Orientaciones para desarrollar el componente Facilitación del Aprendizaje

3.4.1 Two–dimensional Wing Section

A prototype flying–wing is introduced to study the impact of the aerodynamic models on the flight dynamics. The geometry of the flying–wing has a constant chord of 1.0 m and a NACA0012 aerofoil is used to model the wing section.

The strip theory and the UVLM used here, are unable to capture three–dimensional aerodynamic effects around a finite span wing (e.g. wing tip vortices). As a result, the wing is assumed two–dimensional. Moreover, a unit span wing was used for the strip theory and a preliminary study with a varying aspect ratio of 1, 10, and 100 was performed in the UVLM to assess the influence on the response. It was found that the flight response was the same between an aspect ratio of 10 and an aspect ratio of 100 and as a result an aspect ratio of 10 was chosen.

The use of a unit wing span for the CFD results, reduces significantly the computa- tional cost. In this case, a two–dimensional solution of the flow was obtained around a NACA0012 aerofoil. The geometrical and material properties of the wing are given in Table 3.7. The flight response is performed at 50.0 m/s and sea level density. A free response is studied for an initial angle of attack of 1 deg. Herein, only the pitching flight dynamic degree–of–freedom is unconstrained to isolate the impact of the aerodynamic modelling on the response.

Table 3.7: Reference values of the two–dimensional wing section

Parameter Value

Elastic axis 5% chord

Centre of gravity 5% chord

Inertia properties

Mass per unit length 10.0 kg/m Mass moment of inertia (torsional) 10.0 kg·m

Geometry

Chord 1.0 m

Span ∞

Both the horizontal and vertical displacements are constrained and the flying–wing is only free to rotate about the elastic axis (e.a) which is placed at 5% of the chord from the leading–edge. Thin aerofoil theory suggests that the centre of pressure is at one quarter of the chord from the leading–edge and it is expected that this particular wing configuration would be dynamically stable.

Figure 3.13 shows the comparison of the free–to–pitch case for the three different aerodynamic models. The wing section is rotated for 1 degree positive nose up and is let free in both the UVLM and the strip Theory. The wing section in the CFD is not rotated and the flow direction is set to 1 degree angle of attack instead.

A study was performed to verify that the CFD solution was independent of the time step used. Two time steps were used (in physical time: 4.9_·10−₃

s and 1.0_·10−₃

s; in nondimensional time: 2.4_·10−1

and5.0_·10−2

based on the wing chord and freestream speed). Because no significant differences were found, the results presented subsequently are for the larger time step.

It is found that all three coupled computations are dynamically stable and in agree- ment with low speed aerodynamic theories. The differences between the flight responses are attributed to the different aerodynamic models used. Moreover, the oscillatory behaviour of the flight response is caused by the absence of structural damping and stiffness, and the different aerodynamic damping that is introduced by the different aerodynamic theories. However, all three predictions are very close to each other with the three–dimensional UVLM to predict, as expected, slightly better the CFD solution and strip theory, to be closer to the UVLM solution.

Time [s] P it c h a n g le [ d e g ] 0 0.5 1 1.5 -1.5 -1 -.5 0 .5 1 1.5 Strip/Flight UVLM/Flight CFD/Flight

Figure 3.13: Time–domain response of a free–to–pitch two–dimensional wing section; "Strip" denotes two–dimensional thin aerofoil theory (α∞ = 1.0deg,U∞= 50.0 m/s, h= 0.0 m, and

Re= 3.5·106 )

The impact of the aerodynamic models is assessed for the inclusion of additional flight dynamics degrees–of–freedom. The same wing configuration used here has two rigid–body degrees–of–freedom, one in the pitch rotation and one for the plunging mo- tion. The horizontal degree–of–freedom is kept constrained because linear aerodynamic models lack the ability to realistically predict the drag contributions. This allows a direct comparison of the responses computed by linear aerodynamic models with the CFD solution.

The time–domain solution of the angle of attack and the vertical displacement for the same initial condition already discussed above are shown in Figure 3.14. The aeroelastic behaviour predicted by the three coupled models is similar. Moreover, the steady–state response of the potential flow aerodynamics is identical as they both predict a similar steady–state pitching and vertical displacement.

Figure 3.13 and Figure 3.14 show that the wing response reaches the steady–state solution faster in the thin aerofoil aerodynamics which confirms that the aerodynamic damping in this case is larger. The least damped response is for the CFD predictions. Furthermore, the predicted steady–state solution is not always the same as it is shown to depend highly on the aerodynamic model used. For example, potential aerodynamics computed similar steady–state solutions whereas CFD gave different predictions. With no gravity acting on the system, the free–flying wing reaches a steady–state equilibrium when the effective angle of attack is zero. For a two degree–of–freedom system, the effective angle of attack is expressed as

α_{ef f} = θz −

˙

y U∞

(3.10) where y˙ is the velocity component in the vertical direction. From the above equation, it is apparent that the values of rigid–body pitch angle and vertical velocity component need to cancel out each other to yield an effective angle of attack equal to zero, e.g.

αef f = 0. Time [s] P it c h a n g le [ d e g ] 0 0.5 1 1.5 -.5 0 .5 1 1.5 Strip/Flight UVLM/Flight CFD/Flight

(a) Pitch angle

Time [s] V e rt d is p [ m ] 0 0.5 1 1.5 -0.1 0 0.1 0.2 0.3 0.4 0.5 Strip/Flight UVLM/Flight CFD/Flight (b) Vertical displacement

Figure 3.14: Time–domain response of a free–flying two–dimensional wing section; "Strip" denotes two–dimensional thin aerofoil theory (α∞ = 1.0deg,U∞= 50.0m/s,h= 0.0m, and

Re= 3.5·106 )