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Unconventional and Morphable Biologically-Inspired
                   Aircraft Control Elements

I.     Introduction

    Throughout history man has maintained a unique fascination with the natural aviators of our skies...
by the entire Nexstar wing up to the S1210 stall angle. Second, the drag polar displays a favorable box shape
indicating a...
q       Velocity Velocity      Deflection
                                                                  (psf)       (f...
2. Performance
     Current flap deflection is limited to approximately 20 degrees (measured relative to trailing edge pos...
B. Linkage Design through Structural Analysis
    A major design challenge for the outer wing was developing a sweeping me...
The mounting block rests directly inside the fuselage and is constructed of balsa wood to minimize weight. A
small gap exi...
were balanced. For the purposes of this report a list of all configurations would be impractical, however an extensive
the control surface, thus providing a twisting/warping of the surface rather than just a constant span-wise deflection.
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Aiaa Ncsu


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AIAA Student Paper submission from Senior Design for our Biologically Inspired Aircraft

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Aiaa Ncsu

  1. 1. Unconventional and Morphable Biologically-Inspired Aircraft Control Elements Alan Brown, Aaron Bowe, Jamar Dawson, Seung Eun, Alex Gura, Keena Holley, George Locklear, Jeremy Loman, Jason Morgan, and David Vickery1 Department of Mechanical and Aerospace Engineering North Carolina State University Raleigh, NC 27695 Research and experimentation suggest that the modification of conventional aircraft design features by incorporating biologically inspired technologies can result in performance enhancements. Several biologically inspired features were implemented in an effort to improve the performance of a pre-existing aircraft. A Hobbico Nexstar EP trainer aircraft was used as a base model on which to employ unconventional bio-inspired wing and tail designs. The wing design incorporates morphing/variable geometry technologies and was directly inspired by basic anatomy and characteristics of bird wings. Flexible skins with shape memory alloy actuators were used to provide roll control and lift augmentation through continuous deformation of the primary lifting surface airfoil. A sweep-adjustable outer wing consisting of thin overlapping feather surfaces was used to provide yaw control and flight adaptability, though the former was found to be less productive than desired. Inspired by the efficiency and apparent simplicity of bird tails, a gimbaled tail was developed without the use of a vertical stabilizer. A combination of pitch and rotation applied to a simple NACA0012 tail geometry allows for complex, multiple degree of freedom motion in an attempt to mimic bird tail movements and compensate for the vertical stabilizer omission. Loss of flight stability is determined to be excessive compared to the relative improvements in maneuverability. Much additional research is required for the successful integration of both system designs, performance optimization, and quantitative analysis of performance improvements. Additional design features are considered for future applications that may enhance and/or supplement the current design. NOMENCLATURE: α = Angle of attack Cl = Life coefficient Cl,α = Lift coefficient per angle of attack Cm = Pitching moment coefficient Cm,α = Pitching moment per angle of attack Cm,δ = Change in pitching moment per elevator deflection Cmo = Pitching moment when the plane is at an angle of attack of zero δ = Elevator deflection angle h = Center of gravity as percent chord hn = Neutral point as percent chord I = Electric current q = Dynamic Pressure R = Resistance V = Voltage 1 North Carolina State University, Mechanical and Aerospace Engineering, Box 7910, Raleigh, NC 27695
  2. 2. I. Introduction Throughout history man has maintained a unique fascination with the natural aviators of our skies, creatures that have evolved over time to become practically perfected flying machines. Humanity’s earliest attempts at flight were often limited to pure imitation, with little to no understanding of the physical principles that make it possible. As we progressed our fundamental understanding of the science involved, the reality of manned flight was realized. Dominated by rigid, mechanical systems, the evolution of manmade aircraft seems to have little in common with the natural flyers that provided the initial spark of inspiration (due largely to technological limitations of the structures and materials available, etc.) However, recent years have seen major advancements in areas such as material science and microelectronics and allow for a much broader range of design possibilities. Strong, flexible and lightweight materials show an incredible potential for aerospace applications and allow for nature to once again be used as a feasible source of inspiration in designing new technologies. This is especially evident in a growing area of interest in the aerospace industry – morphing. Morphing involves a seamless shape change, unlike the discrete, mechanical morphing typically seen through the use of flaps and ailerons. Many researchers (see