Aerodynamic Shape Optimization of Morphing Wings at Multiple Flight ...

Aerodynamic Shape Optimization of Morphing Wings at Multiple Flight Conditions D. F. Hunsaker * and W. F. Phillips † Utah State University, Logan, Utah 84322-4130 J. J. Joo ‡ Air Force Research Laboratory, Wright-Patterson AFB, OH 45433 It is shown that significant induced drag savings can be realized for aircraft with variable aerodynamic or geometric twist. Variable-twist aircraft can be designed such that a different lift distribution is used during high-loading maneuvers than in efficient forward flight. This allows the aircraft to operate efficiently in steady level flight while meeting specified structural loading constraints. The design methodology outlined in this work is shown to significantly increase the allowable wingspan for a given structural weight, and significantly reduce induced drag during steady level flight. Analytical relations are developed for optimum lift and weight distributions for rectangular wings. Results show that the induced drag for a rectangular wing with variable twist designed for a maximum load factor of 2.5 results in an induced drag reduction at cruise of 22.5 percent over that of a wing with a fixed elliptic lift distribution. Sample numerical optimization results are presented for an aircraft similar to the NASA Ikhana airframe. Optimum camber distributions were obtained for various flight phases including takeoff, climb, cruise, descent, and landing, with an average total aircraft drag reduction of about 10% over most of the flight envelope. It was found that the lift and drag could be adjusted to optimize various flight parameters. Additionally, it was found that roll/yaw coupling could be precisely controlled to produce adverse, proverse, or neutral yaw.

Nomenclature A An Bn B3s b Cσ c ~ CDp C D 0− 2 Di h I L Lc Ls

= = = = = = = = = = = = = = =

beam cross-sectional area Fourier coefficients in the lifting-line solution for the section-lift distribution, Eq. (1) Fourier coefficients in the lifting-line solution for the dimensionless section-lift distribution, Eq. (6) third Fourier coefficient for the section lift distribution used at the structural design limit wingspan shape coefficient for the stress-limited design, Eq. (10) local wing section chord length section parasitic drag coefficient polynomial coefficients for the section parasitic drag equation wing induced drag height of the beam cross-section beam section moment of inertia total wing lift total wing lift at cruise total wing lift at the structural design limit

* †

Assistant Professor, Mechanical and Aerospace Engineering, 4130 Old Main Hill, AIAA Member. Emeritus Professor, Mechanical and Aerospace Engineering, 4130 Old Main Hill, AIAA Senior Member. ‡ Research Engineer, AFRL/RQVC, AIAA Senior Member. 1 American Institute of Aeronautics and Astronautics Copyright © 2017 by Douglas F. Hunsaker and Warren F. Phillips. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Hunsaker, Phillips, and Joo

~ L ~ Lc ~ Ls

~ Mb ng nm Rn S Sb tmax V∞ W Wr Wn Ws ~ Wn ~ Ws z

γ

θ

κW ρ

σ max

F

= = = = = = = = = = = = = = = = = = = = = = =

local wing section lift local wing section lift at cruise local wing section lift at the structural design limit local wing section bending moment limiting load factor at the hard-landing design limit limiting load factor at the maneuvering-flight design limit load-factor ratio, Eq. (17) wing planform area proportionality coefficient between the weight of the wing structure per unit span and the wing section bending moment, Eq. (10) maximum airfoil section thickness freestream airspeed gross weight weight of the non-structural components carried at the wing root total weight of the non-structural components total weight of the wing structure weight of non-structural components per unit span carried within the wing weight of the wing structure per unit span spanwise coordinate relative to the midspan specific weight of the beam material change of variables for the spanwise coordinate, Eq. (1) weight distribution coefficient, Eq. (15) air density maximum longitudinal stress

I. Introduction

IXED-GEOMETRY aircraft optimization is generally a multi-objective optimization problem requiring constraints at various flight conditions to be satisfied simultaneously. For example, the aircraft is typically manufactured with a set amount of twist in the wings. This twist is often designed to minimize drag at the nominal cruise condition. However, the optimum twist distribution for nominal cruise is only the optimum twist distribution at that design lift coefficient. The lift coefficient changes with altitude, velocity, and aircraft weight, and therefore, the optimum twist distribution changes throughout the flight. Because the optimum twist distribution is only optimum at a specific operating condition, the aircraft generally spends a significant amount of time operating in flight conditions that are not optimal. Thus, significant trade studies must be performed in the design of traditional fixed-geometry aircraft in order to select the optimum fixed twist for an airframe. Recent advances in wing morphing technology have initiated interest in aircraft that significantly change shape during flight. These advancements allow an aircraft wing to morph such that the wing geometry can change to the optimal geometry for a given flight condition. Variable geometric and/or aerodynamic twist can be used to implement different lift distributions during different flight phases,1–5 thus minimizing drag over a range of operating conditions. One such morphing wing advancement is the variable-camber compliant wing (VCCW) technology developed at the Air Force Research Lab.3–5 This technology allows the camber of the wing to be varied as a function of span. Current versions of the VCCW employ a series of actuators placed at a number of spanwise locations, which each act independently. As each actuator expands or contracts, the camber of the wing changes due to compliant mechanisms residing in the wing. The current design can produce variations in camber of about 6%. Figure 1 shows a sample VCCW prototype in both low-camber and high-camber configurations.

