Other - Recent Progress in Flapping Wing Aerodynamics and Aeroelasticity

Progress in Aerospace Sciences 46 (2010) 284–327

Contents lists available at ScienceDirect

Progress in Aerospace Sciences

0376-04

doi:10.1

Abbren Corr

E-m

journal homepage: www.elsevier.com/locate/paerosci

Recent progress in flapping wing aerodynamics and aeroelasticity

W. Shyy a,n, H. Aono a, S.K. Chimakurthi a, P. Trizila a, C.-K. Kang a, C.E.S. Cesnik a, H. Liu b

a Department of Aerospace Engineering, University of Michigan, FXB 1320 Beal Avenue, Ann Arbor, MI 48109, USAb Graduate School of Engineering, Chiba University, 1-33 Yayoi-cho, Chiba, Chiba 263-8522, Japan

a r t i c l e i n f o

Available online 13 February 2010

21/$ - see front matter & 2010 Elsevier Ltd. A

016/j.paerosci.2010.01.001

viations: AoA, angle of attack; LEV, leading ed

esponding author. Tel: +1 734 936 0102.

ail address: [email protected] (W. Shyy).

a b s t r a c t

Micro air vehicles (MAVs) have the potential to revolutionize our sensing and information gathering

capabilities in areas such as environmental monitoring and homeland security. Flapping wings with

suitable wing kinematics, wing shapes, and flexible structures can enhance lift as well as thrust by

exploiting large-scale vortical flow structures under various conditions. However, the scaling invariance

of both fluid dynamics and structural dynamics as the size changes is fundamentally difficult. The focus

of this review is to assess the recent progress in flapping wing aerodynamics and aeroelasticity. It is

realized that a variation of the Reynolds number (wing sizing, flapping frequency, etc.) leads to a change

in the leading edge vortex (LEV) and spanwise flow structures, which impacts the aerodynamic force

generation. While in classical stationary wing theory, the tip vortices (TiVs) are seen as wasted energy,

in flapping flight, they can interact with the LEV to enhance lift without increasing the power

requirements. Surrogate modeling techniques can assess the aerodynamic outcomes between two- and

three-dimensional wing. The combined effect of the TiVs, the LEV, and jet can improve the

aerodynamics of a flapping wing. Regarding aeroelasticity, chordwise flexibility in the forward flight

can substantially adjust the projected area normal to the flight trajectory via shape deformation, hence

redistributing thrust and lift. Spanwise flexibility in the forward flight creates shape deformation from

the wing root to the wing tip resulting in varied phase shift and effective angle of attack distribution

along the wing span. Numerous open issues in flapping wing aerodynamics are highlighted.

& 2010 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

2. Equations and parameters of flapping wing dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

2.1. Kinematics of flapping flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

2.2. Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

2.3. Scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

3. Key attribute of unsteady flapping wing aerodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

3.1. Clap and fling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

3.2. Rapid pitch rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

3.3. Wake capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

3.4. Delayed stall of leading edge vortex (LEV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

3.5. Tip vortex (TiV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

3.6. Passive pitching mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

4. Kinematics, wing geometry, Re, and rigid flapping wing aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

4.1. Single wing in forward flight condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

4.2. Single wing in hovering flight condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

4.3. Tandem wing in forward/hovering flight condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

4.4. Implications of wing geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

4.5. Implications of wing kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

4.6. Surrogate modeling for hovering wing aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

ll rights reserved.

ge vortex; MAV, micro air vehicle; MTV, molecular tagging velocimetry

www.elsevier.com/locate/paerosci

dx.doi.org/10.1016/j.paerosci.2010.01.001

mailto:[email protected]

dx.doi.org/10.1016/j.paerosci.2010.01.001

W. Shyy et al. / Progress in Aerospace Sciences 46 (2010) 284–327 285

4.7. Unsteady flow structures around hawkmoth-like model in hover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

4.7.1. Vortex dynamics of hovering hawkmoth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

4.8. Effect of the Reynolds number on the LEV structure and spanwise flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

5. Flapping wing aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

5.1. Chordwise-flexible wing structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

5.2. Spanwise-flexible wing structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

5.3. Combined chordwise-and-spanwise flexible wing structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

1. Introduction

Micro air vehicles (MAVs) have the potential to revolutionizeour sensing and information gathering capabilities in areas suchas environmental monitoring and homeland security. Numerousvehicle concepts, including fixed wing, rotary wing, and flappingwing, have been proposed [1–8]. As the size of a vehicle becomessmaller than a few centimeters, fixed wing designs encounterfundamental challenges in lift generation and flight control. Thereare merits and challenges associated with rotary and flappingwing designs. Fundamentally, due to the Reynolds number effect,the aerodynamic characteristics such as the lift, drag and thrust ofa flight vehicle change considerably between MAVs and conven-tional manned air vehicles [1–8]. And, since MAVs are of lightweight and fly at low speeds, they are sensitive to wind gust[1–9]. Furthermore, their wing structures are often flexible andtend to deform during flight. Consequently, the fluid andstructural dynamics of these flyers are closely linked to eachother. Because of the common characteristics shared by MAVs andbiological flyers, the aerospace and biological science commu-nities are now actively communicating and collaborating. Muchcan be shared between researchers with different training andbackground including biological insight, mathematical models,physical interpretation, experimental techniques, and designconcepts.

In order to handle wind gust, object avoidance, or stationkeeping, highly deformed wing shapes and coordinated wing–tailmovement in the biological flight are often observed. Under-standing the aerodynamic, structural, and control implications ofthese modes is essential for the development of high performanceand robust flapping wing MAVs for accomplishing desirablemissions. Moreover, the large flexibility of the wings leads tocomplex fluid–structure interactions, while the kinematics offlapping and the spectacular maneuvers performed by naturalflyers result in highly coupled nonlinearities in fluid dynamics,aeroelasticity, flight dynamics, and control systems.

Insect wing structures are inherently anisotropic due to theirmembrane–vein configurations, with the spanwise bendingstiffness being approximately 1–2 orders of magnitude largerthan the chordwise bending stiffness in a majority of insectspecies [10,11]. In general, the spanwise flexural stiffness scaleswith the third power of the wing chord, while the chordwisestiffness scales with the second power of the wing chord [10,11].Insect wings exhibit substantial variations in aspect ratio andconfiguration but share a common feature of a reinforced leadingedge. A dragonfly wing has more local variations in its structuralcomposition and is more corrugated than the wing of a cicada or awasp [1,12]. It has been shown in the literature [1–3,12] that wingcorrugation increases both warping rigidity and flexibility.Furthermore, specific characteristic features have been observedin the wing structure of a dragonfly which help prevent fatiguefracture [1,12]. The thin nature of the insect wing skin structuremakes it unsuitable for taking compressive loads, which may

result in skin wrinkling and/or buckling, i.e., large local deforma-tions that will interact with the flow. On the aerodynamics side, ina fixed wing set-up, wind tunnel measurements show thatcorrugated wings are aerodynamically insensitive to the Reynoldsnumber variations, which is quite different from a typical lowReynolds number airfoil [1,4,6,7,12].

As highlighted above, biological flyers showcase desirableflight characteristics and performance objectives [1,12–23]. Thestrategies exhibited in nature have the potential to be utilized inthe design of flapping wing MAVs [1–8,24–27]. In particular, wingflexibility is likely to have a significant influence on the resultingaerodynamics. Based on a literature survey, it was found thatseveral questions in flexible wing aerodynamics have not beenadequately addressed in existing literature, among which, the keyones include: (i) How do geometrically nonlinear effects and theanisotropy of the structure impact the aerodynamics character-istics of the flapping wing? (ii) How can flapping flight bestabilized passively via flexible structures?