Namgoong, et. al. 1, Kornbluh, et. al.2) have begun looking into various morphing technologies such as foldable, warpable, or twistable wings, continuous camber change, etc. This study aims to assess the characteristics and feasibility of several biologically-inspired control surface modifications/additions to a standard RC trainer aircraft (Hobbico Nexstar EP). Primary design criteria involved including unconventional technologies (morphing, etc.) inspired by basic principles of bird flight and anatomy, while also striving to improve the overall performance and efficiency of the aircraft. Entirely new wing and tail systems were developed through the application of the established design criteria just mentioned. The wing design, directly based off basic bird anatomy, incorporates a rigid inner section to serve as the primary lifting surface and an independent, variable-geometry outer wing section for auxiliary control. Avian anatomy also influenced the decision to eliminate a vertical stabilizer, a major design challenge that required innovative solutions and lead to the unique design of a multi-degree of freedom tail surface. II. Continuous Deflection with SMA Actuation Traditional remote controlled aircraft utilize hinged flaps on the wings actuated by servos to provide lift augmentation and roll control. Deflecting the flap effectively changes the airfoil camber, thereby increasing lift production. However, this discrete mechanical change is not typically found in nature. Rather, birds use passive shape modifications through the muscles and feathers in and along the wing to provide control while gliding. The design goal was to replace the servo-hinge system and attempt to mimic the capabilities of avian wings in order to increase aerodynamic efficiency and performance of roll control. A. Continuous Deflection The aerodynamics with which the design is concerned is that influenced by low Reynolds numbers. It has been shown that, except for very thin airfoils, the range of angle of attack within which the aerodynamics is satisfactory decreases with decreasing Reynolds number, but that a modest camber can allow for significant aerodynamic improvements when compared relative to similar airfoils with no camber (higher lift curve slope, greater maximum lift and maximum lift to drag ratio)3. The inner-wing incorporates the highly cambered S1210 high-lift, low Reynolds number airfoil which is necessary in providing sufficient lift with the relatively small planform area. Using the original Nexstar model wing as a comparison, two distinct advantages of our wing design are realized. First, XFoil analysis for the rigid wing suggests significant and necessary improvements in the lift curve slope (slightly more inclined), maximum lift coefficient and lift coefficient at zero angle of attack. Theoretical lift calculations are promising in that the lift Figure 1. Continuous Surface during provided by the inner-wing exceeds or matches that provided (a) rest; (b) deflection
  3. 3. by the entire Nexstar wing up to the S1210 stall angle. Second, the drag polar displays a favorable box shape indicating a minimal drag penalty for increasing lift up to a precipitous increase in drag in the post-stall region. In order to promote drag reduction, the inner-wing control surface camber is adjusted through a continuous surface deflection/morphing rather than using discrete flap actuation. Eliminating any span-wise breaks in surface continuity serves to maintain a smoother flow over the wing, thereby increasing overall lifting efficiency. Although unable to deform the entire inner-wing span, the majority of the surface is variable with a large flap-to-chord ratio (~0.7) covering 85% span. The large flap-to-chord ratio is beneficial in that it provides a greater moment arm for the SMA actuators, reducing the necessary power required, as well as improving the chord-wise uniformity of deformation. Additionally, the relative lack of internal structure afforded by the nature of the design greatly reduces the overall system weight. XFoil analysis also suggests that a larger flap-to-chord ratio requires less deformation/actuation to provide similar performance relative to those of more modest control surfaces. B. SMA Characteristics Shape memory alloys (SMAs) are alloys that remember their original shape and, after deformation through heat application, return to that shape upon cooling. SMA wires contract by several percent of their length when heated and can be easily stretched when cooled back to room temperature. Upon contraction, unlike ordinary thermal expansion, they exert tremendous amount of force for its small size (at least around 25,000 psi). This makes SMA actuators ideal for use in applications that require large movement. SMA actuators have an excellent power to mass ratio and are quite reliable in general, producing enough deformation to cause significant improvement in the static and dynamic characteristics of the inner wing design. Three five-inch long SMA wires (Nickel-Titanium alloy) with 0.005-inch diameter are evenly mounted across each flap. The inner wing is covered with a series of span-wise panels, and the SMA wires are attached to the rear, upper panel, which has its front end pinned at 70% chord. The method of pinning the rear, upper panel to the airfoil underneath the panel upstream of it applies a reaction moment that makes the rear panel act as though it is effectively cantilevered, and by pulling on the cantilevered fiberglass panel from an acute angle, the component of the bias force due to the fiberglass deformation is made to decrease as the contraction of the wire adjusts the curvature of the final panel. Upon applying electric current which leads to heat conduction in the SMAs, the trailing edge of the rear, top panel drops slightly, and in doing so it pushes the back of the rear, lower panel down as well, resulting in a maximum trailing edge deflection of approximately 13˚. 