An efficient numerical lifting line method for practical wing optimization through morphing on the Hydra Technologies UAS-S4
Oliviu Şugar Gabor (a), Andreea Koreanschi (a), Ruxandra Mihaela Botez (a) (a) Department of Automated Production Engineering, École de Technologie Supérieure, 1100 Notre Dame Ouest, Montréal, H3C1K3, Quebéc, Canada,
[email protected],
[email protected],
[email protected]
Abstract In this paper, we describe a practically efficient methodology of calculating the aerodynamic characteristics of an UAS-S4's morphing wing built by Hydra Technologies in Mexico. The method represents a three - dimensional, numerical extension of the classical lifting - line method developed by Prandtl. The finite span wing can have arbitrary sweep, dihedral, camber, twist, as well as various airfoil sections along the wingspan. It uses two - dimensional aerodynamic data from several span wise airfoil sections to calculate the wing characteristics. This particular feature allows for an approximation of the overall wing viscous drag coefficient, in addition to the lift and induced drag distributions predicted by the lifting - line theory. The numerical lifting line method can obtain good quality solutions for only a small fraction of the computational cost of the full CFD solution. This high efficiency, together with the absence of a computational mesh, recommend the method for being used in the UAS wings' optimization process. Introduction In recent years, Unmanned Aerial Systems (UAS) have become increasingly used in both military and civil aviation. Their main goal has been performing long time surveillance flights, at various altitudes and flight speeds, and sometimes in rapidly changing weather conditions. Because of the increasing demand of UAS's, engineers and designers have searched for methods to improve their flight performances, in order to make them more adaptable for various flight missions, to improve their aerodynamic efficiency and to increase their effective range and payload. One answer to all these aircraft design challenges is to use a morphing technique, to provide the aircraft with the capacity of detecting the changes occurring in the airflow around it and to adapt to them by modifying its geometry, usually the wing, during flight. Sofla et al [1] conducted a comprehensive study of the various aircraft morphing solutions proposed by different authors, identifying the part of the aircraft wing subjected to morphing, the design solution that was found in order to achieve the desired modifications, as well as indicating the maturity level of the proposed solution (theoretical analysis, functional prototype, wind tunnel tests, flight tests). The morphing approach used for the optimization of the UAS wing was developed during the CRIAQ 7.1 project [2], [3], [4]. The project took place between 2006 and 2009 and was realized following a collaboration approach between teams form École de Technologie Supérieure (ÉTS), École Polytechnique 1
de Montréal, Bombardier Aerospace, Thales Canada and the Institute for Aerospace Research – National Research Canada (IAR - NRC). The objective of the project was to improve and control the laminarity of the flow past a morphing wing, in order to obtain important drag reductions. In this project, the active structure of the morphing wing combined three main subsystems: a flexible, composite material upper surface, stretching between 3% and 70% of the airfoil chord; a rigid inner surface; an actuator group located inside the wing box, which could morph the flexible skin at two points, located at 25.3% and 47.6% of the chord [2]. An optimizer coupled with a subsonic aerodynamic solver with transition prediction capabilities allowed the flexible skin shape optimization, under the assumption of constant lift coefficient [3]. Very promising results were obtained; the morphing system was able to delay the transition location downstream by up to 30% of the chord, and to reduce the airfoil drag by up to 22%. [4] Prior to performing any aerodynamic optimizations to the UAS's wing by use of the morphing technique described, an aerodynamic solver best suited to perform the necessary calculations must be found. It must give the best possible results in the low speed regimes, it should not need any form of three - dimensional meshing, since the mesh must be regenerated for every change in the wing geometry, and it must not be computationally expensive, in order to be practically implemented in the optimization process. Finite Span Wing Model Prandtl's classical lifting line theory, first published in 1918 [5], represented the first analytical model capable of accurately predicting the lift and induced drag of a finite span lifting surface. The aerodynamic characteristics predicted by the theory were repeatedly proven to be in close agreement with experimental results, for straight wings with moderate to high aspect ratio. The theory was based on the hypothesis that a finite span wing could be replaced by a continuous distribution of vorticity bound to the wing surface, and a continuous distribution of shed vorticity that trails behind the wing, in straight lines in the direction of the free stream velocity. The intensity of these trailing vortices is proportional to rate of chance of the lift distribution along the wing span direction. The trailing vortices induce a velocity, known as downwash, normal to the direction of the free stream velocity, at every point along the span. Because of the downwash, the effective angle of attack at each section in the spanwise direction is different from the geometric angle of attack of the wing, the difference being called the induced angle of attack. Using the effective angle of attack, the downwash produced by the trailing vortices and the two - dimensional Kutta - Joukowski vortex lifting law, Prandtl developed an integral equation that allowed the calculation of the continuous bound vorticity intensity, and thus the calculation of the wing's lift and induced drag. The solution of Prandtl's classical equation is in the form of an infinite sine series for the bound vorticity distribution. Traditionally, the series is truncated to a finite number of terms, and collocation methods are used to determine the sine series coefficients, by requiring that the lifting line equation must be satisfied at a given number of spanwise stations. The method was firstly presented by Glauert [6]. Popular methods of determining the bound vorticity distribution included those developed by Tani [7] and Multhopp [8]. Other authors have proposed modified versions of the original lifting line theory, modifications that increase the quality of the obtained results, increase the applicability or the model or make use of information regarding the wing's airfoil sections [9], [10], [11].
