C_{X,\delta_{e}}\delta_{e}\\ C_{Z}=C_{Z0}+C_{Z,\alpha}\alpha+C_{Z,\bar{q}}\bar{q}+C_{Z,\delta_{e}}\delta_{e}\\C_{m}=C_{m0}+C_{m,\alpha}\alpha+C_{m,\bar{q}}\bar{q}+C_{m,\delta_{e}}\delta_{e} \text{,}$

where $\bar{q}=\frac{qc_{ref}}{2V_{ref}}$ is the normalised pitch rate. The $C_{X}$, $C_{Z}$, and $C_{m}$ are functions of the the angle of attack $\alpha$, the normalised pitch rate $\bar{q}$ and the elevator deflection angle $\delta_{e}$. The aim is to obtain the $C_{X0}$, $C_{Z0}$, and $C_{m0}$ coefficients and the $C_{X,\alpha}$, $C_{X,\bar{q}}$ , $C_{X,\delta_{e}}$, $C_{Z,\alpha}$, $C_{Z,\bar{q}}$, $C_{Z,\delta{e}}$, $C_{m,\alpha}$, $C_{m,\bar{q}}$, and $C_{m,\delta_{e}}$ derivatives using APM.

## Obtaining the 0-coefficients

A simulation of the wing-tail configuration is performed at $\alpha=0$ rad, $p=0$ rad/s, $\delta_{e}=0$ rad. The values for the aerodynamic coefficients are recorded. They are the values of the $0$- coefficients.

## Obtaining the angle of attack and elevator deflection derivatives

As already discussed, a simulation of the wing-tail configuration is performed at $\alpha=0$rad, $p=0$ rad/s, $\delta_{e}=0$rad. The values for the aerodynamic coefficients are recorded. A second simulation of the wing-tail configuration is performed at $\alpha=0.1$ rad, $p=0$rad/s, $\delta_{e}=0$ rad. The choice of value for $\alpha$ is arbitrary. Again the aerodynamic coefficients are recorded. To obtain the $\alpha$- derivatives a simple difference is used:

$C_{Z,\alpha}=\frac{C_{Z}\rvert_{\alpha=0.1}-C_{Z}\rvert_{\alpha=0}}{\Delta\alpha}\text{,}$

where $\Delta\alpha=0.1$rad. The same method is used to obtain the $\delta_{e}$- derivatives.

## Obtaining the pitch rate derivatives

The coordinate system used for the analysis above is fixed to the wing quarter-chord point however, this point does not coincide with the centre of gravity. If one determines the $\bar{q}$- derivatives with respect to this coordinate system they will not be representative of the $\bar{q}$- derivatives with respect to the centre of gravity. To determine the position of the centre of gravity (often driven by a required static margin), the neutral point position must be found. By definition, the stick-fixed neutral point position is where:

$C_{m,\alpha}=0\text{.}$

Using the difference method described above, the value of $C_{m,\alpha}$ is obtained for two different locations of the centre of gravity. The first location is when the centre of gravity coincides with the quarter-chord point of the wing $x_{cg}=0$m and the second location is selected forward of the quarter-chord point of the wing $x_{cg}=0.05$m. The choice of value for $x_{cg}$ is arbitrary. The equation for $C_{m,\alpha}$ as a function of $x_{cg}$ is then:

$C_{m,\alpha}=ax_{cg}+b\\ -2.0012=a0+b \\ -3.5526=a0.05+b \\ a=-31.028 \\ b=-2.0012 \text{.}$

The solution of the equation gives $x_{cg}=-0.0645$, for $C_{m,\alpha}=0$. This is the location of the stick-fixed neutral point, $x_{np}$, ($x_{np}$=$x_{cg}$ for $C_{m,\alpha}=0$). The centre of gravity location for a 15% static margin is then given by:

$SM=0.15=\frac{x_{cg}-x_{np}}{c_{ref}}\\ x_{cg}\rvert_{SM=0.1}=x_{np}+0.15c_{ref} \text{,}$

with $x_{cg}=-0.03852$m. The $\bar{q}$- derivatives can now be obtained with respect to this point. A forced sinusoidal motion is imposed on the wing-tail combination:

$\alpha=Asin(\omega t) \\ \omega=2\pi f\text{,}$

where $A=5$deg is the amplitude of the motion and $f=5$Hz is the frequency of the motion. The video below shows the development of the wing-tail configuration wake during the forced sinusoidal motion.

The values of $C_{m}$ with respect to $\alpha$ make a quasi-steady elliptical hysteresis. Two points exist at which the angular velocity is maximum or minimum. $C_{m,\bar{q}}$ derivative is estimated from the values of the $C_{m}$ at these points

$C_{m,\bar{q}}=\frac{C_{m}\rvert_{q_{max}}-C_{m}\rvert_{q_{min}}}{2kA},\\ k=\frac{\omega c_{ref}}{2V_{ref}}\text{,}$

where $k$ is the reduced frequency. The same method is used to obtain the other $\bar{q}$- derivatives. Below are the wing-tail configuration derivatives estimated by the method.

 $_{0}$ $\frac{\partial}{\partial \alpha}$ $\frac{\partial}{\partial \bar{q}}$ $\frac{\partial}{\partial \delta_{e}}$ $C_{X}$ -0.0114 0.2555 -0.2491 -0.0195 $C_{Z}$ -0.3219 -5.1439 2.7170 -0.8377 $C_{m}$ 0.0556 -0.9575 -20.0581 -2.3205

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