Furthermore, even though the rigid wing aerodynamicshave been explored in more detail than the flexible wingaerodynamics, several questions still remain, among which, thekey ones include: (i) How can the unsteady flow features bemanipulated to enhance performance? As the sizing, flappingkinematics, flapping frequency, and flight speed vary, which fluidphysics mechanisms are important? (ii) How can the observationsfrom high fidelity simulations or experimental studies be distilledinto reduced order models so that they are fast enough to executefor MAV control development? Since all of the above are notnecessarily independent topics, a comprehensive understandingof the role of flapping wing kinematics, aerodynamics, andflexibility is central to the success of future flapping wing MAVdesigns.

As evidenced in the references cited, a number of publicationsexist to address numerous aspects of these issues. Furthermore,recently, many researchers have taken serious efforts in investigatingthese topics. There seems to be a need to consolidate the fastdeveloping information to help update and benefit the community.The purpose of this paper is to complement the recent workpresented by Shyy et al. [1] to review the recent progress in flappingwing aerodynamics and aeroelasticity at low Reynolds numbers,namely, (O(101)–O(104)). In addition to present established informa-tion, open issues in both aerodynamics/aeroelasticity are highlightedso as to encourage future community-wide efforts.

The rest of the paper is organized as follows:Flapping wing kinematics, governing equations, and scaling

laws are presented in Section 2. Unsteady flight mechanismsassociated with flapping wings and frequently encountered in theliterature are described in Section 3. A literature survey focusingon flapping wing aerodynamics and aeroelasticity is presented inSections 4 and 5 while emphasizing computational efforts of theauthors to highlight selected flapping wing physics. Finally,concluding remarks and areas warranting further study are madein Section 6.

Nomenclature

AR aspect ratio, b2/SAs extensional stiffness coefficient of an isotropic plate,

Ehs/(1�n2)b spanCf total aerodynamic force coefficient, Faeroj j=ð0:5rf U

2ref SÞ

CL lift coefficient, FZ=ð0:5rf U2ref SÞ

cm mean chord lengthCT thrust coefficient, FX=ð0:5rf U

2ref SÞ

Ds bending stiffness coefficient of an isotropic plate,Eh3

s =½12ð1�n2Þ�

E Young’s modulus of materialf flapping frequencyfp distributed transverse load on a plateFaero aerodynamic force vector acting on a wing_h resultant velocity due to plunging motion including

rigid-body and wing deformationha plunge amplitudehs thickness of a plate2ha/cm normalized stroke amplitudeIB moment of inertiak reduced frequency, pfcm/Uref

M mass of a flyerMm mass of flight muscle of a flyerO origin of coordinate systemPaero muscle-mass-specific aerodynamic powerQ second invariant of the velocity gradient tensor, �0.5R semi-spanRe Reynolds number, rUrefLref/mS wing planform areaSt Strouhal number, fLrefUref

t timeT period of flapping motionUN forward flight velocity/freestream velocityUtip mean wing tip velocityU velocity vector of fluid

U V W axial, lateral, and vertical velocity

u0 v0 w0 displacement of a point on the mid-surface of aplate considered in the xs–ys plane

Vwing wing velocity vectorx y z wingroot-based coordinate systemxs ys zs global coordinate system for structureX Y Z global coordinate systema geometric angle of attackaa pitch amplitudeae effective angle of attackb stroke plane angley elevation/deviation anglem dynamic viscosity of fluidn Poisson’s ratioP1 effective stiffness, Eh3

s =½12ð1�n2Þrf U2ref c

3m�

P2 effective rotational inertia, IB=ðrf U5ref Þ

r density ratio, rf/rs

rf density of fluidrs density of structuref positional/stroke angle (in 3-D flapping wing) [deg.];

phase lag between translation and rotation of wingmotion (in 2-D and 3-D simplified wing kinematics)

F full positional/stroke amplitudew body angle/ S time averaged valueA dimensionless variable

Subscripts

max maximum valuemin minimum valueL.E. variable with reference to leading edge pointT.E. variable with reference to trailing edge pointtip variable with reference to wing tiprg variable with reference to a point of radius of gyration

for second moment of the wing

W. Shyy et al. / Progress in Aerospace Sciences 46 (2010) 284–327286

2. Equations and parameters of flapping wing dynamics

Aerodynamics associated with flapping wings can be modeledusing the unsteady Navier–Strokes (NS) equations. Nonlinearphysics with multiple variables (velocity, pressure) and movinggeometries are among the aspects of primary interest. Numerousflapping wing kinematics and scaling parameters related to thefluid dynamics and fluid–structure interactions exist. The flightregime of each flyer is characterized by these parameters.

2.1. Kinematics of flapping flight

The kinematics of flapping flight is composed of body and wingmovements. As shown in Fig. 1, assuming that the wing and bodyare rigid, the body kinematics can be represented by the bodyangle (w) (inclination of the body), relative to the horizontal plane(e.g. the ground). The flapping wing kinematics can be describedby three basic positional angles of the wing with the stroke plane(b): (i) flapping about the x-axis in the wing root-fixed coordinatesystem described by the positional or stroke angle (f), (ii)rotation of the wing about the z-axis in the wing root-fixedcoordinate system described by the elevation or deviation angle(y), and (iii) rotation of the wing about the y-axis in the wing root-fixed coordinate system described by the angle of attack (a). The

stroke plane is defined by the wing base and the wing tip of themaximum and the minimum sweep positions. Examples of thetime histories of three insects are shown in Fig. 2. The timehistories of positional angle show approximately first-ordersinusoidal curves. The time histories of elevation angle showsmall amplitude and have twice frequency of main flappingfrequency. On the other hand, the time histories of angle of attackinclude high frequency component of the flapping frequency andsome of insects show asymmetric patterns per stroke. Moreover,the body angle and the stroke plane angle vary in accordance withthe flight speed and flapping wing kinematics of biological flyers[1,28,29].

Due to the complexity of the aerodynamics associated withbio-mimicking kinematics (as shown in Figs. 1 and 2), building adescription of the fundamental factors involved can benefit fromsimplified models. The simplified models referred to later (Section4) in the paper remove the rotational (centripetal and Coriolis)aspects while retaining vortex dynamics important in flappingwing flight. While the combination of pitching and translation(versus flapping about a pivot point) (see Fig. 3) of the entire wingare not found in a biological flyer, the motions used provide abasis for more complex analysis and are feasible mechanicaldesigns.

