昆虫飞行的空气动力学

孙茂

doi:10.6052/1000-0992-14-065

昆虫飞行的空气动力学

doi: 10.6052/1000-0992-14-065

孙茂

北京航空航天大学流体力学研究所, 北京100191

详细信息

通讯作者:
孙茂，1955 年8 月出生

中图分类号: V211
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出版历程
- 收稿日期: 2014-11-05
- 修回日期: 2014-11-24
- 刊出日期: 2015-08-30

Aerodynamics of insect flight

Mao SUN

Institute of fluid mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

More Information

Corresponding author: Mao SUN

摘要

摘要: 昆虫是最早出现、数量最多和体积最小的飞行者. 它们能悬停、跃升、急停、快速加速和转弯, 飞行技巧十分高超. 由于尺寸小, 因而翅膀的相对速度很小, 从而进行上述飞行所需的升力系数很大. 但昆虫翅膀的雷诺数又很低. 它们是如何在低雷诺数下产生高升力的, 是流体力学和生物学工作者都十分关心的问题. 近年来这一领域有了许多研究进展. 该文对这些进展进行综述, 并对今后工作提一些建议. 因2005 年前的工作已在几篇综述文章有了详细介绍, 该文主要介绍2005 年以来的工作. 首先简述昆虫翅的拍动运动及昆虫绕流的基本方程和相似参数; 然后对2005 年之前的工作做一简要回顾. 之后介绍2005 年后的进展, 依次为: 运动学观测; 前缘涡; 翅膀柔性变形及皱褶的影响; 拍动翅的尾涡结构; 翼/身、左右翅气动干扰及地面效应; 微小昆虫; 蝴蝶与蜻蜓; 机动飞行. 最后为对今后工作的建议.
- 昆虫 /
- 飞行 /
- 空气动力学
Abstract: Insects are the earliest, most numerous and smallest fliers in the world. They can hover, fly forward, climb and descend with ease while demonstrating amazing stabilities, and they can also maneuver in impressive ways like no other organisms could. Although the wing of an insect beats at high frequency, the wing's relative velocity is small owing to the small wing length. As a result, the mean lift coefficient of wing required to balance the insect weight is relatively high, about 1.5–2, much higher than that of an airplane at cruising flight. The Reynolds number of insects' wings is small, ranging from about 10 to 3 500. How the required high-lift coefficient is produced at such low Reynolds number? Researchers are very interested in this question and in recent years, significant progress has been made in the area. Works before 2005 have been discussed in detail in several review papers, and in this article, we review the advances made since 2005. We begin with an overview of the flapping kinematics and basic equations of fluid dynamics. It is followed by a summary of the works before 2005. Then we review the advances made since 2005, dealing in turn with measurement of wing motion in freely-flying insects, leading-edge vortex, effect of wing deformation and corrugation, vortex wake of flapping wings, ground effect and aerodynamic interaction between wings and body, flight of tiny insects, flight of butterfly and dragonfly, and maneuvering flight. Finally, we make remarks on the state-of-the-art of this research field and speculate its outlooks in the near future.
- insect /
- flight /
- aerodynamics

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参考文献(124)