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Figure 1. AFRL Variable Camber Compliant Wing prototype in undeformed, deformed (6% camber change) and twist configurations. From Prandtl’s classical lifting-line theory,6,7 an arbitrary spanwise section-lift distribution can be written as a Fourier sine series. Combining the Kutta–Joukowski law8,9 with the classical lifting-line solution for the spanwise section-circulation distribution yields ∞ ~ L (θ ) = 2bρV∞2 ∑ An sin( nθ );

θ ≡ cos −1 ( − 2 z b )

n =1

(1)

For a finite wing with no sweep or dihedral immersed in a uniform flow, Prandtl’s classical lifting-line equation relates this section-lift distribution to the spanwise chord-length and aerodynamic-angle-of-attack distributions. For any given wing planform, Prandtl’s lifting-line equation can be used to obtain the spanwise geometric- and/or aerodynamic-twist distribution required to produce any desired spanwise section-lift distribution.1,2,10–12 From the definitions of wing lift coefficient and aspect ratio, the classical lifting-line solution for the total wing lift coefficient12 can be written in terms of only the first Fourier coefficient in the sine series on the right-hand side of Eq. (1), i.e.,

π b 2A1 L = 2 1 S 2 ρ V∞ S

(2)

2 2 1 2 π b ρ V∞ A1

(3)

This can be rearranged to give

L =

From the definitions of wing drag coefficient and aspect ratio, the classical lifting-line solution for the induced drag coefficient12 can be written as

πb2 Di = 2 1 S 2 ρ V∞ S

∞

∑ nAn2

(4)

n =1

With the application of Eq. (3), Eq. (4) can be rearranged to give

Di =

2( L b ) 2 π ρ V∞2

∞

A2

n =1

1

∑ n An2

(5)

From Eqs. (1) and (3), it is convenient to write an arbitrary dimensionless spanwise section-lift distribution as

~ ∞   bL (θ ) = 4 sin(θ ) + ∑ Bn sin( nθ ) ; L π n=2 

Bn ≡

An , A1

z ≡ −

b cos(θ ) 2

(6)

In steady level flight, the total wing lift L must equal the gross weight W. Thus, using the definition of Bn from Eq. (6), for steady level flight, Eq. (5) requires 3 American Institute of Aeronautics and Astronautics

Hunsaker, Phillips, and Joo

Di =

∞  2(W b ) 2  1 + ∑ nBn2   2  π ρ V∞  n=2 

(7)

For a fixed weight and wingspan, this induced drag is minimized with the section-lift distribution having Bn = 0 for all n > 1. From Eq. (6), this yields the well known elliptic lift distribution introduced by Prandtl,7 i.e.,

~ bL (θ ) b cos(θ ) = 4 sin(θ ); z ≡ − L π 2

or

~ bL ( z ) = 4 1 − (2 z b) 2 π L

(8)

From Eq. (7), the induced drag in steady level flight for the elliptic lift distribution is

Di =

2(W b ) 2 π ρ V∞2

(9)

For a wing with any prescribed lift and wingspan, the elliptic lift distribution given in Eq. (8) will always result in minimum induced drag. Twist distributions that produce the elliptic lift distribution can be obtained analytically for simple wing geometries. For example, Phillips12 provides a closed-form analytical solution for optimum twist distributions of unswept wings as a function of chord distribution, aspect ratio, and lift coefficient. However, for more complex geometries, including wings with sweep, aircraft with multiple lifting surfaces, or wings in ground effect, solutions must be found numerically. Here we consider first the case of a single unswept wing, and evaluate the impact of variable twist technology on the wing structural design. We then show sample optimization computations at various flight conditions for a more complex geometry very similar to that of the NASA Ikhana airframe.

II. Minimizing Induced Drag with Variable Wing Twist In the analysis presented in the previous paper,13 we considered only wing designs restricted to the use of a single lift distribution during all flight phases. It is important to note that, when designing a wing, the weight of the wing structure is not determined by the bending moments encountered in steady level flight. The constraining loads are typically applied to the wing either during a hard landing or during high-load-factor maneuvers. Hence, optimizing a wing design to minimize induced drag during steady level flight, as obtained from Eq. (7) under the restriction of using a single lift distribution during all flight phases, becomes a compromise between minimizing the ratio W b for landing and/or maneuvering requirements and minimizing the function (1 + ∑ nBn2 ) during steady level flight. However, with today’s technology, the designer is not constrained to using a single lift distribution during all flight phases. To demonstrate how variable wing twist could be used to further reduce induced drag during steady level flight, we shall consider the stress-limited design of a rectangular wing under the constraint of prescribed wing loading. For the purpose of simplifying the analysis to obtain a closed form solution, we will assume no constraint on the weight distribution and we will neglect any weight penalty associated with the wing twisting mechanism. In general, for a maximum-stress constraint with spanwise symmetric wing loading, the total weight of the wing structure required to support the bending moment distribution at the structural design limit is given by13

Ws = 2

b 2

∫

z =0

~

| M b ( z) | Sb ( z )

dz ; S b ( z ) ≡

Cσ (t max c ) c ( z )σ max

γ

, Cσ ≡

2 I ( h t max ) Ah 2

(10)

Because we are considering a rectangular wing under the constraint of prescribed wing loading, the chord length is independent of z, i.e., c = S b; and Eq. (10) can be written conveniently as