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With the development of more efficient and powerful computers, several authors have also proposed purely numerical methods for solving Prandtl's lifting line equation. Among these, we can mention McCormick [12], Anderson et al [13] or Katz and Plotkin [14]. However, all the above mentioned numerical methods were based on Prandtl's assumptions of a straight distribution of bound vorticity, and therefore are subjected to all the limitations of the classical lifting line model: a single lifting surface of moderate to high aspect ratio, with no sweep angle and no means of considering the effects of the various wing sections airfoils. The method used in this paper is based on the original work of Phillips and Snyder [15]. Whereas the classical lifting line theory is based on the assumption of a straight lifting surface and the application of the two - dimensional Kutta - Joukowski vortex lifting law for a three dimensional flow, the modern adaptation uses a general horseshoe vortex distribution and uses a fully three - dimensional vortex lifting law. Because of these characteristics, the method has a much wider applicability range compared to the original theory, including multiple lifting surfaces and wings arbitrary camber, sweep and dihedral angle. Also, the method is not based on the assumption of a linear relationship between the lift coefficient and the local angle of attack of any given spanwise section of the wing, thus it can be applied for high geometric angles of attack, to account approximately for the effects of stall. The only limitation of the method is that it is no longer valid for wings with aspect ratios smaller than four, so the constraint of medium to high aspect ratio lifting surface that applies to Prandtl's original theory also applies to its' modern adaptation. Wing Calculation Method In numerical lifting line models, the continuous distribution of bound vorticity over the lifting surface and the continuous distribution of trailing vorticity are approximated using a finite number of horseshoe vortices. The bound portion of the vortices can be aligned with the wing's quarter chord line, thus taking into consideration the local values of the sweep and dihedral angles, while the trailing portions remain aligned with the free stream velocity, as shown in Figure 1.
Figure 1 Horseshoe vortices distributed along the wingspan [15] Each horseshoe vortex is made up of three straight vortex segments. To calculate the velocity induced, at an arbitrary point in space, by a vortex segment, we apply the Biot - Savart law [14]:
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r r Γ r1 × r2 r 1− 2 (1) 2 0 4π r1 × r2 r2 r1 In the above equation Γ is the vortex segment intensity, r1 and r2 are the spatial position vectors from the = V
starting and ending points of the vortex segment to an arbitrary point in space and r0 is the vector along the straight vortex segment. Although the Biot - Savart law is common to any aerodynamics book, its' form is not practical for a numerical implementation, since the denominator goes to zero when the two spatial position vectors are collinear. Phillips [15] proposes a more useful form which can be readily obtained by some simple trigonometric calculations:
V=
Γ ( r1 + r2 )( r1 × r2 ) 4π r1r2 ( r1r2 + r1r2 )
(2)
As mentioned, each horseshoe vortex is made up of three straight vortex segments, one bound to the quarter chord line of the lifting surface, and the other two aligned with the free stream velocity. Figure 2 presents the typical geometry of one horseshoe vortex. (X2, Y2, Z2) r0 r1 (X1, Y1, Z1)
r2 (X, Y, Z)
v∞ v∞ Figure 2 Geometry and position vectors for one horseshoe vortex When we apply equation (2) for all segments of the horseshoe vortex, we can determine the velocity induced at an arbitrary point in space:
v ∞ × r2 v ∞ × r1 Γ Γ ( r1 + r2 )( r1 × r2 ) Γ + − V= 4π r2 ( r2 − v ∞ r2 ) 4π r1r2 ( r1r2 + r1r2 ) 4π r1 ( r1 − v ∞ r1 )
(3)
When considering the distribution of horseshoe vortices over the lifting surface, equation (3) can be used to compute the velocity induced at any point on the lifting surface, by any of the horseshoe vortices, provided that the intensities Γ are known. If we approximate the continuous distribution of bound vorticity with a finite number of N distinct horseshoe vortices, each one having its own intensity Γi , we will need a mathematical system of N equations relating these intensities to some known properties of the wing. In order to find such a relation, we turn to the three dimensional vortex lifting law [16]. Using the same approach as that suggested by
