Wing-tail configuration
Example simulation of a fixed-wing configuration
The wing-tail configuration has a wing aspect ratio
AR=10AR=10
, wing area
Sref=0.3S_{ref}=0.3
m
2^2
, wing chord
cref=0.1732c_{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=0.56l=0.56
m. A body-fixed coordinate system is placed at the wing quarter-chord point. The reference speed is
Vref=25|\mathbf{V}_{ref}|=25
m/s and the reference density, pressure, and viscosity are evaluated at an altitude of
h=0h=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
α=5\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:
CX=CX0+CX,αα+CX,qˉqˉ+CX,δeδeCZ=CZ0+CZ,αα+CZ,qˉqˉ+CZ,δeδeCm=Cm0+Cm,αα+Cm,qˉqˉ+Cm,δeδe,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
qˉ=qcref2Vref\bar{q}=\frac{qc_{ref}}{2V_{ref}}
is the normalised pitch rate. The
CXC_{X}
,
CZC_{Z}
, and
CmC_{m}
are functions of the the angle of attack
α\alpha
, the normalised pitch rate
qˉ\bar{q}
and the elevator deflection angle
δe\delta_{e}
. The aim is to obtain the
CX0C_{X0}
,
CZ0C_{Z0}
, and
Cm0C_{m0}
coefficients and the
CX,αC_{X,\alpha}
,
CX,qˉC_{X,\bar{q}}
,
CX,δeC_{X,\delta_{e}}
,
CZ,αC_{Z,\alpha}
,
CZ,qˉC_{Z,\bar{q}}
,
CZ,δeC_{Z,\delta{e}}
,
Cm,αC_{m,\alpha}
,
Cm,qˉC_{m,\bar{q}}
, and
Cm,δeC_{m,\delta_{e}}
derivatives using APM.

A simulation of the wing-tail configuration is performed at
α=0\alpha=0
rad,
p=0p=0
rad/s,
δe=0\delta_{e}=0
rad. The values for the aerodynamic coefficients are recorded. They are the values of the
00
- coefficients.

As already discussed, a simulation of the wing-tail configuration is performed at
α=0\alpha=0
rad,
p=0p=0
rad/s,
δe=0\delta_{e}=0
rad. The values for the aerodynamic coefficients are recorded. A second simulation of the wing-tail configuration is performed at
α=0.1\alpha=0.1
rad,
p=0p=0
rad/s,
δe=0\delta_{e}=0
rad. The choice of value for
α\alpha
is arbitrary. Again the aerodynamic coefficients are recorded. To obtain the
α\alpha
- derivatives a simple difference is used:
CZ,α=CZα=0.1CZα=0Δα,C_{Z,\alpha}=\frac{C_{Z}\rvert_{\alpha=0.1}-C_{Z}\rvert_{\alpha=0}}{\Delta\alpha}\text{,}
where
Δα=0.1\Delta\alpha=0.1
rad. The same method is used to obtain the
δe\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
qˉ\bar{q}
- derivatives with respect to this coordinate system they will not be representative of the
qˉ\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:
Cm,α=0.C_{m,\alpha}=0\text{.}
Using the difference method described above, the value of
Cm,αC_{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
xcg=0x_{cg}=0
m and the second location is selected forward of the quarter-chord point of the wing
xcg=0.05x_{cg}=0.05
m. The choice of value for
xcgx_{cg}
is arbitrary. The equation for
Cm,αC_{m,\alpha}
as a function of
xcgx_{cg}
is then:
Cm,α=axcg+b2.0012=a0+b3.5526=a0.05+ba=31.028b=2.0012. 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
xcg=0.0645x_{cg}=-0.0645
, for
Cm,α=0C_{m,\alpha}=0
. This is the location of the stick-fixed neutral point,
xnpx_{np}
, (
xnpx_{np}
=
xcgx_{cg}
for
Cm,α=0C_{m,\alpha}=0
). The centre of gravity location for a 15% static margin is then given by:
SM=0.15=xcgxnpcrefxcgSM=0.1=xnp+0.15cref,SM=0.15=\frac{x_{cg}-x_{np}}{c_{ref}}\\ x_{cg}\rvert_{SM=0.1}=x_{np}+0.15c_{ref} \text{,}
with
xcg=0.03852x_{cg}=-0.03852
m. The
qˉ\bar{q}
- derivatives can now be obtained with respect to this point. A forced sinusoidal motion is imposed on the wing-tail combination:
α=Asin(ωt)ω=2πf,\alpha=Asin(\omega t) \\ \omega=2\pi f\text{,}
where
A=5A=5
deg is the amplitude of the motion and
f=5f=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
CmC_{m}
with respect to
α\alpha
make a quasi-steady elliptical hysteresis. Two points exist at which the angular velocity is maximum or minimum.
Cm,qˉC_{m,\bar{q}}
derivative is estimated from the values of the
CmC_{m}
at these points
Cm,qˉ=CmqmaxCmqmin2kA,k=ωcref2Vref,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
kk
is the reduced frequency. The same method is used to obtain the other
qˉ\bar{q}
- derivatives. Below are the wing-tail configuration derivatives estimated by the method.
0_{0}
α\frac{\partial}{\partial \alpha}
qˉ\frac{\partial}{\partial \bar{q}}
δe\frac{\partial}{\partial \delta_{e}}
CXC_{X}
-0.0114
0.2555
-0.2491
-0.0195
CZC_{Z}
-0.3219
-5.1439
2.7170
-0.8377
CmC_{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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