1. Sample Bend Tests The continuous flap is constructed from molded fiberglass panels. In order to determine the most effective skin thickness a series of 4”x4” material coupons were fabricated from 2oz fiberglass and evaluated in a simple bench- top force test. Samples were mounted horizontally along one edge. A force gauge was mounted to the free side perpendicular to the sample. Force was applied until the sample deflected 20 degrees (1.25”). At this point the force required was recorded and converted into an equivalent force exerted by the SMA. Table 1 shows the sample data and force results. The five layer sample was selected for additional testing due to its compromise between material weight and deflection force required. Test Sample Force Equivalent 5" Total Force Deflection Length Weight Reading SMA Force for 18” Flap (deg) (in) (oz) (lbs) (lbs) (lbs) 2 oz X 3 3.375 0.168 20 0.063 0.258 1.302 2 oz X 5 3.375 0.319 20 0.125 0.515 2.319 2 oz X 7 3.375 0.503 20 0.563 2.319 10.276 Table 1. Flap skin sample testing characteristics. Increasing number of fiberglass layers increases pull force required. There is a significant weight increase between 5 and 7 layers. 2. Deflection Testing (Static and Dynamic) To evaluate the mechanism effectiveness using SMAs a test rig was constructed. The test rig had an S1210 airfoil with 10” chord, 24” span, and an 8”x18” aileron. Three SMAs were mounted to the upper surface skin and connected in parallel to an adjustable power source. A static bench-top test was performed first to establish the general effectiveness. The flap deflection values from this test are recorded in figure 2.
  4. 4. q Velocity Velocity Deflection (psf) (fps) (MPH) Rating 0.00 0.00 0.00 Substantial 0.50 20.51 13.98 Substantial 1.00 29.01 19.78 Substantial 1.50 35.53 24.22 Substantial 2.00 41.02 27.97 Substantial 2.50 45.86 31.27 Substantial 3.00 50.24 34.25 Moderate 3.50 54.27 37.00 Low 4.00 58.01 39.55 Low 4.50 61.53 41.95 Concerning Figure 2. Inner wing camber deflection at various Table 2. Inner wing camber deflection at various voltage inputs. Flap deflection angle increases linearly voltage inputs. Flap deflection angle increases with the amount of current input to the SMAs. linearly with the amount of current input to the SMAs. The deflection performance of the SMAs and flap design was then tested in the North Carolina State Subsonic Wind Tunnel in a range of dynamic pressures from 0.0 to 4.5 pounds per square foot (psf). Run dynamic pressure, wind speed, and flap deflection at maximum amperage were collected to establish the flight performance envelope and can be found in table 2. Flap deflection performed excellently in the 0-2.5 psf dynamic pressure range, while dropping performance in speeds higher than 3 psf. The team is continuing examination of the flight performance envelope to ensure consistent control throughout flight. 3. Load Cell Testing In order to have a better understanding of the actuation, a test for characterization of SMA is done. The experimental set up will be composed of a SMA wire, load cell, linear variable differential transformer (LVDT), and a thermocouple. They will be used so that a stress-strain relationship of the SMA wires mounted on the wing can be determined. The wire is heated using electrical resistive heating and the temperature is monitored using a thermocouple (K-Type) attached directly to the SMA wire. The load is measured using a load cell (10lb capacity) while the contraction due to the heating of the wire is measured using the LVDT. The core of the LVDT is mounted on the SMA wire, and the body of the LVDT is fixed at the base of the set-up. The set-up does not have to be enclosed by a thermal chamber to prevent the surrounding conditions from influencing the temperature of the SMA wire, because the wire mounted inside the wing would be performed independent of the surrounding temperature. C. Design Evaluation 1. Power Usage Because the SMAs require a different power source than the servos, it is necessary to measure how much power SMAs would draw for a desirable flight time. The SMAs used in this project have electrical resistance of 1.8 ohms per inch. The approximate current draw at room temperature is 0.25 amps. This is approximate since room temperatures, air currents, and heat sinking of the test area varies. The voltage draw is found using Ohm's law, V = IR (1) Since six of the SMAs are used, the total current draw would be six times of the current draw of each single wire, which would be 1500mA. Also, under assumption that the all of the SMAs are fully actuated during flight time of 30 minutes, the ampere-hour would be, (2) It can be concluded that, given the basic characteristics of the SMAs, the battery necessary for proper flight should have minimal voltages of 2.25 and that batteries with higher mAh will provide extra margin for longer flight time.