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Saffman [16] or by Phillips [15], the non - viscous force acting on a horseshoe element of the lifting surface is equal to:
Fi =ρΓi Vi × l i (4) In the above equation, ρ is the air density, Γi is the unknown intensity of the horseshoe vortex, Vi is the local airflow velocity and l i is a spatial vector along the bound segment of the horseshoe vortex, aligned in the direction of the local circulation. The significant difference between equation (4) and the two - dimensional Kutta - Joukowski vortex lifting law used in the classical lifting line theory is the presence of the local airfoil velocity, instead of using the free stream velocity. This allows for introduction of three - dimensional effects such as wing sweep and dihedral angle in the numerical lifting line model, and also captures more accurately the influence of local wing section airfoil characteristics. In order to calculate the local velocity, with each horseshoe vortex we have to associate a control point at which the induced velocities will be determined. We consider the control points to be situated on the wing surface, at equal distance between the two semi-infinite trailing legs of the horseshoe vortex, at the three quarter chord point, as measured from the local section leading edge. The importance of choosing the three quarter chord point has been revealed by various authors [8], [14], [10]. Thus, the local velocity at each control point becomes: N
Vi = V∞ + ∑ Γ j v ij
(5)
j =1
We denote by V∞ the free stream velocity, while v ij represents the velocity induced by the horseshoe vortex j , considered to be of strength equal to unity, at the control point i , and is given by equation(3). From wing theory, we know that the magnitude of the non - viscous aerodynamic force acting on an Ai area section of the wing located at a control point i can be written as:
Fi=
1 ρV∞2 Ai CLi 2
(6)
The lift coefficient CLi is that of the local airfoil situated at the wingspan section corresponding to control point i and it depends on the local effective angle of attack. If the lift coefficient can be determined using other means, such as experimentally determined lift curves or using a two - dimensional airfoil calculation solver, then, by replacing the local velocity given by equation(5) into equation (4), and then equating the modulus of the three - dimensional vortex lifting force with the expression give in equation (6), we obtain the following relation:
N
j =1
1 2
ρΓi V∞ + ∑ Γ j v ij × l i =ρV∞2 Ai CL
i
(7)
By writing equation (7) for all control points along the wingspan, we obtain a nonlinear system of N equations for the calculation of the unknown horseshoe vortex intensities. The system can be solved in an iterative fashion, using Newton's method. However, in order to avoid the necessity of analytically 5
calculating all the entries in the system's Jacobian matrix, we use Broyden's quasi - Newton method [17], in which the Jacobian is replaced by an approximation that is updated at each iteration. Also, the right hand side of the nonlinear system must be updated each iteration, since the local lift coefficient depends on the local angle of attack, and each change in the horseshoe vortices intensities, changes that are inevitable in an iterative process, will cause a change in the local angles of attack values, as follows:
Vi ni (8) Vi ci As before, Vi represents the local airflow velocity, ni is a unit length vector in the direction normal to the
α i = tan −1
chord of the local airfoil and ci is a unit length vector in the direction of the chord of the local airfoil. Once the intensities of the horseshoe vortices have been determined, the total non - viscous aerodynamic force can be readily calculated by summing up all the individual forces given by equation(4), resulting: N Γ + Γ ρ V v ∑ i ∞ ∑ j ij × l i =i 1 = j 1
F=
N
(9)
The calculation method presented, like that developed in [11], can be used to approximate the profile drag coefficient of the wing, since the spanwise lift distribution will not only verify all the constraints imposed by the numerical lifting line theory, but also those imposed by several wingspan airfoil characteristics. As presented in [11], the profile drag coefficient of the wing is given by: b /2
