Modelling Continuously Morphing Aircraft for ... - Michael I Friswell

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Figure 11: Scheme of the m-th panel. Figure 12: Velocity of point a ′′ induced by the finite vortex Γ m of limit aa ′ . panel. This leads to the calculation of the influence matrices C x , C y and C z which provide, given the vector collecting the vortex strengths of each panel Γ = [. . . , Γ m , . . .] T , the vectors of the x, y and z component of the induced velocities w x = C x Γ w y = C y Γ (67) w z = C z Γ For example, w x collects the x component of the induced velocity of each panel so that w xm = ∑ n Cx mnΓ n . The boundary condition of no-flow-through at the control point of the m-th panel is −w xm sinδ m cosφ m − w ym cosδ m sin φ m + w zm cosδ m cosφ m + U ∞ sin(α − δ m )cosφ m = 0 (68) where δ m and φ m are the direction cosines of the normal to the panel in the x and y directions respectively. Considering Eqs. (67), and defining the following quantities Eq. (68) can be expressed in matrix form as Λ x = diag([. . . , sin δ m cosφ m , . . .] T ) Λ y = diag([. . . , cosδ m sin φ m , . . .] T ) Λ z = diag([. . . , cosδ m cosφ m , . . .] T ) b = [. . . , sin(α − δ m )cosφ m , . . .] T (69) (Λ x C x + Λ y C y + Λ z C z )Γ = U ∞ b (70) and the vortex strengths are found by solving the linear system of equations Γ = U ∞ (Λ x C x + Λ y C y + Λ z C z ) −1 b = U ∞ A −1 b(α) (71) Note that the A matrix in Eq. (71) is a non-linear function of the geometric parameters while the b vector also depends on the angle of attack. Once the vortex strengths are obtained, the loads on each panel are calculated by applying the generalised Kutta-Joukowsky law 40 at the mid-point of the quarter-chord bound vortex, in the form f m = ρ Γ m ( u∞ + w 1/4 ) × l (72) where l is the vector along the quarter-chord line of the panel, u ∞ is the free stream velocity expressed in global coodinates and w 1/4 is the induced velocity at the mid-point of the quarter-chord bound vortex. Expression (72) provides loads in the global coordinate system and takes into account the induced drag effect. 16 of 23 American Institute of Aeronautics and Astronautics
Figure 13: Scheme of the aerodynamic load distribution for the wing example. c 0.3 m b 1.8 m S 0.54 m 2 Λ LE 30 ( ◦ ) φ 5 ( ◦ ) AR 6 - λ 1 - N p 10 - Table 4: Geometric properties of the test case wing. IV.B. Validation A test case has been selected in order to validate the aerodynamic tool. The results are compared with AVL which is a publicly available vortex-lattice based code developed by Drela. 41 The geometric features of the wing under investigation (see Fig. 13) are shown in Tab. 4. The wing consists of a 30 ◦ leading edge sweep with 5 ◦ of dihedral, φ, and unitary taper ratio, λ, with constant chord c = 0.3 m. The wing section is modeled as a symmetric flat plate and the wing is discretised with a total number of 10 panels. Note that the vortex lattice code also allows the user to model non-symmetric configurations, which is typical for an elastic morphing wing producing lateral maneuvers. Thus no wall condition has been implemented to simulate the symmetry and, as a consequence, N p panels over the wing means the algebraic linear system given by Eq. (71) has dimension N p × N p . The same wing configuration has been reproduced with the AVL code, and the same number of panels along the wing span has been considered. The present analysis compares values of lift coefficient C L , induced drag coefficients C Di and moment coefficients C m , for a range of angle of attack between −2 ◦ and 6 ◦ . The results are shown in Figs. 14, 15 and 16, and an excellent correlation is obtained between the two approaches. For the sake of clarity, the lift coefficient comparison of Fig. 14 take into account the lift slope given by other theories which are 1. profile wing slope (AR → ∞), a 0 = 2π 2. finite wing slope 37 for elliptical lift distribution (Oswald number e = 1), valid for small values of sweep angle. a 0 a = 1 + a 0 /AR 3. Kuchemann 37 wing slope, which corrects the finite wing slope in the case of non-zero sweep angle Λ. a = a 0 cos(Λ) ( √ 1 + z 2 + z) (73) where z = a 0 cos(Λ)/(πAR). As expected the lift coefficients given by the Kuchemann theory, AVL and the vortex lattice under validation are very well correlated. For the induced drag coefficient (Fig. 15), again a good correlation can be observed with a maximum relative error of 7% which corresponds to an angle of attack of 6 ◦ . In terms of moment coefficient, which is evaluated about the leading edge of the root chord, the maximum relative error is 7% when the angle of attack is 6 ◦ . 17 of 23 American Institute of Aeronautics and Astronautics
Page 1 and 2: AIAA Guidance, Navigation and Contr
Page 3 and 4: Figure 2: Generic trapezoidal wing
Page 5 and 6: II.C. First-Order Shear Deformation
Page 7 and 8: the double integration over the win
Page 9 and 10: Vertical displacement [m] (ΛLE = 0
Page 11 and 12: % Λ LE = 0 ◦ Λ LE = 30 ◦ ε C
Page 13 and 14: where b ∆T = [α 1 , α 2 , 0] T
Page 15: 2 x 10−5 Vertical displacement [m
Page 19 and 20: U ∞ 30 m/s α 3 ( ◦ ) ρ 1.22 k
Page 21 and 22: is given in Fig. 18. Moreover, two
Page 23: 22 S. Tizzi, “Numerical Procedure

Modelling Continuously Morphing Aircraft for ... - Michael I Friswell

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