  5. 5. 2. Performance Current flap deflection is limited to approximately 20 degrees (measured relative to trailing edge position) as seen in the static and wind tunnel tests. While this exceeds typical deflection amounts, this does not match the flap deflection possible using servo actuation. However, due to the increased size of the flap the effective influence is still greater than a traditional hinge flap system. Under no aerodynamic loads the SMAs require approximately one second to actuate full deflection and two seconds to return to the non-deformed position. However, even with aerodynamic loading, this transition time is remained constant for deflection and return. This is longer than the sub-second servo actuation time. III. Morphing Planform through Outer Wing Sweep A. Feather Design and Rationale The general shape of the outer-wing incorporates three thin overlapping feathers with a sharp leading edge to aid in the creation of leading edge vortices (LEVs). Relative to the rigid inner-wing, the feathered outer-wing will have a slightly positive angle of attack (4-5 degrees) to provide a greater range of lift up to the generally shallow stall angle of the inner-wing. The unswept outer-wing trailing edge contour from root to tip forms a favorable quarter- elliptical curve designed to maximize area change between swept and unswept wing configurations (approximately 40% decrease in surface area). The decrease in area leads to a proportional decrease in lift, as well as a reduction in the resultant moment arm of the lift force, increasing the effectiveness for roll control and stability. The LEVs serve to augment the lift generation of the outer-wing. It has been shown that large angles of attack result in larger and more unstable vortex flow, decreasing the benefits in performance. Additionally, the swept wing configuration will result in generally higher quality vortex generation and sustainability than the unswept wing, therefore improving lift generation. However, this is counterproductive in that a decrease in lift is desired through the reduction in surface area and may lead to a decrease in the overall effectiveness of outer-wing roll control. The outer wing sections were also designed to allow for independent, asymmetric sweep actuation in order to provide a means of yaw control. The difference in drag produced for the swept and unswept configurations was intended to provide enough moment for adequate yaw control, however, it was determined that this difference is not sufficient enough to have a significant effect. A major design feature of the morphing outer-wing is that it expands the aerodynamic performance envelope of the aircraft. For example, a symmetrically un-swept configuration provides greater lift and is more suitable for slow glides and turns. Alternatively, a symmetrically swept wing configuration provides a reduction in drag and is more suitable for higher speed performances and sharper, more agile turns. In general, the variable wing-sweep design allows for the potential of energy saving flight adaptation within the expanded range of flight performance. Also, it is interesting to consider that the outer wing presence reduces the effects of wingtip vortices on the rigid inner-wing section, allowing the inner-wing to be thought of as “quasi-two-dimensional” thus providing an extra advantage in aerodynamic efficiency. It is unknown; however, if/how the transition between inner and outer-wing flow fields, especially possible interactions concerning the outer-wing LEVs, will affect the overall aerodynamic performance of the entire wing system. Figure 3. Feather Mount Block and Mechanism Figure 4. Feather Mount Block and Mechanism
  6. 6. B. Linkage Design through Structural Analysis A major design challenge for the outer wing was developing a sweeping mechanism that could be mounted inside of the inner wing and not fail under aerodynamic loading. An aluminum mounting block serves as the junction between the feathers and rigid wing (Figure 3). Through a series of mechanical linkages, the feathers are actuated using a single degree of freedom. A servo is used to translate a linkage arm connecting all three feather mounts, resulting in a graduated sweeping motion in all feathers. To withstand the high loads exerted on these joints, the mounting block is constructed out of machine aluminum. An ANSYS structural analysis was used to determine stress peaks in the mount material (Figure 4) and several design modifications have since reduced the weight of each aluminum block to .055 pounds. Using the flight condition of a maximum 5g force exerted on the feathers (0.99 lbs/in2), the maximum stress was 8.69e5 psi with a factor of safety of 6.74. The design mimics the sweeping motion exhibited by some birds during flight. The sweeping mechanism transmits the movement Figure 5. (a) Swept; (b) Unswept Feather Position from the servo to the swing arm which actuates the feather via a cord link. The feathers can be swept back to a maximum of 50 degrees and are designed for continuous