[1]	Altshuler D L, Dickson W B, Vance J T, Roberts S P, Dickinson M H. 2005. Short-amplitude high frequencywing strokes determine the aerodynamics of honeybee flight. PNAS., 102: 18213-18218.
[2]	Ansari S A, Phillips N, Stabler G, Zbikowski R, Knowles K. 2009. Spanwise flow on an impulsively-startedrotating wing at low Reynolds numbers. In: Proceedings of 39th AIAA Fluid Dynamics Conference, SanAntonio, Texas, AIAA-2009-4032: 1–9.
[3]	Ansari S A, Zbikowski R, Knowles K. 2006. Aerodynamic modeling of insect-like flapping flight for microair vehicles. Prog. Aerosp. Sci., 42: 129-172.
[4]	Aono H, Liang F, Liu H. 2008. Near-and far-field aerodynamics in insect hovering flight: An integratedcomputational study. J. Exp. Biol., 211: 239-257.
[5]	Ansari S A. 2004. A nonlinear, unsteady, aerodynamic model for insect-like flapping wings in the hover withmicro air vehicle applications. [PhD Thesis]. Shrivenham: Cranfield University.
[6]	Berman G J, Wang Z J. 2007. Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech., 582:153-168.
[7]	Bergou A J, Ristroph L, Guckenheimer J, Cohen I, Wang Z J. 2010. Fruit flies modulate passive wingpitching to generate in-flight turns. Phys. Rev. Lett., 104: 148101.
[8]	Betts C R, Wootton R J. 1988. Wing shape and flight behaviour in butterflies (Lepidoptera: papilionoideaand hesperioidea): A preliminary analysis. J. Exp. Biol., 138: 271-288.
[9]	Birch J M, Dickinson M H. 2001. Spanwise flow and the attachment of the leading-edge vortex on insectwings. Nature, 412: 729-733.
[10]	Birch J M, Dickinson M H. 2003. The influence of wing-wake interactions on the production of aerodynamicforces in flapping flight. J. Exp. Biol., 206: 2257-2272.
[11]	Birch J M, Dickson W B, Dickinson M H. 2004. Force production and flow structure of the leading edgevortex on flapping wings at high and low Reynolds numbers. J. Exp. Biol., 207: 1063–1072.
[12]	Bomphrey R J, Taylor G K, Thomas A L R. 2009. Smoke visualization of free-flying bumblebees indicatesindependent leading-edge vortices on each wing pair. Exp Fluids, 46: 811–821.
[13]	Brodsky A K. 1991. Vortex formation in the tethered flight of the peacock butterfly Inachis Io L. (Lepidoptera,Nymphalidae) and some aspects of insect flight evolution. J. Exp. Biol., 161: 77-95.
[14]	Card G, Dickinson M H. 2008. Performance trade-offs in the flight initiation of Drosophila. J. Exp. Biol.211: 341-353.
[15]	Carr Z R, Chen C, Ringuette M J. 2013. Finite-span rotating wings: three-dimensional vortex formationand variations with aspect ratio. Exp. Fluids, 54: 1–26.
[16]	Chen M W, Zhang Y L, Sun M. 2013. Wing and body motion and aerodynamic and leg forces duringtake-off in droneflies: J. R. Soc. Interface, 10: 20130808.
[17]	Chen M W, Sun M. 2014. Wing/body kinematics measurement and force and moment analyses of the takeoffflight of fruitflies. Acta Mechanica Sinica, 30: 495-506.
[18]	Davis W R, Kosichi B B, Boroson D M, Kostishack D F. 1996. Micro air vehicle for optical surveillance.The Lincoln Laboratory J., 9: 197-217.
[19]	Dickinson M H, G¨otz K G. 1993. Unsteady aerodynamic performance of model wings at low Reynoldsnumbers. J. Exp. Biol., 174: 45-64.
[20]	Dickinson M H, Lehman F O, Sane S P. 1999. Wing rotation and the aerodynamic basis of insect flight.Science, 284: 1954-1960.
[21]	Du G, Sun M. 2008. Effects of unsteady deformation of flapping wings on its aerodynamic forces. Appl.Math. Mech. Engl. Ed., 29: 731-741.
[22]	Du G, Sun M. 2010. Effects of wing deformation on aerodynamic forces in hovering hoverflies. J. Exp. Biol.,213: 2273-2283.
[23]	Du G, Sun M. 2012. Aerodynamic effects of corrugation and deformation in flapping wings of hoveringhoverflies. J. Theor. Biol., 300: 19-28.