The motions of the wing can be described by sinusoidaltranslation (along the X-axis) and pitching (about the Y-axis) with

z

y

Stroke plane

O

Z

X

Y

z

�

�

�

�

x

Stroke plane

Yaw

Pitch

Roll

O

Body

Wing

Fig. 1. Schematic diagrams of coordinate systems and kinematics of flapping wing

and body movement. The local wingroot-fixed and the global space-fixed

coordinate systems are shown. The local wingroot-fixed coordinate system

(x,y,z) is fixed at the center of the stroke plane (origin O at the wing root) with x

direction normal to the stroke plane, the y direction vertical to the body axis, and

the z direction parallel to the stroke plane. Definitions of the positional or stroke

angle (f), the angle of attack (a), the elevation or deviation angle (y), the body

angle (w), and the stroke plane angle (b) are indicated in the figure.


the same frequency but not necessarily in phase. Three indepen-dent design variables are introduced: the normalized strokeamplitude with reference to a point of radius of gyration forsecond moment of the wing (2ha=cmrg ) dictating the magnitude ofthe translation, the pitching amplitude (aa) which then definesthe angle of attack (a), and the phase lag (f) which specifies thetiming of the rotation relative to the translation. In synchronizedrotation, the rotation takes place at the ends of translation and theleading edge is the same for both the forward and backstroke.Changing phase lag will cause the rotation to start early(advanced rotation) or after the translation has switched direction(delayed rotation). This will affect the angle of attack duringsubsequent interactions with the wake as well when the wing hasachieved maximum translational velocity at the middle of thestroke.

In unsteady flows, there is no single angle of attack at eachwing section since the flow direction varies along the chord as aresult of the induced flow. Therefore, a very useful quantityknown as effective angle of attack is defined in the literature for

the case of simple harmonic motion. It is given by the followingexpression [30,31]:

ae ¼ tan�1 �_h

U1

!�a ð1Þ

where the effective angle of attack ae is measured at referencepoints of the airfoil, a is the angle of attack, UN is freestream velocityor flight velocity, and h is the velocity due to plunge motion,respectively. It may be noted that in the unsteady motion of flexiblewing structures, the velocities ( _h) not only include the rigid bodyplunge and pitch motions, respectively but also those due to bendingand twist deformations. In such a case, the angle of attack (a) can bedefined simply as the angle between the line joining the leading andtrailing edges of an airfoil section and the direction of freestream. Ifthe structure behaves as a beam, the effective angle of attack willremain the same at each point along the chord (the angle of attackstill varies along the chord due to chordwise variation of inducedflow). It is clearly not the case if the structure behaves like a plate ora shell. Effective angles of attack vary through the chord due toplate-like deformations. The angle at three-quarter chord then maybecome a representative sectional effective angle of attack [30].Fig. 4 illustrates the effective angle of attack due to prescribedplunge motion (reference point is three-quarter chord from thetrailing edge [31]), and prescribed plunge motion (reference point isthe leading edge) with chordwise deformation [32].

2.2. Governing equations

The governing equations of fluid are the unsteady, incompres-sible 3-D NS equations and the continuity equation, which areexpressed in vector form as follows:

@U

@tþUUrU¼�

1

rf

rpþmrf

r2U;

rUU¼ 0

ð2Þ

where rf is the fluid density, m is the dynamic viscosity coefficient,U¼ ðU V W Þ is the velocity vector of the fluid, t is the time,X¼ ðX Y Z Þ is the position vector of the fluid based on theinertial frame, r is the gradient operator with respect to X, and p

is the pressure, respectively.Assuming an isotropic plate-like flapping wing structure that is

loaded in the transverse direction, the governing equations ofmotion can be written as

As@2u0

@x2s

þAs@2v0

@xs@ysþð1�nÞAs

@2u0

@y2s

¼ 0 ð3aÞ

As@2u0

@xs@ysþð1�nÞAs

@2v0

@x2s

þAs@2v0

@y2s

¼ 0 ð3bÞ

Ds@4w0

@x4s

þ2Ds@4w0

@x2s @y2

s

þDs@4w0

@y4s

�rshs@2w0

@t2¼ fp ð3cÞ

where u0, v0, and w0 are displacement in the xs, ys, and zs

direction, respectively, of a point on the mid-surface of the plateconsidered in the xs–ys plane. And, the coefficients As=Ehs/(1�n2)and Ds ¼ Eh3

s = 12ð1�n2Þ� �

correspond to the extensional andbending stiffnesses, respectively, rs is the density of the platematerial, hs is the thickness of the plate, E and n is Young’smodulus of material and Poisson’s ratio, and fp is the distributedtransverse load on the plate. The first two equations presentedabove correspond to the in-plane motion and the last onecorresponds to the out-of-plane motion. For the purpose ofscaling, only the equation of the out-of-plane motion is con-sidered in Section 2.3. More details of the plate equations aregiven in Refs. [33,34].

Fig. 2. Time histories of the three angles associated with flapping wing motion of hovering flight for (A) a hawkmoth model [42]; (B) a honeybee model [42]; and (C) a fruit

fly model [42]. Positional/stroke angle, elevation/deviation angle, and angle of attack of the wing are indicated by solid (green), dash-dotted (blue), and dashed (orange)

lines, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. Schematics diagram for simplified wing kinematics. The stroke plane angle

(b) is equal to zero. The global coordinate system (X,Y,Z) is fixed at the center of the

stroke plane with Z direction normal to the stroke plane, the Y direction

perpendicular to the wing axis, and the X direction parallel to the stroke plane.

a and 2ha=cmrg are the angle of attack and normalized stroke amplitude with

reference to a point of radius of gyration for second moment of the wing [16],

respectively.


2.3. Scaling laws

Scaling laws are useful to reduce the number of parameters, toclearly identify characteristic properties of the system underconsideration, and to indicate which combination of parameters

becomes important under a given condition. From the view pointof fluid–structure interaction, several dimensionless parametersarise during the non-dimensionalization process of the fluid andstructural dynamics equations using a set of suitable referencescales. Depending upon the problem at hand and the type ofequations used to model the physical phenomena involved, theresultant set of scaling parameters could vary. In flapping wingflight, three dimensionless parameters related to the fluiddynamics, wing kinematics, and three other dimensionlessparameters relevant to the fluid–structure interaction are high-lighted next (see Table 1):

(i)
The Reynolds number (Re) represents a ratio between inertialforces and viscous forces.Given a reference length (Lref) and a reference velocity (Uref),the Reynolds number (Re) is defined as
Re¼rf Lref Uref

m: ð4Þ

In flapping wing flight, since lift and thrust are generated byflapping wings, the mean chord length of the wing (cm), isused as the reference length (Lref) and the inverse of theflapping frequency (1/f) is often utilized as the reference time(Tref). On the other hand, the reference velocity needs to beselected carefully considering the flight conditions and wingkinematics.In hovering flight, and the mean wingtip velocity of theflapping wing can be used as the reference velocity, writtenas Uref=Utip=oR, where o is the mean angular velocity of the

Fig. 4. Illustrations to show effective angle of attack due to prescribed plunge motion and chordwise deformation of (A) ae is the effective angle of attack of an airfoil due to

plunge motion (extracted from Ref. [31] ) (B) aeL:E:and aL.E. are the effective angle of attack for the chordwise flexible wing case and the angle of attack due to the wing

deformation. The effective angle of attack decreases with geometric angle of attack (aL.E.) due to the wing deformation extracted from Ref. [32].

Table 1Dimensionless parameters and scaling dependency for flapping wings.