= CD 0
1 1 N ( ) ( ) c y c y dy ≈ ∑ cD 0 ci ∆yi D0 S − b∫/2 S i =1 i
(10)
In the above equation, S represents the total wing area, b is the wing span, cD 0i is the profile drag coefficient of wingspan section i , as calculated from the available experimental data or calculated by the two - dimensional solver, while ci is the wing's chord at the given control point section. Two - dimensional flow solver The code used for the calculation of the two - dimensional aerodynamic characteristics of the wing's control sections is XFOIL, version 6.96, developed by Drela and Youngren [18]. The XFOIL code was chosen because it has proven its precision and effectiveness over time, and because it reaches a converged solution very fast. The inviscid calculations in XFOIL are performed using a linear vorticity stream function panel method. A Karman - Tsien compressibility correction is added, allowing good predictions all the way to transonic flow. For the viscous calculations, XFOIL uses a two - equation lagged dissipation integral boundary layer formulation, and incorporating the e N transition criterion. The flow in the boundary layer and in the wake is interacted with the inviscid potential flow by using the surface transpiration model. [18]
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Results The wing optimization procedure shall focus on replacing the conventional, rigid wing of the Hydra Technologies S4 Éhecatl Unmanned Aerial System with a morphing wing. The S4 was designed and build in Mexico by Hydra Technologies, and it was created as an aerial unmanned surveillance system, directed towards providing security and surveillance capabilities for the Armed Forces, as well as civilian protection in hazardous situations. It is a high performance vehicle, capable of reaching altitudes of 4500 m and cruising speeds of over 100 knots. The purpose of replacing the conventional wing with a morphing one is the ability to dynamically change its shape during flight, reduce the wing's drag coefficient, and thus, reduce engine fuel consumption. This will grant the S4 UAS extended flight times and a longer effective range, improving the cost efficiency of its operation. In this paper, we focus on validating the proposed numerical lifting line model, by comparing the aerodynamic characteristics determined by the model with the results obtained by a widely used commercial CFD solver. The test case chosen is the conventional, rigid wing of the Hydra Technologies S4 UAS, considered at several free - stream velocities and angles of attack within the S4's flight envelope. We chose to perform the calculations at four values of the Mach number, Mach = 0.05, 0.10, 0.15, 0.20 and for a range of angles of attack between -2 degrees and 8 degrees. Figure 3 shows the variation of the wing lift coefficient CL , total drag coefficient CD and pitching moment coefficient about the leading edge of the root chord CM , with the angle of attack, for a Mach number of 0.05. 1.00E+00
CL, CD, CM
8.00E-01 6.00E-01
CL LLM
4.00E-01
CL FLU
2.00E-01
CD LLM
0.00E+00
CD FLU
-2.00E-01
-2
0
2
4
6
CM LLM CM FLU
-4.00E-01 -6.00E-01
8
Angle of Attack
Figure 3 Aerodynamic coefficients comparison for Mach = 0.05 Figure 4 shows the variation of the wing lift coefficient CL , total drag coefficient CD and pitching moment coefficient about the leading edge of the root chord CM , with the angle of attack, for a Mach number of 0.10.
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1.00E+00
CL, CD, CM
8.00E-01 6.00E-01
CL LLM
4.00E-01
CL FLU
2.00E-01
CD LLM
0.00E+00
CD FLU
-2.00E-01
-2
0
2
4
6
8
CM LLM CM FLU
-4.00E-01 -6.00E-01
Angle of Attack
Figure 4 Aerodynamic coefficients comparison for Mach = 0.10 Figure 5 shows the variation of the wing lift coefficient CL , total drag coefficient CD and pitching moment coefficient about the leading edge of the root chord CM , with the angle of attack, for a Mach number of 0.15. 1.00E+00
CL, CD, CM
8.00E-01 6.00E-01
CL LLM
4.00E-01
CL FLU
2.00E-01
CD LLM
0.00E+00
CD FLU
-2.00E-01
-2
0
2
4
6
CM LLM CM FLU
-4.00E-01 -6.00E-01
8
Angle of Attack
Figure 5 Aerodynamic coefficients comparison for Mach = 0.15 Figure 6 shows the variation of the wing lift coefficient CL , total drag coefficient CD and pitching moment coefficient about the leading edge of the root chord CM , with the angle of attack, for a Mach number of 0.20.