[24]	Dudley R. 1990. Biomechanics of flight in neotropical butterflies: Morphometrics and kinematics. J. Exp.Biol., 150: 37-53.
[25]	Dudley R. 1991. Biomechanics of flight in neotropical butterflies: Aerodynamics and mechanical powerrequirements. J. Exp. Biol. 159: 335-357.
[26]	Dudley R. 2000. The Biomechanics of Insect Flight: Form, Function, Evolution. Princeton: PrincetonUniversity Press.
[27]	Dudley R, Ellington C P. 1900a. Mechanics of forward flight in bumblebees: I. Kinematics and morphology.J. Exp. Biol., 148: 19-52.
[28]	Dudley R, Ellington C P. 1990b. Mechanics of forward flight in bumblebees: II. Quasi-steady lift and powerrequirements. J. Exp. Biol., 148: 53-88.
[29]	Eldredge J D, Toomey J, Medina A. 2010. On the roles of chord-wise flexibility in a flapping wing withhovering kinematics. J. Fluid Mech. 659: 94-115
[30]	Ellington C P. 1984a. The aerodynamics of hovering insect flight. I. The quasi-steady analysis. Phil. Trans.R. Soc. Lond. B, 305: 1-15.
[31]	Ellington C P. 1984b. The aerodynamics of hovering insect flight. II. Morphological parameters. Phil.Trans. R. Soc. Lond. B, 305: 17-40.
[32]	Ellington C P. 1984c. Aerodynamics of hovering insect flight. III. Kinematics. Phil. Trans. R. Soc. Lond.B, 305: 41-78.
[33]	Ellington C P 1984d The aerodynamics of hovering insect flight. V. A vortex theory. Phil. Trans. R. Soc.Lond. B, 305: 115–144.
[34]	Ellington C P. 1991. Aerodynamics and the origin of insect flight. Adv. Insect Physiol., 23: 171-210.
[35]	Ellington C P. 1995. Unsteady aerodynamics of insect flight. Symp. Soc. Exp. Biol., 49: 109-129.
[36]	Ellington C P. 1999. The novel aerodynamics of insect flight: Applications to micro-air vehicles. J. Exp.Biol., 202: 3439-3448.
[37]	Ellington C P, Machin K E, Casey T M. 1990. Oxygen consumption of bumblebees in forward flight. Nature,347: 472.
[38]	Ellington C P, Van Den Berg C, Willmott A P, Thomas A L R. 1996. Leading-edge vortices in insect flight.Nature, 384: 626-630.
[39]	Ennos A R. 1988. The importance of torsion in the design of insect wings. J. Exp. Biol., 140: 137-160.Ennos A R. 1989. The kinematics and aerodynamics of the free flight of some Diptera. J. Exp. Biol., 142:49-85.
[40]	Fry S N, Sayaman R, Dickinson M H. 2003. The aerodynamics of free-flight maneuvers in Drosophila.Science, 300: 495-498.
[41]	Fry S N, Sayaman R, Dickinson M H. 2005. The aerodynamics of hovering flight in Drosophila: J. Exp.Biol., 208: 2303-2318.
[42]	Fung Y C. 1969. An Introduction to the Theory of Aeroelasticity. John Wiley & Sons, Inc., New York,Chapman & Hall, Ltd., London.
[43]	Garmann D J, Visbal M R. 2014. Dynamics of revolving wings for various aspect ratios. J. Fluid Mech.,748: 932–956.
[44]	Garmann D J, Visbal M R, Orkwis P. 2013. Three-dimensional flow structure and aerodynamic loading ona revolving wing. Phys. Fluids, 25: 034101-034127.
[45]	Harbig R R, Sheridan J, Thompson M C. 2013. Reynolds number and aspect ratio effects on the leading-edgevortex for rotating insect wing planforms. J. Fluid Mech., 717: 166–192.
[46]	Huang H, Sun M. 2012. Forward flight of a model butterfly: Simulation by equations of motion coupledwith the Navier–Stokes equations. Acta Mechanica Sinica, 28: 1–12.
[47]	Ishihara D, Horie T, Denda M. 2009. A two-dimensional computational study on the fluid–structure interactioncause of wing pitch changes in dipteran flapping flight. J. Exp. Biol., 212: 1-10.
[48]	Jardin T, Farcy A, David L. 2012. Three-dimensional effects in hovering fapping flight. J. Fluid Mech.,702: 102–125.
[49]	Kim D, Gharib M. 2010. Experimental study of three-dimensional vortex structures in translating androtating plates. Exp. Fluids, 49: 329–339.
