# Wing-tail 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 ](https://81269019-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MV8htLPXN-_fWc_anY_%2F-MVfb2THhjc8DbQE2_tT%2F-MVfbklO0WszKasUf74a%2Fwing_tail_cp.png?alt=media\&token=5a75f9a8-65ea-4a12-9ce6-6494ad37449e) 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. {% embed url="" %} Wing-tail configuration wake development {% endembed %} 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 {% file src="" %}