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.

# Obtaining the 0-coefficients

A simulation of the wing-tail configuration is performed at
$\alpha=0$
$p=0$
$\delta_{e}=0$
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
$\alpha=0$
$p=0$
$\delta_{e}=0$
rad. The values for the aerodynamic coefficients are recorded. A second simulation of the wing-tail configuration is performed at
$\alpha=0.1$
$p=0$
$\delta_{e}=0$
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.
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

# Files

wing_tail_configuration.zip
375KB
Binary