[50]	Lan S L, Sun M. 2001. Aerodynamic properties of a wing performing unsteady motions at low Reynoldsnumber. Acta. Mechanica, 149: 135-147.
[51]	Lentink D, Dickinson M H. 2009. Rotational accelerations stabilize leading edge vortices on revolving flywings. J. Expl Biol., 212: 2705–2719.
[52]	Liang B, Sun M. 2013. Aerodynamic interactions between wing and body of a model insect at forward flightand in maneuvers. J. Bionic Eng., 10: 19-27.
[53]	Lighthill M J. 1973. On the Weis-Fogh mechanism of lift generation. J. Fluid Mech., 60: 1-17.
[54]	Liu H, Ellington C P, Kawachi K, Van Den Berg C, Willmott A P. 1998. A computational fluid dynamicstudy of hawkmoth hovering. J. Exp. Biol., 201: 461-477.
[55]	Liu H, Aono H. 2009. Size effects on insect hovering aerodynamics: An integrated computation study.Bioinsp. Biomm. 4: 015002.
[56]	Liu Y P, Sun M. 2008. Wing kinematics measurement and aerodynamics of hovering drone-flies. J. Exp.Biol., 211: 2014-2025.
[57]	Lu Y, Shen G X. 2008. Three-dimensional flow structures and evolution of the leading-edge vortices on aflapping wing. J. Exp. Biol., 211: 1221–1230.
[58]	Luo G Y, Sun M. 2005. The effects of corrugation and wing planform on the aerodynamic force productionof sweeping model insect wings. Acta Mechanica Sinica, 21: 531-541.
[59]	Ma K Y, Chirarattananon P, Fuller S B, Wood R J. 2013. Controlled flight of a biologically inspired,insect-scale robot. Science, 340: 603-607.
[60]	Maxworthy T. 1979. Experiments on the Weis-Fogh mechanism of lift generation by insects in hoveringflight. Part 1. Dynamics of the “fling”. J. Fluid Mech., 93: 47-63.
[61]	Meng X G, Sun M. 2013. Aerodynamic effects of wing corrugation at gliding flight at low Reynolds numbers.Physics of Fluids, 25 : 071905.
[62]	Miller L A, Peskin C S. 2005. A computational fluid dynamics of “clap and fling” in the smallest flyinginsects. J. Exp. Biol., 208: 195-212.
[63]	Miller L A, Peskin C S. 2009. Flexible clap and fling in tiny insect flight. J. Exp. Biol., 212: 3076-3090.
[64]	Mou X L, Liu Y P, Sun M. 2011. Wing motion measurement and aerodynamics of hovering true hoverflies.J. Exp. Biol., 214: 2832-2844.
[65]	Muijres F T, Elzinga M J, Melis J M, Dickinson M H. 2014. Flies evade looming targets by executing rapidvisually directed banked turns. Science, 344: 172-177.
[66]	Nakata T, Liu H. 2012a. A fluid-structure interaction model of insect flight with flexible wings. J. Comput.Phys., 231: 1822-1847.
[67]	Nakata T, Liu H. 2012b. Aerodynamic performance of a hovering hawkmoth with flexible wings: A computationalapproach. Proc. R. Soc. B., 279: 722-731.
[68]	Newman D J S, Wootton R J. 1986. An approach to the mechanics of pleating in dragonfly wings. J. Exp.Biol., 125: 361-372.
[69]	Ozen C A, Rockwell D. 2012. Three-dimensional flow structure on a rotating wing. J. Fluid Mech., 707:541–550.
[70]	Pesavento U, Wang Z J. 2004. Navier–Stokes solutions, model of fluid forces, and center of mass elevation.Phys. Rev. Lett., 93: 116-164.
[71]	Rees C J C. 1975. Form and function in corrugated insect wings. Nature, 256: 200-203.
[72]	Sane S P. 2003. The aerodynamics of insect flight. J. Exp. Biol., 206: 4191-4208.
[73]	Sane S P, Dickinson M H. 2002. The aerodynamic effects of wing rotation and a revised quasi-steady modelof flapping flight. J. Exp. Biol., 205: 1087-1098.
[74]	Shyy W, Liu H. 2007. Flapping wings and aerodynamic lift: The role of leading-edge vortices. AIAAJournal, 45: 2817–2819
[75]	Shyy W, Trizilla P, Kang C K, Aono H. 2009. Can Tip Vortices enhance lift of a flapping wing? AIAAJournal, 2: 289–293.
