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.Wing-tail configuration pressure coefficient

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.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.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.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.Wing-tail configuration wake development

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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Obtaining the 0-coefficients

Obtaining the angle of attack and elevator deflection derivatives

Obtaining the pitch rate derivatives

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