Dimensionless parameter Based on flapping wing velocity Uref=Utip=ARcmUf Based on flight speed Uref=UN

Length Frequency Length Frequency Velocity

Reynolds number: Re=qfUrefcm/l c2m

f cm Independent UN

Reduced frequency: k=(pfcm)/Uref Independent Independent cm f U�11

Strouhal number: St=(2fha)/Uref Independent Independent cm f U�11

Effective stiffness: P1 ¼Ds=ðqf U2ref c3

mÞ c�2m

f�2 Independent Independent U�21

Effective rotational inertia: P2 ¼ IB=ðqf c5mÞ Independent Independent Independent Independent Independent

Density ratio: q¼ qs=qf Independent Independent Independent Independent Independent

Note that Ds indicates the flexural rigidity which is defined as the force couple required to bend a structure to a unit curvature. In the case of the isotropic plate,

Ds ¼ Eh3s =ð12ð1�n2ÞÞ.


wing about x-axis (o=2Ff, where F is the full strokeamplitude, measured in radians) and R is the wing semi-span. Therefore the Reynolds number (Re) for hovering flightcan be rewritten as

Re¼rf Lref Uref

m¼rf ARFfc2

m

m; ð5Þ

where the aspect ratio AR=b2/S, with the wing planform area(S) being the product of the wing span (b) and the meanchord length (cm). Note that the Reynolds number isproportional to the flap amplitude, the flapping frequency,square of the mean chord length, and the aspect ratio of thewing (see Table 2).In forward flight, there are multiple candidates for thereference velocity, for example, the mean wing tip velocityand the forward flight velocity (UN). If the reference velocity

is chosen as the forward flight velocity, the Reynolds numbercan be represented as

Re¼rf Lref Uref

m ¼rf cmU1

m ð6Þ

In comparison with the Reynolds number based on the meanwing tip velocity, the Reynolds number based on the forwardflight velocity is proportional to the mean chord length andflight velocity, and not related to the flapping frequency andthe flapping amplitude.

(ii)
The Strouhal number (St) describes the relative influence offorward flight (UN) versus the flapping speeds [1]. TheStrouhal number (St) characterizes the vortex dynamics ofthe wake and shedding behavior of vortices of a flappingwing in forward flight [1,35]. For flapping wing flight, theStrouhal number is defined based on the flapping frequency,

Table 2Morphological, flight, scaling, and non-dimensional parameters of selected biological flyers.

Parameter Chalcid Wasp(Encarsia formosa)

Fruit fly(Drosophila melanogaster)

Honeybee(Apis mellifica)

Hawkmoth(Manduca sexta)

Rufous Hummingbird(Selasphorus rufus)

Mean chord length: cm (mm) 0.33 0.78 3.0 18.3 12

Semi-span: R (mm) 0.70 2.39 10.0 48.3 54.5

Aspect ratio: AR 4.24 6.12 6.65 5.3 9

Total mass: M (g) 2.6�10�7 0.96�10�3 0.1 1.6 3.4

Flapping frequency: f (Hz) 370 218 232.1 26.1 41

Flapping amplitude: U (rad) 2.09 2.44 1.59 2.0 2.02

Mean wing tip velocity: Urip (m/s) 1.08 2.54 7.38 5.04 8.66

Flight speed: UN (m/s) 0.0 0.0 0.0 0.0 0.0

Normalized stroke amplitude: 2ha=cmtip 4.4 7.48 5.3 5.28 9.17

Normalized stroke amplitude: 2hacmrg4.4 4.3 3.05 3.04 5.27

Reynolds number: Re 23 126 1412 5885 6628

Reduced frequency: k 0.355 0.212 0.297 0.296 0.172


the travel distance of full flapping amplitude (RF), and thespeed of forward fight, namely,

St¼fLref

Uref¼

fRFU1¼

fARcmF2U1

: ð7Þ

This definition offers a measure of propulsive efficiency offlapping wing flyer in forward flight.

(iii)
The reduced frequency (k) provides a better measure ofunsteadiness associated with a flapping wing than theStrouhal number by comparing spatial wavelength of theflow disturbance with the chord length [35]. It is defined bythe angular speed of the flapping wing (2pf), mean chordlength (cm), and the reference velocity (Uref), namely,
k¼pfcm

Urefð8Þ

The reduced frequency based on the mean wingtip velocitycan be formulated as

k¼pfcm

Uref¼

pFAR

ð9Þ

Note that the reduced frequency is inversely proportional tothe flapping amplitude and aspect ratio of the wing and notrelated to flapping frequency.On the other hand, the reduced frequency based on theforward or cruising flight speed can be rewritten as

k¼pfcm

U1ð10Þ

The reduced frequency is proportional to flapping frequencyand the mean chord length, and inversely flight speed. Therelationship between the Strouhal number and the reducedfrequency based on the forward flight speed is

St

k¼

ARF2p ð11Þ

The density ratio (r) describes the ratio between the
(iv) equivalent structural density and fluid density.
r¼ rs

rf

ð12Þ

The effective stiffness (P1) describes the ratio between
(v) elastic bending forces and aerodynamic (or fluid dynamic)forces.
P1 ¼Eh3

s

12ð1�n2Þrf U2ref c3

m

ð13Þ

If an isotropic shear deformable plate is considered, an
(vi) additional dimensionless parameter [33,34], the effective
rotational inertia (P2), will be

P2 ¼IB

rf c5m

; ð14Þ

where IB is the mass moment of inertia. This describes theratio between rotational inertia forces and aerodynamic (orfluid dynamic) forces.

As a summary, assuming that the geometric similarity ismaintained, the scaling laws for rigid and flexible flapping wingaerodynamics for two sets of reference velocities are listed inTable 1.

Furthermore, in order to understand the effect of the above-scaling parameters on the governing equations, non-dimensionalforms of the governing equations are presented based on thereference velocity as follows:

If the flapping wing velocity is chosen as the velocity scale,then the resulting non-dimensional form of the NS equations andthe equation of out-of-plane motion of the isotropic plate are

k

p@U

qtþUUrU¼�rpþ

1

Rer

2U ð15Þ

P1@4w0

@x4s

þ2@4w0

@x2s @y2

s

þ@4w0

@y4s

!�rhs

k

p

� �2 @2w0

@t2¼ f p ð16Þ

where the over-bar designates the dimensionless variable. This formof the equation separates the reduced frequency, the Reynoldsnumber, the density ratio, and the effective stiffness, making itconvenient to study the effects of these parameters. For biologicalflyers, the flapping frequency ranges 10–600 Hz, and the wing lengthvaries from 0.3 to 600 mm yielding the Reynolds number from O(104)to O(101) [1,36]. In this flight regime, unsteady effects, inertia,pressure, internal, and viscous forces are all important.

3. Key attribute of unsteady flapping wing aerodynamics

Natural flyers utilize flapping mechanisms to generate lift andthrust. These mechanisms are related to formation and shedding ofthe vortices into the flow, varied wing shape, and structural flexibility.Therefore, understanding the vortex dynamics, the vortex–winginteraction, and fluid–structure interaction is very important. A briefintroduction to some of the key unsteady mechanisms associatedwith flapping wings that are frequently encountered in the literatureis given next. Specific topics related to the interplay between the


unsteady aerodynamics and kinematics, Reynolds number, and winggeometry will be discussed in detail in Section 4.

3.1. Clap and fling

The earliest unsteady lift generation mechanism to explain howinsects fly, found by Weis-Fogh [16], was the clap-and-fling motion ofa chalcid wasp, Encarsia formosa. Based on the steady-stateapproximation, the lift generated by the chalcid wasp was insufficientto stay aloft. To explain this, he observed that a chalcid wasp clapstwo wings together and then flings them open about the horizontalline of contact to fill the gap with air. During the fling motion, the airaround each wing acquires circulation in the correct direction togenerate additional lift. A schematic of this procedure is shown inFig. 5. As illustrated in Fig. 6, Lehmann et al. [37] elucidated this clap-and-fling mechanism with PIV flow visualizations and forcemeasurements using a dynamically scaled robotic wing model. Alsonumerical investigations further demonstrated lift enhancement dueto the clap-and-fling mechanisms at low Reynolds numbers [38–42].The relative benefit of clap-and-fling lift enhancement stronglydepended on stroke kinematics and could potentially increase theperformance by reducing the power requirements [43,44]. The clap-and-fling mechanism is beneficial in producing a mean lift coefficientto keep a low weight flyer aloft: numerous natural flyers, such ashawkmoths, butterflies, fruitflies, wasps, and thrips enhance theiraerodynamic force production with the clap-and-fling mechanism[16,45–49].