[76]	Shyy W, Aono H, Chimakurthi S K, Trizila P, Kang C K, Cesink C E S, Liu H. 2010. Recent progress inflapping wing aerodynamics and aeroelasticity. Prog. Aerosp. Sci., 46: 284.
[77]	Shyy W, Berg M, Ljungqvist D. 1999. Flapping and flexible wings for biological and micro air vehicles.Prog. Aerosp. Sci., 35: 455.
[78]	Shyy W, Lian Y, Tang J, Viieru D, Liu H. 2008. Aerodynamics of Low Reynolds Number Fliers. New York:Cambridge University Press.
[79]	Srygley R B, Thomas A L R. 2002. Unconventional lift-generating mechanisms in free-flying butterflies.Nature, 420: 660-664.
[80]	Sun M. 2005. High-lift generation and power requirements of insect flight. Fluid Dynamics Research, 37:21-39
[81]	Sun M, Du G. 2003. Lift and power requirements of hovering insect flight. Acta Mechanica Sinica, 19:458-469.
[82]	Sun M, Lan S L. 2004. A computational study of the aerodynamic forces and power requirements of dragonfly(Aeschna juncea) hovering. J. Exp. Biol., 207: 1887-1901.
[83]	Sun M, Tang J. 2002. Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion.J. Exp. Biol., 205: 55-70.
[84]	Sun M, Wu J H. 2004. Large aerodynamic forces on a sweeping wing at low Reynolds number. ActaMechanica Sinica, 20: 24–31.
[85]	Sun M, Yu X. 2003. Flow around two airfoils performing fling and subsequent translation and translationand subsequent flap. Acta Mechanica Sinica, 19: 103-117.
[86]	Sun M, Yu X. 2006. Aerodynamic force generation in hovering flight in a tiny insect. AIAA Journal, 44:1532-1540.
[87]	Sunada S, Kawachi K, Watanabe I. 1993. Performance of a butterfly in take-off flight. J. Exp. Biol., 183:249-227.
[88]	Sunada S, Takashima H, Hattori T, Yasuda K, Kawachi K. 2002. Fluid-dynamic characteristics of a bristledwing. J. Exp. Biol., 205: 2737–2744.
[89]	Tanaka S. 1995. Thrips’ flight. Part 1. In: Symposia 95 of Exploratory Research for Advanced Technology,Japan Science and Technology Corporation, Tokyo, 27–34.
[90]	Usherwood J R, Ellington C P. 2002a. The aerodynamics of revolving wings. I. Model hawkmoth wings. J.Exp. Biol., 205: 1547-1564.
[91]	Usherwood J R, Ellington C P. 2002b. The aerodynamics of revolving wings. II. Propeller force coefficientsfrom mayfly to quail. J. Exp. Biol., 205: 1565-1576.
[92]	Usherwood J R, Lehmann F. 2008. Phasing of dragonfly wings can improve aerodynamic efficiency byremoving swirl. J. R. Soc. Interface, 5: 1303–1307.
[93]	Vanella M, Fitzgerald T, Preidikman S, Balaras E, Balachandran B. 2009. Influence of flexibility on theaerodynamic performance of a hovering wing. J. Exp. Biol., 212: 95-105.
[94]	Vogel S. 1967a. Flight in Drosophila. II. Variations in stroke parameters and wing contour. J. Exp. Biol.,46: 383-392.
[95]	Vogel S. 1967b. Flight in Drosophila. III. Aerodynamic characteristics of fly wings and wing models. J.Exp. Biol., 46: 431-443.
[96]	Walker S M, Thomas A L R, Taylor G K. 2010. Deformable wing kinematics in free-flying hoverflies. J. R.Soc. Interface, 7: 131-142.
[97]	Wang H, Zeng L J, Liu H, Yin C Y. 2003. Measuring wing kinematics, flight trajectory and body attitudeduring forward flight and turning maneuvers in dragonflies. J. Exp. Biol., 206: 745-757
[98]	Wang H, Zeng L J, Yin C Y. 2002. Measuring the body position, attitude and wing deformation of a freeflightdragonfly by combining a comb fringe pattern with sign points on the wing. Measurement Scienceand Technology, 13: 903-908.
[99]	Wang Z J. 2004. The role of drag in insect hovering. J. Exp. Biol., 207: 4147-4155.
[100]	Wang Z J. 2005. Dissecting insect flight. Annu. Rev. Fluid Mech., 37: 183-210.
[101]	Wang Z J, Russell D. 2007. Effect of forewing and hindwing interactions on aerodynamic forces and powerin hovering dragonfly flight. Phys. Rev. Lett., 99: 148101.