Ws = 2 Sb

b 2

~

∫ | M b ( z ) | dz;

z =0

Sb ≡

2 I ( h t max ) Cσ (t max c )σ max W , Cσ ≡ b γ (W S ) Ah 2

4 American Institute of Aeronautics and Astronautics

(11)

Hunsaker, Phillips, and Joo

Because there is no constraint on the weight distribution, the bending moments and structural weight are minimized for any given wingspan with the weight distribution specified by13

~ ~ L ( z) ~ Wn ( z ) = (W − Wr ) − Ws ( z ) L

Wr =

ng − 1 W nm + n g

(12)

(13)

Assuming that the lift is positive over the entire semispan, the bending-moment distribution at the structural design limit is given by13

~ M b ( z) =

κW

where

~ L ( z ′) ( z ′ − z ) dz ′, for z ≥ 0 L z ′= z b 2

κW Wr ∫

ng − 1  Wr ≥ W nm , nm + n g ≡  ( n g − 1) W − n g , Wr < n g − 1 W Wr nm + n g 

(14)

(15)

Hence, using Eq. (13) in Eqs. (15) and (14) yields κ W ≡ nm and

~ M b ( z ) = nm RnW

~ Ls ( z ′ ) ∫ Ls ( z ′ − z ) dz ′, for z ≥ 0 z ′= z b 2

(16)

~ where Ls and Ls are the section lift distribution and total lift used at the structural design limit; and Rn ≡

ng − 1 nm + n g

(17)

Using the change of variables and Fourier series expansion for an arbitrary lift distribution, which are given in Eq. (6), Eq. (16) can be written equivalently as π

∞   ′ + θ Bn sin( nθ ′) [cos(θ ) − cos(θ ′)] sin(θ ′) dθ ′, for θ ≥ π sin( ) ∑ ∫  2 n=2  θ ′=θ 

~ n R M b (θ ) = m n Wb

π

(18)

Using the same change of variables in Eq. (11) yields

Ws =

π ~ γ (W S ) b2 | M b (θ ) | sin(θ ) dθ Cσ (t max c )σ max W θ =∫π 2

(19)

Using Eq. (18) in Eq. (19) gives

Ws =

nm Rn (W S )γ b 3 π π Cσ (t max c )σ max θ =∫π

π

∞   ′ + sin( θ ) Bn sin( nθ ′) [cos(θ ) − cos(θ ′)] sin(θ ) sin(θ ′) dθ ′dθ ∑ ∫  n=2  2 θ ′=θ 

Using the trigonometric identity sin(2θ ) = 2 cos(θ ) sin(θ ), Eq. (20) can be written

5 American Institute of Aeronautics and Astronautics

(20)

Hunsaker, Phillips, and Joo

Ws =

π

nm Rn (W S )γ b 3 2π Cσ (t max c )σ max θ =∫π

π

∞   sin(θ ′) + ∑ Bn sin( nθ ′) [sin( 2θ ) sin(θ ′) − sin( 2θ ′) sin(θ )]dθ ′dθ n=2  2 θ ′=θ 

∫

(21)

For any spanwise symmetric lift distribution, the Fourier coefficients Bn are exactly zero for all even n. Integration of the odd terms in Eq. (21) is easily shown to give exactly zero for all terms with n > 3. Finally, integration of the remaining terms gives

Ws =

nm Rn (W S )γ b 3 (1 + B3 s ) 32Cσ (t max c )σ max

(22)

where B3s is the n =3 Fourier coefficient for the section lift distribution used at the structural design limit. Because the gross weight is the sum of the weight of the non-structural components and the weight of the wing structure, from Eq. (22), the ratio of the gross weight to the wingspan can be written as 2 W = Wn + (1 + B3 s ) nm Rn (W S )γ b b b 32Cσ (t max c )σ max

(23)

From Eq. (23), we find that this ratio of gross weight to wingspan is minimized when the wingspan is given by

b =

3

16Cσ (t max c )σ maxWn (1 + B3 s ) nm Rn (W S )γ

(24)

Using the optimum wingspan from Eq. (24) in Eq. (23) gives the minimum possible ratio of the gross weight to the wingspan for the stress-limited design of a rectangular wing with any given all-positive spanwise symmetric lift distribution and the weight distribution specified by Eqs. (12) and (13). From Eq. (7) we see that this minimum ratio of gross weight to wingspan will result in minimum possible induced drag for any given lift distribution, air density, and airspeed. Because B3s is the only Fourier coefficient from Eq. (6) that contributes to the required weight of a rectangular wing with all-positive spanwise symmetric lift and the weight distribution specified by Eqs. (12) and (13), the ratio W b can be minimized for the critical landing and maneuvering requirements by using the dimensionless lift distribution

~ bLs (θ ) = 4 [sin(θ ) + B3 s sin(3θ )]; Ls π

z ≡ −

b cos(θ ) 2

(25)

Using Eq. (25) with Eq. (18), the bending-moment distribution at the structural design limit can be written as π ~ n R M b (θ ) = m n Wb ∫ [sin(θ ′) + B3 s sin(3θ ′)][cos(θ ) − cos(θ ′)] sin(θ ′) dθ ′, for θ ≥ π π 2 θ ′=θ

(26)

Using the trigonometric identity sin(2θ ) = 2 cos(θ ) sin(θ ), the integral in Eq. (26) can be evaluated to give a dimensionless bending-moment distribution at the structural design limit