3.2. Rapid pitch rotation

At the end of each stroke, flapping wings can experience rapidpitching rotation, which can enhance the aerodynamic force

Fig. 5. Sectional schematic of wings approaching each other to clap (A–C) and fling apart

flow lines and dark blue arrows show induced velocity. Light blue arrows show net for

their leading edges touch initially (B), and the wing rotates around the leading edge. As t

in the form of stopping vortices (C), which dissipate into the wake. The leading edge vo

giving and additional thrust. (D–F) Fling. The wings fling apart by rotating around the tr

between the two wing sections, giving an initial boost in circulation around the wing sys

are mutually annihilated as they are of opposite circulation. (For interpretation of the r

this article.)

generation [50]. The phase difference between the translation andthe rotation can be utilized as a lift controlling parameter: similarto the Magnus effect, if the wing flips before the stroke ends, thenthe wing undergoes rapid pitch-up rotation in the correcttranslational direction enhancing the lift. This is called theadvanced rotation. On the other hand, in delayed rotations, if awing rotates back after the stroke reversal, then when the wingstarts to accelerate it pitches down resulting in reduced lift [50].In a follow-up study Sane and Dickinson [51] related the lift peakat the stroke ends to be proportional to the angular velocity of thewing using the quasi-steady theory. The numerical studies [1,52]showed an increase in the vorticity around the wing due to rapidpitch-up rotation of the wing led to augmentation of the liftgeneration.

3.3. Wake capture

The wake capture mechanism is often observed during awing–wake interaction. When the wings reverse their transla-tional direction, the wings meet the wake created during theprevious stroke, by which the effective flow velocity increases andadditional aerodynamic force peak is generated. Lehmann et al.[37], Dickinson et al. [50], and Birch and Dickinson [53] examinedthe effect of wake capturing of several simplified fruit fly-likewing kinematics using a dynamically scaled robotic fruit fly wingmodel at Re=1.0–2.0�102. They compared the force measure-ment data with the quasi-steady approximation then isolated theaerodynamic influence of the wake. Results demonstrated thatwake capture force represented a truly unsteady phenomenondependent on temporal changes in the distribution and magni-tude of vorticity during stroke reversal. The sequence of the wakecapture mechanism is illustrated in Fig. 7. Wang [54] and Shyyet al. [1,55] further elucidated the wake capture mechanism and

(D–F) adopted from Ref. [18] and originally described in Ref. [16]. Black lines show

ces acting on the airfoil. (A–C) Clap. As the wing approach each other dorsally (A),

he trailing edges approach each other, vorticity shed from the trailing edge rolls up

rtices also lose strength. The closing gap between the two wings pushes fluid out,

ailing edge (D). The leading edge translates away and fluid rushes in to fill the gap

tem (E). (F) A leading edge vortex forms anew but the trailing edge starting vortices

eferences to color in this figure legend, the reader is referred to the web version of

Fig. 7. Illustrations of wake capture mechanism [1,55]. (A) Supination, (B) beginning of upstroke, and (C) early of upstroke. At the end of the stroke, (A), the wake shed in

the previous stroke denoted by CWV is en route of the flat plate. As the flat plate moves into the wake (B and C) the effective flow velocity increases and additional

aerodynamic force is generated. The color of contour indicates spanwise-component of vorticity. CWV and CCWV indicate clock-wise and counter clock-wise vortex.

Fig. 6. Experimental visualization of clap-and-fling mechanism by two-wings (M-T) using robotic wing models adopted from Ref. [37]. The leading edge of the dorsal wing

surface is marked by a white half circle. Fluid flow velocities are plotted in color/shades; red/light indicates large velocity magnitudes and arrows represent the velocity

vectors at each point in the fluid; longer arrows signify larger velocities. (For interpretation of the references to color in this figure legend, the reader is referred to the web

version of this article.)


lift augmentation of the instantaneous lift peak using 2-Dnumerical simulations. The effectiveness of the wake capturemechanism was a function of wing kinematics and flow structuresaround the flapping wings [1,37,50,53]. A different view on theeffect of wake capture existed as well. Jardin et al. [56] used aNACA0012 airfoil under asymmetric flapping wing kinematicssuch that in the downstroke the interaction of the previously shedwake with the leading edge vortex (LEV) formation was reduced.In the most cases they considered this reduced effect of wakecapture led to a closely attached downstroke LEVs. Compared to asynchronized wing rotation they saw enhanced downstrokeaerodynamic loading.

3.4. Delayed stall of leading edge vortex (LEV)

Ellington et al. [57–60] suggested that the delayed stall of LEVcan significantly promote lift associated with a flapping wing. TheLEV created a region of lower pressure above the wing and henceit would enhance lift. Multiple follow-up investigations [61–64]were conducted for different insect models, resulting in a betterunderstanding on the role played by the LEV and its implicationson lift generation. When a flapping wing travels several chordlengths, the flow separates from the leading and trailing edges, as

well as at the wing tip, and forms large organized vortices knownas a leading edge vortex (LEV), a trailing edge vortex (TEV), and atip vortex (TiV). In flapping wing flight, the presence of LEVs isessential to delay stall and to augment aerodynamic forceproduction during the translation of the flapping wings as shownin Fig. 8 [18]. Fundamentally, the LEV is generated and sustainedfrom the balance between the pressure-gradient, the centripetalforce, and the Coriolis force in the NS equations. The LEVgenerates a lower pressure area in its core, which results in anincreased suction force on the upper surface. Employing 3-D NScomputations, Liu and Aono [42] and Shyy and Liu [65]demonstrated that a LEV is a common flow feature in flappingwing aerodynamics at Reynolds numbers O(104) and lower, whichcorrespond to the flight regime of insects and flapping wingMAVs. However, main characteristics and implications of the LEVon the lift generation varied with changes in the Reynoldsnumber, the reduced frequency, the Strouhal number, the wingflexibility, and flapping wing kinematics. Milano and Gharib [66]measured the forces generated by pitching rectangular flat plateat approximately Re=4.0�103 and observed the trajectoriesyielding maximum average lift based on a genetic algorithm.Results showed the optimal flapping produces LEVs of maximumcirculation and that a dynamic formation time that described thevortex formation process of about 4 is associated with production

FNormal FResult

FSuction

Lift FResult = FSuction + FNormal

Drag

Fig. 8. Leading edge suction analogy adopted from Ref. [18]. (A) Flow around the

blunt wing. The sharp diversion of flow around the leading edge results in a

leading edge suction force (dark blue arrow), causing the resultant force vector

(light blue arrow) to tilt towards the leading edge and perpendicular to free

stream. (B) Flow around a thin airfoil. The presence of a leading edge vortex causes

a diversion of flow analogous to the flow around the blunt leading edge in (A) but

in a direction normal to the surface of the airfoil. This results in an enhancement of

the force normal to the wing section. (For interpretation of the references to color

in this figure legend, the reader is referred to the web version of this article.)


of a maximum-circulation vortex [67,68]. Rival et al. [69]investigated experimentally the formation process of LEVsassociated with several combinations of pitching and plungingSD7003 airfoils in forward flight using PIV at Re=3.0�104. Resultssuggested that by carefully tuning the airfoil kinematics, thusgradually feeding the LEV over the downstroke, it was to someextent possible to stabilize the LEV without the necessity of aspanwise flow.