[102]	Wang X X, Wu Z N. 2010. Stroke-averaged lift forces due to vortex rings and their mutual interactions fora flapping flight model. J. Fluid Mech., 654: 453-472.
[103]	Wang X X, Wu Z N. 2012. Lift force reduction due to body image of vortex for a hovering flight model. J.Fluid Mech., 709: 648-658.
[104]	Weis-Fogh T. 1972. Energetics of hovering flight in hummingbirds and in Drosophila. J. Exp. Biol., 56:79-104.
[105]	Weis-Fogh T. 1973. Quick estimates of flight fitness in hovering animals, including novel mechanism for liftproduction. J. Exp. Biol., 59: 169-230.
[106]	Weis-Fogh T, Jensen M. 1956. Biology and physics of locust flight. I. Basic principles of insect flight. Acritical review. Philos. Trans. R. Soc. B: Biol. Sci., 239: 415-458.
[107]	Willmott A P, Ellington C P. 1997a. The mechanics of flight in the hawkmoth Manduca Sexta. I. Kinematicsof hovering and forward flight. J. Exp. Biol., 200: 2705-2722.
[108]	Willmott A P, Ellington C P. 1997b. The mechanics of flight in the hawkmoth Manduca sexta. II. Aerodynamicconsequences of kinematic and morphological variation. J. Exp. Biol., 200: 2723-2745.
[109]	Wilson J. 2001. Micro warfare. Popular Mechanics, 2: 62.Wojcik C J, Buchholz J H J. 2014. Vorticity transport in the leading-edge vortex on a rotating blade. J.Fluid Mech., 743: 249-261.
[110]	Wootton R J. 1981. Palaeozoic insects. Annu. Rev. Ent., 26: 319-344.
[111]	Wu J H, Sun M. 2004. Unsteady aerodynamic forces of a flapping wing. J. Exp. Biol., 207: 1137-1150.Wu J H, Sun M. 2005. The influence of the wake of a flapping wing on the production of aerodynamic forces.Acta Mechanica Sinica, 21: 411-418.
[112]	Wu T Y. 2011. Fish swimming and bird/insect flight. Annu. Rev. Fluid Mech., 43: 25-48.
[113]	Yamamoto M, Isogai K. 2005. Measurement of unsteady aerodynamic forces for a mechanical dragonflymodel. AIAA Journal, 43: 2475-2480.
[114]	Yokoyama N, Senda K, Iima M, Hirai N. 2013. Aerodynamic forces and vortical structures in flappingbutterfly’s forward flight. Physics of Fluids, 25: 021902.
[115]	Young J, Walker S M, Bomphrey R J, Taylor G K, Thomas L R. 2009. Details of insect wing design anddeformation enhance aerodynamic function and flight efficiency. Science, 325: 1549-1552.
[116]	Yu X, Sun M. 2009. A computational study of the wing-wing and wing-body interactions of a model insect.Acta Mechanica Sinica, 25: 421-431.
[117]	Yu Y L, Tong B G. 2005. A flow control mechanism in wing flapping with stroke asymmetry during insectforward flight. Acta Mechanica Sinica, 21: 218-227.
[118]	Yu Y L, Tong B G, Ma H Y. 2003. An analytical approach to theoretical modeling of highly unsteadyviscous flow excited by wing flapping in small insects. Acta Mechanica Sinica, 19: 508-516.
[119]	Yu Y L, Tong B G, Ma H Y. 2005. Unsteady flow mechanics revisited in insect flapping flight. ActaMechanica Sinica, 37: 257-265.
[120]	Zhang J, Lu X Y. 2009. Aerodynamic performance due to forewing and hindwing interaction in glidingdragonfly flight. Physical Review E, 80: 017302-017305.
[121]	Zhang Y L, Sun M. 2010. Wing kinematics measurement and aerodynamics of free-flight maneuvers indrone-flies. Acta Mechanica Sinica, 26: 371-382.
[122]	Zanker J M. 1990. The wing beat of Drosophila melanogaster. I. Kinematics. Phil. Trans. R. Soc. Lond.B, 327: 1-18.
[123]	Zhao L, Huang Q, Deng X Y, Sane S P. 2010. Aerodynamic effects of flexibility in flapping wings. J. R.Soc. Interface, 7: 485-497.
[124]	Zhao L, Deng X Y, Sane S P. 2011. Modulation of leading edge vorticity and aerodynamic forces in flexibleflapping wings. Bioinsp. Biomim., 6: 036007.flapping wings. Bioinsp. Biomim., 6: 036007.