Welcome back. Today, I hope to share the introduction to the senior design project I completed last semester at USC. Hopefully it is enjoyable, and please comment if there are any questions. Until next week, please take care.
Introduction:
Centimeter-scale micro air vehicles (MAVs) can play valuable roles in the military and private sectors, from espionage and surveillance, to surveying dense rain forests and construction sites [1]. To extend the range and endurance of these MAVs, there has been growing interest in finding new methods of propulsion that can be more efficient than the small propellers typically used on drones. Birds and insects have served as inspiration for these studies [2] because their flapping wings can produce high amounts of lift and thrust that allow them to migrate for thousands of miles. Experimental [3] and computational [4] studies have found that flapping wings can improve both lift and thrust at small scales. A primary reason is that the lift-to-drag ratio L/D is a function of the chord-based Reynolds number,
Re = Uc/?, (1)
where U is the mean forward flight speed, c is the mean chord length, and ? is the kinematic viscosity of air. As demonstrated in [6], L/D decreases as Re decreases. Typical MAVs operate at low chord-based Reynolds numbers Re = 102-103 [5], and therefore the lift-to-drag ratio of a small propeller mounted on their wings would be small. To increase L/D, small flying devices can generate lift at larger length scales, and hence at higher Re. One way of accomplishing this is to flap large wings instead of spinning small propellers. When wings are oriented to produce thrust, this also allows for improvements in propulsive efficiency, defined as
? = TU/P, (2)
where P is the mean input power required for flapping and T is the mean thrust force.
Replicating MAV conditions in low-cost experimental setups can be challenging, so recent studies of flapping flight [7] [8] [9] have considered basic rectangular or semicircular wing geometries in pure sinusoidal flapping motion. Thiria and Godoy-Diana [8] studied semicircular wings with distributed flexibility across the planform at Reynolds numbers Re = 1000-3000. To study more natural self-propelled flight, their wings were mounted on a merry-go-round system, which consisted of a fixed central post and a rotating arm that moved under the thrust generated by the flapping wings. In agreement with prior work by Lewin and Haj-Hariri [10], propulsive efficiency was a function of the Strouhal number,
St = fA/U, (3)
where f was the flapping frequency and A was the peak-to-peak flapping amplitude of the leading-edge at midspan. It turned out that propulsive efficiency had a non-monotonic dependence on Strouhal number, and was largest at St= 0.4 ± 0.1. More flexible wings made from thinner polyvinyl chloride also had higher propulsive efficiencies than stiffer wings at St = 0.3-0.8. Contemporary studies by Vanella et al.[4] and Eldredge et al. [7] simplified chordwise flexibility to a single spring-loaded hinge located at mid-chord that could passively deflect with the flow. Two-dimensional simulations of these hinged wings in hovering motion showed that the hinged wings could exhibit a 10 ± 5% increase in lift by encouraging leading-edge vortices to remain attached [7]. Wan et al. [11] studied the effect of changing the hinge location for wings in hover. Results from 2D computations suggested that placing the hinge at the 3/4 chord could improve L/D by more than 10% for stroke amplitudes less than c/3 by creating stronger leading-edge and trailing-edge vortices. However, it remains unclear whether 3D wings would have the highest propulsive efficiency with a hinge at 3/4 chord, since Wan et al. [11] did not account for finite wing effects and also did not study propulsion directly.
The goal of the present study was to experimentally investigate the propulsive efficiency of self-propelled hinged wings in forward flight. There were two questions of interest. The first was whether wings with a single chordwise hinge could have greater propulsive efficiency compared to rigid wings or wings with distributed flexibility. Although Vanella et al. [4], Eldredge et al. [7], and Wan et al. [11] had shown that hinged wings could have greater lift than rigid wings, their studies were restricted to hovering flight in 2D simulations. The second question was how the propulsive efficiency of hinged wings would depend on Strouhal number. Since Lewin and Haj-Hariri [10] found St to be a significant parameter that determined efficiency, it was of particular interest to see whether hinged wings could have superior performance over a range of Strouhal numbers. To address these questions, the merry-go-round setup from Thiria and Godoy-Diana [8] was adapted to study three hinged wings, one rigid wing, and one wing with distributed flexibility. Although simulations by Wan et al. [11] found superior lift performance with a hinge location at the 3/4 chord, the present study considered hinges at midchord, after preliminary trials found that midchord hinges could yield superior forward velocity for Mylar and polycarbonate wings.