Tarascio et al. [70] and Ramasamy and Leishman [71]visualized the flow fields around a biologically inspired flappingwing at Re=1.0�104 by a mineral oil fog strobed with a lasersheet and PIV. They presented the presence of a shed dynamicstall vortex that spans across most of the wing span and multipleshedding LEVs on the top surface of the wing during each wingstroke. Also they provided several observations related to the roleof turbulence at low Reynolds numbers. Poelma et al. [72]measured the time-dependent 3-D velocity field around aflapping wing at Re=256 based on maximum chord. A dynami-cally scaled fruitfly wing in mineral oil with hovering kinematicsextracted from real insect movements was used. They presentedrefined 3-D structures of LEVs and suggested including thecounter-rotating TEVs to get a complete picture for productionof circulation. Lu and Shen [73] highlighted the detailedstructures of LEVs for a flapping wing in hover at Re=1.6�103.They used phase-lock-based multi-slice digital stereoscopic PIV toshow that the spanwise variation along the LEV was time-dependent. Their results demonstrated that the observed LEVsystems were a collection of four vortical elements: one primaryvortex and three minor vortices, instead of a single conical ortube-like vortex as reported or hypothesized in previous studies[50,57]. Recently, Pick and Lehmann [74] used 3-D three-components multiple-color-plane stereo PIV techniques to obtaina 3-D velocity field around a flapping wing. The need for 3-D PIV isevident since the critical flow features in understanding flappingwing aerodynamics, such as LEVs and unsteady wakes behind aninsect body, are inherently 3-D in nature. Compared to theprevious findings, they reported similar structure of the LEV butstronger outward axial flow inside the LEV of up to 80% of themaximum in-plane velocity. On the other hand, Liang et al. [75]presented results based on direct numerical simulation toinvestigate wake structures of hummingbird hovering flight andassociated aerodynamic performance. They reported that theamount of lift produced during downstroke is about 2.95 times ofthat produced in upstroke. Two parallel vortex rings were formedat the end of the upstrokes. There is no obvious leading edgevortex can be observed at the beginning of the upstroke. Althoughonly rigid wing structures were considered, the results wereclaimed to be in good agreement with PIV measurements ofWarrick et al. [76] and Altshuler et al. [77].

3.5. Tip vortex (TiV)

Tip vortices (TiVs) associated with fixed finite wings are seento decrease lift and induce drag [78]. However, in unsteady flows,TiVs can influence the total force exerted on the wing in threeways (see Figs. 8 and 9): (i) creating a low-pressure area near thewing tip [55,79–81], (ii) an interaction between the LEV and theTiV [55,79–81], and (iii) constructing wake structure bydownward and radial movement of the root vortex and TiV [71].First two mechanisms ((i) and (ii)) also were observed forimpulsively started flat plates normal to the motion with lowaspect ratios: Riguette et al. [82] presented experimentally thatthe TiVs contributed substantially to the overall plate force byinteracting with the LEVs at Re=3.0�103. Taira and Colonius [83]utilized the immersed boundary method (IBM) to highlight the

3-D separated flow and vortex dynamics for a number of lowaspect ratio flat plates at different angles of attacks. At Re of3.0�102–5.0�102. They showed that the TiVs could stabilize theflow and exhibited nonlinear interaction with the shed vortices.Stronger influence of downwash from the TiVs resulted inreduced lift for lower aspect ratio plates.

For flapping motion in hover, however, depending on thespecific kinematics, the TiVs could either promote or make littleimpact on the aerodynamics of a low aspect ratio flapping wing.Shyy et al. [55] demonstrated that for a flat plate with AR=4 atRe=64 with delayed rotation kinematics, the TiV anchored thevortex shed from the leading edge increasing the lift compared toa 2-D computation under the same kinematics. On the other hand,under different kinematics with small angle of attack andsynchronized rotation, the generation of TiVs was small and theaerodynamic loading was well approximated by the analogous2-D computation. They concluded that the TiVs could eitherpromote or make little impact on the aerodynamics of a lowaspect ratio flapping wing by varying the kinematic motions [55].

3.6. Passive pitching mechanism

Wing torsional flexibility can allow for a passive pitchingmotion due to the inertial forces during wing rotation at strokereversals [84–88]. There were three modes of passive pitchingmotions which were similar to those suggested by rigid roboticwing model experiments [50]; (1) delayed pitching, (2) synchro-

Fig. 9. Drag coefficients of the aspect ratio 6, and 2 plates during a starting-up translation as a function of T (the number of chord lengths the plate has traveled, see

definition of Ref. [79]) adopted from Ref. [79]. (A) Overall view; (B) Detailed view of (A). Both (A) and (B) show drag coefficients for the free tip aspect ratio of 6 plate

(continuous line) and for the same plate with the tip grazing a raised bottom (dash-dotted line); the dashed line is drag coefficient for the aspect ratio of 2 plate with the tip

free. Low aspect ratio of the wing reduces the drag coefficient significantly due to interactions between a TiV and a LEV.


nized pitching, and (3) advanced pitching. It was found that theratio of flapping frequency and the natural frequency of the wingwere important to determine the modes of passive pitchingmotions of the wing [88,89]. If the flapping frequency was lessthan the natural frequency of the wing, then the wing experiencedan advanced pitching motion, which led to lift enhancement byintercepting the stronger wake generated during the previousstroke [89]. Moreover, it was shown for 2-D flows, the LEVsproduced by the airfoil motion with passive pitching seemed toattach longer on the flexible airfoil than on a rigid airfoil [88].

4. Kinematics, wing geometry, Re, and rigid flapping wingaerodynamics

This section presents a literature survey on flapping wingaerodynamics using experimental, theoretical, and computationalapproaches. Selected computational efforts of the authors areused to highlight implications of flow structures on the perfor-mance of rigid flapping wings. Note that the Reynolds numbersshown for hovering studies in this literature review may differfrom those in the referenced studies as consistent definitions(average wing velocity) are used for the sake of comparison in thispaper.

Experimentally, numerous previous efforts on flow visualiza-tion around biological flyers have been made, including smokevisualizations [47,49,90–92] and PIV [76,77,93–101] measure-ments. The advance of such technologies has enabled researchersto obtain not only 2-D but also 3-D flow structures aroundbiological flyers [93,98–101] and/or scaled models [72–74] withreasonable resolution in space. At the same time, measurementsof wing and body kinematics have been conducted using high-speed cameras [102–110], laser techniques (a scanning projectedline method [111], a reflection beam method [112], a fringeshadow method [113], and a projected comb fringe method[114]), and a combination of high-speed cameras and a projectedcomb-fringe technique with the Landmarks procedure [115].Advancement in measurement techniques also enabled quantifi-cation of flapping wing and body kinematics along with the 3-Ddeformation of the flapping wing. Recently, data on the instanta-neous wing kinematics involving camber along the span, twisting,and flapping motion have been reported (a hovering honeybee

[116]; a hovering hover fly and a tethered locust [117,118], afree-flying hawkmoth [119]). These efforts help in establishingmore complex and useful computational models [120,121].Furthermore, the in-vivo measurement of aerodynamic forcesgenerated by biological flyers in free-flight is a very challengingresearch topic.