~ M b (θ ) sin( 2θ )  sin(θ ) sin(3θ ) n R  = m n  π − θ + −  cos(θ ) + Wb π  2 2 4  4 12  sin( 2θ ) sin( 4θ )  sin(θ ) sin(5θ )   π + B3 s  − + −  cos(θ ) + , for θ ≥ 4 8  4 20   2 

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(27)

Hunsaker, Phillips, and Joo

Starting with Eq. (19) and continuing with our assumption that the section lift distribution remains positive over the entire semispan, a dimensionless total structural weight can be written in terms of the integral of this dimensionless bending-moment distribution, i.e.,

~ π Cσ (t max c )σ maxWs M b (θ ) = ∫ Wb sin(θ ) dθ (W S )γ b 3 θ =π 2

(28)

Using Eqs. (18) and (25) in Eq. (28) yields π Cσ (t max c )σ maxWs n m Rn = π θ =∫π (W S )γ b 3

π

∫ [sin(θ ′) + B3 s sin(3θ ′)][cos(θ ) − cos(θ ′)]sin(θ ) sin(θ ′) dθ ′dθ

(29)

2 θ ′=θ

The integrals in Eq. (29) can be evaluated to give

Cσ (t max c )σ maxWs n R = m n (1 + B3 s ) 32 (W S )γ b 3

(30)

Because steady level flight is a 1-g maneuver, the bending-moment distribution for steady level flight is13

~ M b ( z) =

 L~c ( z ′) ~  ~ W L − Wn ( z ′) − Ws ( z ′)  ( z ′ − z ) dz ′, for z ≥ 0 c  z ′= z  b 2

∫

(31)

~ where Lc and Lc are the section lift distribution and total lift used at cruise. Because the weight distribution was specified to minimize the bending-moment distribution for the critical landing and maneuvering requirements, using Eq. (12) with the section lift distribution used at the structural design limit, and applying this relation to Eq. (31), the bending-moment distribution for steady level flight can be written as ~ M b ( z) =

~  L~c ( z ′) Ls ( z ′ )  ( ) ( z ′ − z ) dz ′, for z ≥ 0 − − W W W r ∫  Lc Ls  z ′= z  b 2

(32)

Using Eqs. (13), (17), and (25) in Eq. (32), applying the change of variables given in Eq. (6), and using the elliptic lift distribution at cruise, the bending-moment distribution for steady level flight can be written equivalently as

~ M b ( z ) = Wb

π

π

∫ sin

θ ′=θ

2

(θ ′)[cos(θ ) − cos(θ ′)]dθ ′ π

− Wb (1 − Rn ) ∫ [sin(θ ′) + B3 s sin(3θ ′)][cos(θ ) − cos(θ ′)] sin(θ ′) dθ ′, for θ ≥ π π 2 θ ′=θ

(33)

The integrals in Eq. (33) can be evaluated to give the dimensionless bending-moment distribution for steady level flight with the elliptic lift distribution, i.e.,

~ sin(θ ) sin(3θ )  sin( 2θ )  M b ( z) R  = n  π − θ + −  cos(θ ) + 4  4 12  Wb π  2 2  sin( 2θ ) sin( 4θ )  sin(θ ) sin(5θ )  1 − Rn π − B3 s  − − +  cos(θ ) +  , for θ ≥ 2 20 4 8 4 π    

(34)

For B3 s < −1 3, the bending-moment distribution obtained from Eq. (27) will be negative over some portion of the wingspan. Hence, the absolute value of this bending-moment distribution should be used for computing the 7 American Institute of Aeronautics and Astronautics

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wing section weight. At each section of the wing, the wing structure and section weight must be sufficient to support the largest of the two bending moments computed from the absolute value of Eqs. (27) and (34). The induced-drag savings possible through this design method is a function of the Fourier coefficient B3s, the maximum maneuvering flight load factor, and the load-factor ratio given in Eq. (17). For the case of B3 s < −1 3, nm = 2.5, ng = 2.5, and Rn = 0.3, induced drag can be reduced by about 22.5% over that of a fixed elliptic lift distribution. In contrast, maximum induced-drag savings for wings with fixed-twist are on the order of 11% compared to wings with fixed elliptic lift distributions.13

III. Structural Advantages of Variable Lift Distributions Perhaps the single largest benefit of a morphing wing geometry is that the lift distribution can be altered to alleviate structural loads at various flight conditions. For example, current federal regulations require that the wing structure be designed such that a manned aircraft can withstand a maneuver of 2.5 g’s. Therefore, current practice follows a pattern of designing the aircraft to minimize drag at a cruise lift condition, and then multiplying that lift distribution by 2.5 for structural design. However, with current morphing technology, different lift distributions could be used at different points in the flight envelope. This would allow a different lift distribution to be used during maneuvering flight than during cruise. Consider, for example, the normalized lift distributions generated from Eq. (25) and shown in Fig. 2. The solid line represents a sample elliptic lift distribution at 1 g. This is the lift distribution that minimizes the induced drag, and very similar lift distributions are generally sought for efficient aircraft. The dashed line shows the same lift distribution at 2.5 g’s. This lift distribution also minimizes drag for the given lift coefficient and wingspan, and, due to current regulation, is the lift distribution this wing structure would be designed to if the wing were designed to minimize drag for the given span and a fixed lift distribution. The dotted line represents an alternate lift distribution at 2.5 g’s. This lift distribution was first suggested by Prandtl in 193314, and was obtained by minimizing the induced drag for a given lift and moment of inertia of lift. Figure 3 shows the corresponding normalized bending-moment distributions along the wing computed from Eq. (27). Again, the solid line represents the bending moment at 1.0 g for an elliptic lift distribution, the dashed line represents the design bending moment for a 2.5-g maneuver with an elliptic lift distribution, and the dotted line represents the bending moment for Prandtl’s 193314 lift distribution at 2.5 g’s. The structural weight required to support the bending moment at each spanwise location is proportional to the bending moment. Therefore, Fig. 3 shows that the structural weight currently required for aircraft is significantly larger at the root than the required structural weight to support 1-g flight. However, if the elliptic lift distribution could be used during 1-g flight and Prandtl’s lift distribution could be used during maneuvering flight, the structural requirements of the wing would be significantly reduced.