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1.00E+00
CL, CD, CM
8.00E-01 6.00E-01
CL LLM
4.00E-01
CL FLU
2.00E-01
CD LLM
0.00E+00
CD FLU
-2.00E-01
-2
0
2
4
6
CM LLM CM FLU
-4.00E-01 -6.00E-01
8
Angle of Attack
Figure 6 Aerodynamic coefficients comparison for Mach = 0.20 Conclusions We have described a practically efficient methodology of calculating the aerodynamic characteristics of an UAS's morphing wing. Because the method uses two - dimensional aerodynamic data from several span wise airfoil sections to calculate the wing characteristics, it can accurately include the effects of changing the airfoil shapes through morphing. Before integrating the numerical lifting line method into an optimization process, we have compared the results obtained by the present method with the results obtained by a widely used commercial CFD solver. The test case was the conventional, rigid wing of the Hydra Technologies S4 UAS, considered at several free - stream velocities and angles of attack within the S4's flight envelope. The numerical lifting line method can obtain good quality solutions for only a small fraction of the computational cost of the full CFD solution. This high efficiency, together with the absence of a computational mesh, recommends the method for being used in the UAS wings' optimization process. Acknowledgements We would like to thank to the Hydra Technologies team in Mexico for their continuous support, mainly to Mr Carlos Ruiz, Mr Eduardo Yakin and Mr Alvaro Guttierez Prado. References [1] A. Sofla, S. Meguid, K. Tan and W. Yeo, "Shape morphing of aircraft wings: Status and challenges," Materials and Design, vol. 31, pp. 1284 - 1292, 2010. [2] V. Brailovski, P. Terriault, D. Coutu, T. Georges, E. Morellon, C. Fischer and S. Berube, "Morphing laminar wing with flexible extrados powered by shape memory alloy actuators," in Proceedings of SMASIS08 ASME Conference on Smart Materials, Adaptive Structures and Intelligent Systems, Ellicott City, Maryland, USA, 2008.
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[3] L. Pages, O. Trifu and I. Paraschivoiu, "Optimized laminar flow control on an airfoil using the adaptable wall technique," in Proceedings of the Canadian Aeronautics and Space Institute Annual General Meeting, Toronto, Canada, 2007. [4] C. Sainmont, I. Paraschivoiu, D. Coutu, V. Brailovski, E. Laurendeau, M. M. Y. Mamou and M. Khalid, "Boundary layer behaviour on an morphing wing: simulation and wind tunnel tests," in Canadian Aeronautics and Space Institute AERO09 Conference, Toronto, Canada, 2009. [5] L. Prandtl, "Tragflugel Theorie," Nachrichten von der Gesellschaft der Wisseschaften zu Gottingen, Vols. Geschaeftliche Mitteilungen, Klasse, pp. 451 - 477, 1918. [6] H. Glauert, The Elements of Aerofoil and Airscrew Theory, Cambridge: Cambridge University Press, 1927. [7] I. Tani, "A simple method of calculating the induced velocity of a monoplane wing," Rep. No. 111, Vols. IX, 3, Tokyo Imperial University, 1934. [8] E. Multhopp, "Die Berechnung der Auftriebsverteilung von Tragflugein," Luftfahrtforschung Bd. 15, vol. 4, pp. 153 - 169, 1938. [9] R. Jones, "Correction of the lifting line theory for the effect of the chord," NACA Technical Note No. 817, 1941. [10] J. Weissinger, "The lift distribution of swept back wings," NACA Technical Note No. 1120, 1947. [11] J. Sivells and R. Neely, "Method for calculating wing characteristics by lifting line theory using nonlinear section lift data," NACA Technical Note No. 1269, 1947. [12] B. McCormick, "The lifting line model," in Aerodynamics, Aeronautics and Flight Mechanics, 2nd Edition, New York, Wiley, 1995. [13] J. Anderson, S. Corda and D. Van Wie, "Numerical lifting line theory applied to drooped leading edge wings below and above stall," Journal of Aircraft, vol. 17, no. 12, pp. 898 - 904, 1980. [14] J. Katz and A. Plotkin, "Lifting line solution by horseshoe elements," in Low speed Aerodynamics, from Wing Theory to Panel Methods, New York, McGraw - Hill, 1991. [15] W. Phillips and D. Snyder, "Modern adaptation of Prandtl's classic lifting line theory," Journal of Aircraft, vol. 37, no. 4, 2000. [16] P. Saffman, "Vortex force and bound vorticity," in Vortex Dynamics, Cambridge, Cambridge University Press, 1992. [17] C. Broyden, "A class of methods for solving nonlinear simultaneous equations," Mathematics of Computation, vol. 19, pp. 577 - 593, 1965. [18] M. Drela and D. Youngren, "XFOIL, version 6.96 documentaion," 2001.
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