Various models have been developed in an effort to under-stand flapping wing phenomena where the variables are known,controllable, and repeatable. Detailed discussion regarding theexperimental and numerical methodologies utilized to examineflapping wing-related studies is beyond the scope of the currenteffort. Suffice it to say, numerous computational techniques-based moving meshes [122–124] or stationary meshes (cut cell orimmersed boundary) [125,126] have been developed. The physi-cal models include NS as well as simplified treatments [127,128].Some of the experimental methods employed are introduced inthe paragraph above.

In the following section recent progress regarding rigidflapping wing aerodynamics is presented. First, studies forforward flight will be described. Then studies for hovering flightwill be presented. Explorations on the implications of wingkinematics and wing shape will be touched upon after that. Thiswill be followed by a highlight focusing on the unsteadyaerodynamics of 2-D and 3-D hovering flat plates, Re=O(102)based on a surrogate modeling approach. Finally, the fluiddynamics related to the LEV, the TEV, and the TiV will bepresented, including the authors’ computational efforts to high-light the vortex dynamics of a hovering hawkmoth at Re=O(103)and the effect of Reynolds number (size of flyers) on the LEVstructures and spanwise flow.

4.1. Single wing in forward flight condition

Von Ellenrieder et al. [129] studied the impact of variation ofStrouhal number (0.2oSto0.4), pitch amplitude (01oaao101),and phase angle (651ofo1201) between pitching and plungingmotion on 3-D flow structures behind a plunging/pitching finite-span NACA0012 wing using dye flow visualization at Re=164. Theresults demonstrated that the variation of these parameters hadobservable effects on the wake structure. However, they observeda representative pattern of the most commonly seen flowstructures and proposed a 3-D model of the vortex structure


behind a plunging/pitching wing in forward flight. Godoy-Dianaet al. [130,131] investigated the vortex dynamics associated witha pitching 2-D teardrop shaped airfoil (2.21oaao16.91,0.1oSto0.5) in forward flight at Re=1.2�103 using PIVmeasurements. Their results illustrated the transition from thevon Karman vortex streets to the reverse von Karman vortexstreets that characterize propulsive wakes. Furthermore, thesymmetry breaking of this reverse von Karman vortex patterngave rise to an asymmetric wake which was intimately related tothe time-averaged aerodynamic force production. Lee et al. [132]numerically investigated aerodynamic characteristics of unsteadyforce generation by a 2-D pitching and plunging 5% thick ellipticairfoil with inclined stroke plane at Re=6.8�102. They showedthat the thrust was generated due to correct alignment of thevortices at the end of the upstroke and there was a monotonicdecrease in thrust as the rotational center of the pitching motionwas moved from the leading edge towards the trailing edge(0.1oXo0.5).

Anderson et al. [133] considered harmonically oscillatingNACA0012 airfoils in a water tunnel to measure the thrust. Aftera parametric study (01oaao601, 0.25oha/cmo1.0,301ofo1101) to find the optimum flow condition for the thrustgeneration (aa=301, ha/cm=0.75, f=751) at Re=4.0�104 andSt=0.05–0.6, they proceeded to show the presence of a reversevon Karman vortex street formed by the vortices shed from theleading and trailing edges for St in range of 0.3 and 0.4 atRe=1.1�103. Triantafyllou and co-workers [134–136] performedparametric investigations using experiments on the performance ofa pitching/plunging NACA0012 airfoil in forward flight at Re

between 2.0�104 and 4.0�104, and St between 0.1 and 0.45.Systematic measurements of the fluid loading showed a uniquepeak efficiency of more than 70% for optimal combinations of theparameters (e.g. ha/c=0.75, aa=151, and f=901 at St=0.25 gives anefficiency of 73%) and observed that higher thrust can be expectedwhen increasing the Strouhal number and/or the maximum of theangle of attack. Then a parametric range where the efficiency andhigh thrust conditions were achieved together would fall in theparameter domain they considered. Lai and Platzer [137] used LDVand dye injection techniques to visualize the velocity field and thewake structures of an oscillating NACA0012 airfoil in water at Re

ranging from 5.0�102 to 2.1�104. The transition from drag tothrust was seen to depend on the non-dimensional plunge velocity(kha which is proportional to St), i.e. for kha40.4 the consideredairfoil was thrust-producing. Cleaver et al. [138] performed forceand supporting PIV measurements on a plunging NACA0012 airfoilat a Reynolds number of 1.0�104, at pre-stall, stall, and post-stallangles of attack. The lift coefficient for pre-stall and stall angles ofattack at larger plunge amplitudes showed abrupt bifurcationswith the branch determined by initial conditions. With thefrequency gradually increasing, very high positive lift coefficientswere observed, this was termed mode A. At the same frequencywith the airfoil impulsively started, negative lift coefficients wereobserved, this was termed mode B. The mode A flow field isassociated with trailing edge vortex pairing near the bottom of theplunging motion causing an upwards deflected jet, and a resultantstrong upper surface leading edge vortex. The mode B flow field isassociated with trailing edge vortex pairing near the top of theplunging motion causing a downwards deflected jet, and aresultant weak upper surface leading edge vortex. The bifurcationwas not observed for plunge small amplitudes due to insufficienttrailing edge vortex strength, nor at post-stall angles of attack dueto greater asymmetry in the strength of the trailing-edge vortices,which creates a natural preference for a downward deflected modeB wake.

Lewin and Haj-Hariri [139] conducted 2-D numerical investiga-tions for fluid dynamics associated with elliptic-like plunging

airfoils at Re=5.0�102 and compared and contrasted the findingswith ideal flow theories. They mentioned that for high-frequencyplunging, the vorticity was shed primarily from the trailing edge,and therefore better matched the inviscid models. Though for highkha the LEV and its secondary vortices were occasionally shed intothe wake, which would differentiate the physics from those foundin inviscid models. The monotonic trend found in ideal fluidmodels of efficiency decreasing as reduced frequency increases,was not seen in the numerical models where the maximumefficiency was found for an intermediate range of reducedfrequencies. Lua et al. [140] experimentally examined the wakestructure formation of a 2-D elliptic airfoil undergoing a sinusoidalplunging motion at Re=1.0�103 and k=0.1–2.0. The resultsshowed the type of wake structure produced depends on not onlythe reduced frequency, but also the normalized stroke amplitude.Bohl and Koochesfahani [141] focused on quantifying, via MTV, thevortical structures in the wakes of a sinusoidally pitchingNACA0012 airfoil with low pitching amplitude, aa=21, atRe=1.3�104. The reduced frequency was set to a relatively highrange, between 4.1 and 11.5, to generate thrust. They found thatthe transverse alignment of the vortices switched at k=5.7, i.e. thevortices of positive circulation switched from below to above thevortices of negative circulation. The mean streamwise velocityprofile herewith changed from velocity deficit (wake) to velocityexcess (jet). However, this switch from the vortex array orientationcould not be used to determine the crossover from drag to thrust.Von Ellenrieder and Pothos [142] conducted PIV measurementsbehind a 2-D plunging NACA0012 airfoil, operating at St between0.17 and 0.78, and Re=2.7�104. Their results showed that forStrouhal numbers larger than 0.43, the wake became deflectedsuch that the average velocity profile was asymmetric about themean heave position of the airfoil. Jones and Babinsky [143]studied the fluid dynamics associated with a three-dimensional2.5% thick waving flat plate. The spanwise velocity gradient andwing starting and stopping acceleration that exist on an insect-likeflapping wing are generated by rotational motion of a finite spanwing. The flow development around a waving wing at Re=6.0�104