Figure 2. Possible optimum lift distributions including an elliptic lift distribution at 1 g, an elliptic lift distribution at 2.5 g’s, and Prandtl’s 193314 lift distribution at 2.5 g’s. 8 American Institute of Aeronautics and Astronautics

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Eq. (27)

Eq. (34)

Figure 3. Section bending moment distributions for an elliptic lift distribution at 1 g from Eq. (34), an elliptic lift distribution at 2.5 g’s from Eq. (27), and Prandtl’s 193314 lift distribution at 2.5 g’s from Eq. (27). The structural savings from using a different lift distribution during maneuvering flight than during cruise flight can be used to improve the efficiency of the airframe in one or both of two ways: 1) this weight reduction could be used to decrease the weight of the overall aircraft, and therefore, the required lift and associated drag; 2) this structural savings could be used to increase the wingspan of the aircraft without increasing the airframe weight. Initial calculations show that the wingspan of a rectangular wing designed for a fixed elliptic lift distribution can be increased by about 13.6% for a morphing wing, without increasing the wing structural weight. This increases the aspect ratio, and in turn, decreases the induced drag during cruise, as demonstrated in the following example.

IV. Numerical Optimization of a Long-Endurance Airframe As an example of numerical shape optimization for morphing wings, we consider here an airframe very similar to the NASA Ikhana airframe15, but with a main wing aspect ratio of about 22 and a main wing taper ratio of about 0.35. The mission of this long-endurance airframe was assumed to be such that the aircraft lifts off at sea level, climbs to an altitude of 15,000 ft, cruises at altitude at 84 mph, and then descends back to sea level for touch down. The total mission time is 8.8 hours, over which the aircraft is assumed to burn 635 lbs of fuel at a constant rate. The weight of the aircraft and payload without fuel is 1160 lbs. The aircraft was assumed to employ a NACA 4-digit airfoil with 10% thickness that varies linearly from the root to the tip. The fuselage and landing gear were aligned with the horizontal at cruise, and the fuselage drag was estimated by calculating the drag on a cylinder whetted area with a radius near the average radius of the Ikhana fuselage. This configuration was assumed as the baseline configuration, and results pertaining to this airframe will be referred to as the baseline configuration results. A VCCW configuration of this aircraft was also analyzed with the following changes relative to the baseline configuration. Due to the estimated structural advantages described in the previous section, the wingspan was increased by 13.6%. The wing model was constructed such that the airfoil camber could be specified at 5 equally spaced locations along each semispan of the wing. The camber was assumed to vary linearly between each of these 5 control points, which would be actuated individually. The change in weight of the airframe due to the actuators required to morph the wing was assumed to be negligible. The aerodynamic model was developed in MachUp16, an open-source tool for predicting the aerodynamics of multiple lifting surfaces. MachUp uses a modern numerical lifting-line algorithm to calculate span loading, and employs a viscous model based on the 2D airfoil viscous drag as a function of wingspan.16 This algorithm is generally accepted to be accurate for wings of aspect ratio greater than 4.0.17 This particular implementation of the numerical lifting-line algorithm is constructed in such a way as to allow the user to input a family of airfoils and their properties. The tool then allows the user to identify which airfoils to use at particular locations along the wing. MachUp linearly interpolates to evaluate airfoil properties between control locations for which the properties are 9 American Institute of Aeronautics and Astronautics

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specified. MachUp uses JavaScript Object Notation (JSON) file structures as well as simple formatted text files for input/output, which makes it ideal for interfacing with other software tools such as JavaScript, Python, or Matlab. For the optimization work, Python was used to interface with MachUp, and the freely available scipy.optimize routine was employed using the SLSQP solver. This solver provided a gradient-based optimization routine able to satisfy boundary limitations and constraints. The optimization algorithm was allowed to alter the airfoil at each of the 5 spanwise control points, and could vary the airfoil at each of these five locations within the family of airfoils ranging from a NACA -2410 to a NACA 8410. The NACA -2410 airfoil data was obtained by using a NACA 2410 airfoil flipped about the horizontal axis. This provided a possible 10% camber change through this family of airfoils, ranging from airfoils with some negative camber to airfoils with significant positive camber. Viscous drag is estimated in MachUp by applying a section parasitic drag coefficient as a function of the section lift coefficient according to the relation

~ ~ ~ C D p = C D 0 + C D1C L + C D 2 C L2

(35)