This experiment intended to compute propulsive efficiency using Eq. (2) and Strouhal number using Eq. (3), with the amplitude A measured as the peak-to-peak arc length displacement of the leading-edge at midspan. However, in the event that T or P could not be measured, it was known that U could be used as a proxy for propulsive efficiency, since U was directly proportional to ?, and U was expected to vary more than T or P at the flapping frequencies of interest (10-25 Hz) [8]. Aside from changing the wing geometries, the only other independent variable was the flapping frequency, f. Adjusting the flapping frequency changed the Strouhal number, as shown in Eq. (3), and also changed U, which caused ? and Re to change. To be comparable to the results in [8], the intended Reynolds number range was Re = 1000-3000 and the intended Strouhal number range was St = 0.3-0.7. These estimates were made using the forward flight speeds achieved by [8], but the speeds achieved by the actual experimental setup were difficult to predict because they were implicitly determined by f and friction in the apparatus. Indeed, the actual Reynolds numbers achieved in this experiment were Re = 54-890 and the actual Strouhal numbers were St = 5-60. Therefore, this study did not promise to produce results that were directly comparable to flapping-wing MAVs. Radavidther, it was a study into the basic physics of flapping flight that could inspire future experiments in MAV conditions.
References:
[1] Mueller, T. J., Fixed and flapping wing aerodynamics for micro air vehicle applications, American Institute of Aeronautics and Astronautics, 2001.
[2] Shyy, W., Aono, H., Chimakurthi, S., Trizila, P., Kang, C.-K., Cesnik, C., and Liu, H., “Recent progress in flapping wing aerodynamics and aeroelasticity,” Progress in Aerospace Sciences, Vol. 46, No. 7, 2010, pp. 284–327.
[3] David, M. J., Govardhan, R., and Arakeri, J., “Thrust generation from pitching foils with flexible trailing edge flaps,” Journal of Fluid Mechanics, Vol. 828, 2017, pp. 70–103.
[4] Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E., and Balachandran, B., “Influence of flexibility on the aerodynamic performance of a hovering wing,” Journal of Experimental Biology, Vol. 212, No. 1, 2009, pp. 95–105.
[5] Finio, B., Perez-Arancibia, N., and Wood, R., “System identification and linear time-invariant modeling of an insect-sized flapping-wing micro air vehicle,” 09 2011, pp. 1107–1114.
[6] Vogel, S., “Living in a physical world: VI. Gravity and life in the air,” Journal of Biosciences, Vol. 31, 04 2006, pp. 13–25.
[7] Eldredge, J. D., Toomey, J., and Medina, A., “On the roles of chord-wise flexibility in a flapping wing with hovering kinematics,” Journal of Fluid Mechanics, Vol. 659, 2010, pp. 94–115.Thiria
[8] Thiria, B. and Godoy-Diana, R., “How wing compliance drives the efficiency of self-propelled flapping flyers,” Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, Vol. 82, No. 1, 2010, pp. 015303.
[9] Ramananarivo, S., Godoy-Diana, R., and Thiria, B., “Rather than resonance, flapping wing flyers may play on aerodynamics to improve performance,” Proceedings of the National Academy of Sciences, Vol. 108, No. 15, 2011, pp. 5964–5969.
[10] Lewin, G. C. and Haj-Hariri, H., “Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow,” Journal of Fluid Mechanics, Vol. 492, 2003, pp. 339–362.
[11] Wan, H., Dong, H., and Huang, G. P., “Hovering hinge-connected flapping plate with passive deflection,” AIAA journal, Vol. 50, No. 9, 2012, pp. 2020–2027.
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