collection and to minimize gaps in the swept and unswept positions, as shown in Figure 5. IV. Tail Design The major design challenge for the tail section was how to successfully eliminate the vertical stabilizer while also improving maneuverability and efficiency using unconventional means. The initial design allowed for a 3 degree of freedom system: pitch deflection, axial rotation, and surface area expansion/contraction. Unfortunately, structural and mechanical issues involved in providing the area change were determined to be too complicated and the idea was dismissed. The remaining degrees of freedom (pitch and rotation) are realized through the use of the block mechanism, shown in Figure 6. Figure 6. (a) Tail Maneuver Mechanism; (b) Universal Joint Movement A. Tail Motion Mechanism A micro servo connected to lever arms on either side of the block is used to provide the pitch actuation. The servo moves the lever arms up and down which are connected to a carbon fiber shaft that provides the support connection and means of rotation for the tail. Another micro servo is located at the end of the carbon fiber shaft and is attached via washer and servo rods to produce an axial rotation about the shaft axis. A universal joint is used to provide the necessary combination of pitch and rotation, the simultaneous implementation of which allows for complex motions to be achieved.
  7. 7. The mounting block rests directly inside the fuselage and is constructed of balsa wood to minimize weight. A small gap exists between the fuselage and the tail leading edge to provide the necessary clearance through a full range of motion. Figure 7 shows the block inside the fuselage and the actuation mechanism performing a combination pitch and roll displacement. Figure 7. (a) Tail with Pitch and Roll; (b) Isometric view of Tail in Horizontal Position B. Planform Area Expansion The area expansion of the tail was desired for three reasons, the first being that an area increase would provide more lift for the aircraft. This would help with the extra weight that was added to the plane from both the wing and tail. Also the plane would be able to fly at a slower stall speed which is desirable particularly for experimental aircraft. Secondly, an area decrease would be used during standard cruise to reduce drag, increasing the efficiency and range of the aircraft. Finally, an area increase/decrease can be used to counteract adverse effects from tail pitch and rotation. Figure 8. (a) Swept; (b) Unswept Tail Configurations Figure 8 above shows a model of the tail with an area decrease (a) and an area increase (b). This movement is achieved by SMAs pulling on a torsional spring. As the SMA contracts it twists the connecting rod attached to a feather, sweeping the feather outwards. Unfortunately, the manufacturing of this part proved to be more complicated than anticipated leading to unacceptable weight gain. At this time only a bench top model was created and the tail on the plane does not have the area increase/decrease capabilities. C. Coupled Tail Control Analysis The aircraft maneuverability was increased with the elimination of the vertical tail at the detriment of stability. The aircraft stability was analyzed in AVL to obtain the forces and moments for all different deflections of the tail and wing control elements. A table was then created for different flight profiles (Table 3) and then a configuration for the control surfaces was chosen. For example, the case of climbing would require no sweep of the outer feathers, full flap deflection and positive tail pitch. The forces and moments were then compared for these configurations to set the control surfaces to the desired position. The control surfaces would be deflected until the forces or moments
  8. 8. were balanced. For the purposes of this report a list of all configurations would be impractical, however an extensive example is given below. For this example the Outer-Wing Aileron Tail Tail Pitch flight profile of cruise will Sweep Deflection Rotation be analyzed. During cruise Steady Level it was decided that the outer NONE NONE NONE NONE Flight feathers should be swept Banked Left back and the ailerons and RIGHT RIGHT UP LEFT Turn tail are in their neutral position. This is done to Banked LEFT LEFT UP RIGHT minimize the drag and wing Right Turn tip vortices, which in return Climbing NONE BOTH UP NONE minimizes power consumption and maximizes endurance. Descending BOTH NONE DOWN NONE Using the equations below5 the trim alpha was found to Table 3. Plane Flight Cases. Configuration of control surfaces to optimize be 1.5 degrees. performance during specified maneuvers. Cm  Cmo  Cl  (h  hn ) (3) Where the pitching moment (Cm) was set to zero, Cmo was found by graphing the pitching moments from AVL, Cl,α and the neutral point (hn) was found from AVL, and the range for h was given by the Nexstar flight manual. Next, the forces were analyzed in AVL. For cruise, the lift generated should balance the weight of the plane, but at this angle of attack and 45 degree sweep, the lift