The parasitic drag coefficients required for Eq. (35) along with the other required properties of the airfoils used in this analysis are shown in Table 1. The section drag is integrated within MachUp to obtain the overall viscous wing drag. This drag is added to the induced drag calculated by MachUp to obtain the total wing drag reported here. This set up was used to optimize the airfoil camber as a function of span for several flight conditions. Figure 4 shows the resulting change in total drag relative to the baseline configuration due to the VCCW wing geometry and optimization. Each of these flight conditions is discussed in detail. Camber

Table 1. Aerodynamic properties for the family of airfoils used in this study. αL0 (rad) CL,α (1/rad) CmL0 Cm,α (1/rad) CD0 CD1

CD2

NACA -2410

0.03678

6.19764

0.05246

0.03258

0.00569

0.00450

0.01040

NACA 0010

0.00000

6.43365

0.00000

0.00000

0.00513

0.00000

0.00984

NACA 2410

-0.03678

6.19764

-0.05246

0.03258

0.00569

-0.00450

0.01040

NACA 4410

-0.07457

6.28807

-0.10446

0.01404

0.00839

-0.01036

0.00969

NACA 6410

-0.11158

6.12955

-0.15481

0.03094

0.01233

-0.01571

0.01040

NACA 8410

-0.14469

6.29908

-0.19912

-0.01202

0.01918

-0.02706

0.01611

Figure 4. Change in total wing drag and total aircraft drag at each flight condition for the VCCW airframe relative to the baseline. 10 American Institute of Aeronautics and Astronautics

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A. Ground Roll and Rotation During ground roll and rotation, the optimization tool was set to minimize total drag with the aircraft at a constant angle of attack equal to the cruise angle of attack of the baseline configuration. Because the fuselage was assumed to be horizontal during the cruise phase of the baseline airframe, using this angle of attack during ground roll models a ground-roll phase with the landing gear and fuselage aligned with the horizontal. Ground effect was included by mirroring the aircraft geometry across the ground plane.16,18 This allows the suppression of the wing vorticity due to the presence of the ground to be accurately modeled during takeoff. Figure 4 shows that during ground roll and rotation, the total drag of the wing was reduced by over 50% relative to the baseline configuration, which equates to a reduction in total drag for the total airframe of about 30% relative to the baseline configuration. This can offer a significant reduction in the balanced field length. Although minimizing the total drag during ground roll may be a possible first step in optimization during takeoff, it is likely not the most useful. The aircraft balanced field length is a function of the lift, drag, and rolling friction of the airframe from zero velocity up to the liftoff velocity. The aircraft drag and rolling friction are both functions of the lift. Therefore, in order to minimize the balanced field length of the airframe, the effects of lift, drag, and rolling friction must all be included in the model. This is an area of future research, which would be a simple extension to the work presented here. B. Liftoff, Climb, Cruise, and Descend Total drag was also minimized during the liftoff, climb, cruise, and descend phases. Ground effect was included during the liftoff phase directly following rotation, and neglected for the subsequent phases. Notice that during these phases, the total wing drag is about 15% less than the baseline configuration, and the total aircraft drag is about 10% less than the baseline configuration. This reduction in drag can have a significant effect on climb rate and/or fuel burn during climb. During the cruise phase, a reduction in total drag of 10% can be equated roughly to a similar increase in endurance or range. A 10% increase in aircraft range is significant, especially for long-endurance airframes such as the NASA Ikhana. The VCCW airframe allows the drag to be minimized at each point during cruise. The optimum camber distribution will be a function of the current weight coefficient, which is a function of velocity, density, and aircraft weight. As the aircraft burns fuel, the lift coefficient changes, which, in turn, changes the optimum camber distribution. During the descent phase, it is not always desirable to minimize drag. In fact, it is often beneficial to vary drag such that the glide slope of the aircraft can be precisely controlled. The optimization tool was used to evaluate the range of total drag that could be produced by the wing during the descent phase, while holding the lift coefficient constant. It was found that the total wing drag could be varied from about -20% to +50% relative to the baseline configuration, which equates to a change in total aircraft drag of -10% to +25% relative to the baseline configuration. The optimization tool could be used to select any drag production within this range during flight. The ability to select a desired drag coefficient while maintaining a desired lift coefficient is a significant benefit of the VCCW design. C. Flare, Touchdown, and Ground Roll During the flare phase, the optimization tool was used to minimize drag, and the influence of the ground was included. Again, the drag could be adjusted for a given lift coefficient and could be used similar to flaps and/or spoilers. At touchdown, the optimization tool was used to maximize total wing drag during ground roll. Results show that the total wing drag was increased by over 35%, and the total aircraft drag was increased by over 20% relative to the baseline configuration. This significantly affects the ground roll distance required for the aircraft to come to a complete stop. However, the required distance is, again, a complex function of lift, drag, and rolling friction. Future work in this area could be performed to study the optimum camber distributions to minimize ground roll rather than to maximize drag. D. Maximum Wing Loading Perhaps the largest advantage of the VCCW technology is the ability to change the lift distribution in high-g maneuvering flight. During such a maneuver, which could be encountered during maneuvering flight or turbulence, the VCCW technology allows the wing to move the lift inboard and minimize the structure required to support the lift. An optimization case was set to minimize the root-mean-square of the lift distribution relative to Prandlt’s 1933 lift distribution.14 This produced a camber distribution that, in turn, produces a lift distribution along the wing to match the distribution suggested by Prandtl. It was found that the resulting lift distribution produces a total wing drag increase of 5.8% and a total aircraft drag of 3.7% relative to the baseline configuration at the cruise lift 11 American Institute of Aeronautics and Astronautics

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coefficient. Hence, this configuration could be used during maneuvering flight or in turbulent atmospheric conditions to minimize the structural impact to the airframe with only a minimal impact to the aircraft flight time. In an ideal situation, the turbulent atmospheric conditions or times of maneuvering flight would only represent a small portion of the overall flight time of the mission. E. Lateral Control One significant benefit of the VCCW technology is the ability to alter the lift distribution such that it can be used for lateral control. Current technology relies on articulated control surfaces, such as ailerons, placed outboard along the wing, which are sized to produce adequate rolling moments for maneuvering flight. Most current aileron designs suffer from a natural phenomenon referred to as adverse yaw, in which a yawing moment is produced opposite to the direction the pilot wishes to turn due to the aileron input. Adverse yaw is a result of increasing drag as lift is increased on one semispan of the wing. On the opposite wing where the lift is decreased due to aileron deflection, drag is usually decreased. This creates a yawing moment in the opposite direction as that intended for the turn caused by the rolling moment. Adverse yaw is one reason vertical stabilizers and rudders are required on conventional airframes.19 On the baseline configuration, the articulated ailerons extend from about mid-semispan to the tip, and have a chord fraction of about 0.3. For a positive 5-degree aileron deflection, a rolling moment coefficient of -0.037 and an adverse yawing moment of 0.0003 are produced. Two optimization cases were run to evaluate the effect of the VCCW technology. First, the optimization tool was set to minimize the change in camber distribution at a given flight condition while constraining the rollingmoment coefficient to match that of the 5-degree aileron deflection from the baseline configuration and constraining the yawing moment to be zero. This is called a pure rolling case. Results of this optimization solution produced an optimum camber distribution, which results in a rolling-moment coefficient that matches that of the articulated control surfaces, but produces zero yawing moment. The total wing drag of this configuration was 30% less than that of the baseline configuration with the deflected ailerons. A second optimization case was set to minimize the change in camber distribution at a given flight condition while constraining the rolling-moment coefficient to -0.037, and constraining the yawing-moment coefficient to -0.0003. Results of this camber distribution produced proverse yaw, in which the yawing-moment coefficient was the same magnitude of the adverse yaw condition produced from the baseline configuration. Figure 5 shows the lift distributions of the baseline configuration with a 5-degree aileron deflection, the pure roll optimization solution, and the proverse yaw optimization solution. Each of these lift distributions produce the same rolling moment coefficient, but varying yawing moment coefficients from adverse to proverse yaw. Results of the two optimization cases show that the VCCW technology could be used to produce yawing moments ranging from significant proverse yaw to significant adverse yaw. Such roll and yaw control coupling could be very significant for aircraft, in that the size of the vertical stabilizer and/or rudder could possibly be reduced. The empennage drag on the baseline configuration is estimated to be between 4% and 7% of the total drag on the airframe. Reducing the size of the empennage would provide an additional endurance benefit of this morphing wing technology.

Figure 5. Lift distributions producing the same rolling moment coefficient, but resulting in varying levels of adverse, neutral, and proverse yaw. 12 American Institute of Aeronautics and Astronautics

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It should be noted that the large drag benefits realized from the VCCW technology and presented here rely on the fact that this technology allows a larger wingspan for a given wing structural weight than the baseline configuration. The increased span has a significant impact on the induced drag, which can be optimized at nearly every point in the flight envelope. If the VCCW technology is applied to the baseline configuration without an increased span and without decreasing the structural weight of the wing, optimization results show a total aircraft drag decrease of only 1-2% over the flight envelope. To obtain the largest benefits of this technology, the structural properties of the wing must be designed to take advantage of the load alleviation at high-g maneuvering. Even if the increased wingspan or reduced wing structure cannot be realized on a given airframe, an aircraft could benefit from the roll/yaw control coupling, the ability to vary the drag coefficient for a given lift coefficient, and the ability to minimize balanced-field length and/or landing distance, which are all possible from this morphing-wing technology.

V. Conclusions It has been shown that significant drag savings can be realized for aircraft that have the ability to alter lift distributions during flight. This is due to the fact that the structural requirements at each wing section are a function of the local bending moment, which is a function of the lift distribution, mass distribution, and aircraft acceleration. Aircraft that can alter lift distributions during flight have the ability to control the bending moment distribution along the wing for different flight phases and operating conditions. Thus, a lift distribution can be used during highloading maneuvers to minimize the structural requirements of the wing, and an alternate lift distribution can be used during cruise to minimize induced drag. This can provide significant weight and induced drag savings if the weight of the morphing mechanism does not severely add to the wing structural weight. Initial results show that for a rectangular wing, induced drag during cruise can be reduced by as much as 22.5% by using the design methods discussed here. This is about twice the induced drag savings that can be realized from wings that have fixed twist distributions.16 The available induced drag savings are a function of the Fourier coefficient B3s, the maximum maneuvering flight load factor, and the load-factor ratio given in Eq. (17). Example optimization results were presented for an airframe very similar to the NASA Ikhana using this VCCW technology. Using the MachUp software along with an optimization tool in Python, optimum camber distributions were obtained for various flight phases. It was found that the optimized aircraft produced about 10% less drag over most of the mission profile compared to the baseline configuration. During takeoff and landing, the lift distribution could be adjusted to minimize or maximize drag. During descent, a range of drag values could be obtained for a given lift coefficient, thus allowing the descent rate to be controlled. Rolling moments could be generated with neutral yaw, adverse yaw, or proverse yaw. This roll/yaw coupling can have a significant impact on vertical tail sizing and performance of future aircraft. The methods demonstrated in this work could be applied to other airframes to obtain optimum geometric or aerodynamic twist distributions for various flight phases. Future work will focus on a better understanding of generic roll/yaw coupling, and a general algorithm for minimizing balanced field length.