force is too low and thus it will be necessary to have the wings in an un-swept configuration. This will allow for more lift to be produced, however, the trim angle of attack must be raised and so the tail will be pitched up. This is found by setting the trim alpha such that the lift balances the weight of the plane and solving for elevator deflection using the equation: Cm  Cmo  Cm   Cm  (4) Thus, all of the desired settings can be realized. This was done for all the different flight profiles. This is important because now these configurations can be given to a team of electrical engineers that are designing the radio controller. This radio controller will be programmed so that the pilot can fly the aircraft like any other aircraft, even though the control surface deflections will be different. V. Conclusions and Further Work It has been determined that SMA actuated, continuous deformation flaps are a viable means of roll control under low to modest aerodynamic loading based on the deflection performance observed in testing. Potential benefits in power consumption and performance were not found to be as dramatic as originally hypothesized, though still remain noteworthy. The SMA actuated wing is a successful proof of concept prototype for biologically inspired morphing flight systems. To further understand the effectiveness of SMA actuated deflection, additional study into various configurations and force testing should be implemented. Also, it will be necessary to Figure 9. Implementation of described design elements. continue gathering data on lift and moment forces of the flap in wind tunnel and flight tests in an effort to quantify and increase performance margins. Potential future design modifications may involve independently deflecting either the inner or outer (relative to wing root) portion of
  9. 9. the control surface, thus providing a twisting/warping of the surface rather than just a constant span-wise deflection. This would improve the control resolution by allowing the flap to essentially be “partially deflected” while also providing the option of morphing the surface in various distinct ways. In addition, the option to selectively recruit individual SMAs instead of an all-or-none approach would greatly reduce energy consumption. The sweeping outer-wing was mechanically successful. The design exceeded all structural expectations although aerodynamic performance was not as strong as desired. It is believed that more sophisticated feather designs which contain a slight degree of camber will contribute to increased aerodynamic performance and improve the roll and yaw control exercised by the feathers. Additional research should be done to determine effective methods for implementing camber and other forms of morphing into sweeping feather designs. It was found that the elimination of the vertical stabilizer in the tail design provided more complications than benefits. While the maneuverability of the plane was greatly increased, the loss of dynamic stability (lack of yaw control) proved to be too great. The roll rates and rates of sideslip are likely too high for a pilot to overcome, although this will not be known for certain until flight tests are done. A future design incorporating area expansion may aid in eliminating some of the stability issues and may also benefit from a larger section of variable sweep on the wing. Serious research should also be done concerning the benefits of incorporating biologically-inspired leading edge tubercles on either/both wing and tails surfaces6. The greatest challenge seems to involve the successful integration of both the wing and tail systems from a stability and control standpoint, and thus future work should focus on extensive quantitative control analysis. Overall, the novelty and feasibility of these unconventional control elements provide a promising first step toward realizing the benefits of biological design features and morphing technologies. VI. Acknowledgements The North Carolina State University Bioflight design team would like to express gratitude for the North Carolina Space Grant Consortium for their financial support. VII. References 1. Namgoong, H., Crossley, W. A. , and Lyrintzis, A. S.: Aerodynamic Optimization of a Morphing Airfoil Using Energy as an Objective, AIAA Paper 2006-1324, 2006. 2. Kornbluh, R., Pelrine, R., Eckerle, J., and Joseph, J., “Electrostrictive Polymer Artificial Muscle Actuators,” Proceedings of the 1998 IEEE International Conference on Robotics & Automation, Leuven, Belgium, 1998, pp. 2147-2154. 3. Viieru, D., Tang, J., Lian, Y., Liu, H., and Shyy, W.: Flapping and Flexible Wing Aerodynamics of Low Reynolds Number Flight Vehicles, AIAA Paper 2006-503, 2003. 4. Elsayed, M., Scarano, F., and Verhaagen, N. G.: Leading-Edge Shape Effect on the Flow over Non-Slender Delta Wings, AIAA Paper 2008-344, 2008. 5. Etkin, B., and Reid, L. D., Dyanmics of Flight Stability and Control, 3rd ed., Wiley, New York, 1996. 6. Watts, P., and Fish, F.E.: The Influence of Passive, Leading Edge Tubercles on Wing Performance.