Acknowledgement This work was partially performed during a summer faculty fellowship sponsored by the Air Force Research Laboratory. The authors wish to express their appreciation for the support of AFRL. This paper has been cleared for public release, Case Number: 88ABW-2016-6088.

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W. F., Alley, N. R., and Goodrich, W. D., “Lifting-Line Analysis of Roll Control and Variable Twist,” Journal of Aircraft, Vol. 41, No. 5, 2004, pp. 1169–1176. 2Phillips, W. F., and Alley, N. R., “Predicting Maximum Lift Coefficient for Twisted Wings Using Lifting-Line Theory,” Journal of Aircraft, Vol. 44, No. 3, 2007, pp. 898–910. 3Miller, S. C., Rumpfkeil, M. P., and Joo, J. J., “Fluid-Structure Interaction of a Variable Camber Compliant Wing,” AIAA Paper 2015-1235, 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida, 5–9 January 2015. 4Joo, J. J., Marks, C. R., and Zientarski, L., “Active Wing Shape Reconfiguration using a Variable Camber Compliant Wing System,” 20th International Conference on Composite Materials, Copenhagen, Denmark, 19–24 July 2015. 5Marks, C. R., Zientarski, L., and Joo, J. J., “Investigation into the Effect of Shape Deviation on Variable Camber Compliant Wing Performance,” AIAA Paper 2016-1313, 24th AIAA/AHS Adaptive Structures Conference, San Diego, California, 4–8 January 2016.

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6Prandtl, L., “Tragflügel Theorie,” Nachricten von der Gesellschaft der Wissenschaften zu Güttingen, Ges-chäeftliche Mitteilungen, Klasse, 1918, pp. 451–477. 7Prandtl, L., “Applications of Modern Hydrodynamics to Aeronautics,” NACA TR-116, June 1921. 8Kutta, M. W., “Auftriebskräfte in Strömenden Flüssigkeiten,” Illustrierte Aeronautische Mitteilungen, Vol. 6, No. 133, 1902. 9Joukowski, N. E., “Sur les Tourbillons Adjoints,” Traraux de la Section Physique de la Societé Imperiale des Amis des Sciences Naturales, Vol. 13, No. 2, 1906. 10Phillips, W. F., “Lifting-Line Analysis for Twisted Wings and Washout-Optimized Wings,” Journal of Aircraft, Vol. 41, No. 1, 2004, pp. 128–136. 11Phillips, W. F., Fugal, S. R., and Spall, R. E., “Minimizing Induced Drag with Wing Twist, Computational-Fluid-Dynamics Validation,” Journal of Aircraft, Vol. 43, No. 2, 2006, pp. 437–444. 12Phillips, W. F., “Incompressible Flow over Finite Wings,” Mechanics of Flight, 2nd ed., Wiley, Hoboken, NJ, 2010, pp. 46–94. 13Hunsaker, D. F., Phillips, W. F., and Joo, J. J., “Designing Wings with Fixed Twist for Minimum Induced Drag,” 55th AIAA Aerospace Sciences Meeting, AIAA Paper 2017-1419, Grapevine, Texas, 9–13 January 2017. 14Prandtl, L., “Über Tragflügel kleinsten induzierten Widerstandes,” Zeitschrift für Flugtechnik und Motorluftschiffahrt, Vol. 11, 1933, pp. 305–306. 15Boyle, D., "Flight Performance and Profile Analysis for Atmospheric Greenhouse Gas Measurements with the NASA Ikhana UAS", AIAA-2011-1560, Infotech@Aerospace 2011, St. Louis, Missouri, 29–31 March 2011. 16Hodson, J., and Hunsaker, D. F., “Wing Optimization in MachUp using Dual Number Automatic Differentiation,” 55th AIAA Aerospace Sciences Meeting, AIAA Paper 2017-0033, Grapevine, Texas, 9–13 January 2017. 17Phillips, W. F. and Snyder, D. O., “Modern Adaptation of Prandtl’s Classic Lifting-Line Theory,” Journal of Aircraft, Vol. 37, No. 4, 2000, pp. 662–670. 18Phillips, W. F. and Hunsaker, D. F., “Lifting-Line Predictions for Induced Drag and Lift in Ground Effect,” Journal of Aircraft, Vol. 50, No. 4, 2013, pp. 1226–1233. 19Bowers, A. H., Murillo, O. J., Jensen, R., Eslinger, B., and Gelzer, C., “On Wings of the Minimum Induced Drag: Spanload Implication for Aircraft and Birds,” NASA/TP—2016–219072, March 2016.

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