was studied using high-speed PIV to capture the unsteady velocityfield. Vorticity field computations and a vortex identificationscheme reveal the structure of the 3-D flow-field, characterizedby strong leading edge and tip vortices. A transient high lift peakapproximately 1.5 times the quasi-steady value occurred in thefirst chord-length of travel, caused by the formation of a strongattached leading edge vortex. This vortex then separated from theleading edge resulting in a sharp drop in lift. As weaker leadingedge vortices continued to form and shed lift values recovered toan intermediate value. They also reported that the wing kinematicshad only a small effect on the aerodynamic forces produced by thewaving wing if the acceleration is sufficiently high. Calderon et al.[144] presented an experimental study on a plunging rectangularwing with aspect ratio of 4, at low Reynolds numbers of 1�104–3�104. Time-averaged force measurements were presented as afunction of non-dimensional frequency, alongside PIV measure-ments at the mid-span plane. In particular, they focused on theeffect of oscillations at low amplitudes and various angles of attack.The presence of multiple peaks in lift was identified for this 3-Dwing, thought to be related to the natural shedding frequency ofthe stationary wing. Wing/vortex and vortex/vortex interactionswere identified which may also contribute to the selection ofoptimal frequencies. Lift enhancement was observed to becomemore notable with increasing plunging amplitude, to lowerreduced frequencies, with increasing angle of attack. Despite thehighly 3-D nature of the flow, lift enhancements up to 180% werepossible.

Under Research and Technology Organization (RTO) arrange-ment of the North Atlantic Treaty Organization (NATO) there was


a community-wide effort organized, which offered a wide rangeof experimental and computational data for both SD7003 airfoiland flat plate, with kinematics promoting different degrees offlow separation. The detailed information can be found in [145].Here we present samples of the information collected in thisgroup endeavor. Ol et al. [146] compared PIV flow fieldmeasurements of a pitching and plunging SD7003 airfoil atRe=6.0�104, k=0.25, and St=0.08 with computed result by Kanget al. [147]. They considered two kinematic motions, a shallow-stall case and a deep-stall case where the maximum effectiveangle of attack was larger than the former case. In the shallow-stall case where the flow was moderately attached overall, thecomputed result was able to approximate the flow field measuredusing PIV well. For the deep-stall case the flow separated justbefore the middle of the downstroke (i.e. maximum effectiveangle of attack). The numerical solution showed vortical struc-tures similar to the PIV, but at a later phase of motion. However,the instantaneous lift over a motion cycle obtained from bothmethods compared well, indicating that the differences in detailsof the flow structures do not necessarily lead to large differencesin the forces integrated over the airfoil, as long as the large-scaleflow structures remain similar.

For Re(104) and higher, turbulence influences the developmentof the flow structures and forces. Ol et al. [145], Baik et al. [148]investigated the fluid physics at Re=O(104) of a pitching andplunging SD7003 airfoil and flat plate, experimentally using PIVfocusing on the second-order turbulence statistics. They observedlaminar boundary layer and laminar-to-turbulence transition. In acompanion paper Kang et al. [147] used RANS computations withSST turbulence model [149] to simulate the same cases [146] toinvestigate the implications of the turbulence modeling. Bylimiting the production of turbulence kinetic energy they observedthat leading edge separation was dependent on the level of eddyviscosity for the SD7003 airfoil, and hence turbulence, in the flow.Regarding the computed lift, they concluded that the large scalevortical structures in the flow were the contributing factors. Baiket al. [150] conducted an experimental study of a pitching andplunging flat plate at Re=1.0�104 constrained to motions enfor-cing a pure sinusoidal effective angle of attack. The effect of non-dimensional parameters governing pitching and plunging motionincluding Strouhal number (St), reduced frequency (k), and theplunge amplitude (ha) was investigated for the same effective angleof attack kinematics. The formation phase of the LEV was found tobe dependent on k: the LEV formation is delayed for higher k value.It was found that for cases with the same k the velocity profilesnormal to the airfoil surface closely follow each other in all casesindependent of pitch rate and pivot point effect. Of course, eventhough the flow structures with constant k seemed little affectedby Strouhal number and plunging amplitude, the time history offorces along the horizontal (thrust) and normal (lift) directions canbe substantially altered because the geometric angle-of-attack,viewed from the ground.

Visbal et al. [151] computed the unsteady transitional flowover a plunging 2-D and 3-D SD7003 airfoil with high reducedfrequency (k=3.93) and low plunge amplitude (ha/cm=0.05) usingimplicit large Eddy simulations at Re=1.0�104 and 4.0�104. Theresults showed that the generation of dynamic-stall-like vorticesnear the leading edge was promoted due to motion-induced highangles of attack and 3-D effects in vortex formation around thewing. Radespiel et al. [152] compared the flow field over a SD7003airfoil with and without plunging motion at Re=6.0�104. Using aNS solver along with the linear stability analysis to predicttransition from laminar-to-turbulent flow, they concluded thattransition and turbulence can play an important role in theunsteady fluid dynamics of flapping airfoils and wings at theinvestigated Reynolds numbers.

4.2. Single wing in hovering flight condition

Ellington and co-workers [57–60,62,103,104] did pioneeringresearch on flapping wing aerodynamics at Re=4.0�103–7.0�103. They built a scaled-up robotic hawkmoth wing modeland visualized the flow field of hovering hawkmoth wingmovements [57–60,62,103,104] using smoke visualization tech-niques. They observed that the presence of the LEV at high angleof attack during the downstroke, and suggested that ‘delayed stall’of the LEV was responsible for high lift production by hoveringhawkmoths (see Section 3.4). Dickinson and co-workers[21,22,37,50,51,72,153–159] made original contributions to theunderstanding the flapping wing aerodynamics at lower Reynoldsnumber regimes (Re=1.0�102–1.5�103). They utilized a dyna-mically scaled robotic fruit fly model wing in an oil tank andconducted systematic experiments to relate the prescribedsimplified fly-like kinematics to the resulting aerodynamic forces.They categorized the aerodynamic loading into three parts: forcesdue to (i) translation (delayed stall of the LEV [50,72,153–155],(ii) rotation [50,51,156–157], and (iii) interaction with the wakes(wake capture [50,55]). Furthermore, Fry et al. [158] investigatedthe aerodynamics of hovering/tethered fruit flies using a dyna-mically scaled robotic fruit fly wing model at Re=1.2�102.Altshuler et al. [159] studied the aerodynamics of a hoveringhoneybee using a dynamically scaled robotic honeybee wingmodel at Re=1.0�103. Their results showed that aerodynamicforce enhancement due to wake capturing and rotational forceswere important in both fruit fly and honeybee hovering. Sunadaet al. [160] measured fluid dynamic forces generated by a bristledwing model with four different wing kinematics using scale-uprobotic wings at Re=10. The results demonstrated that fluiddynamic forces acting on the bristled wing were a little smallerthan those on the solid wings.

Nagai et al. [161] used a mechanical bumblebee wing modeland measured the resulting forces with strain gauges and flowstructures using PIV for a hovering and forward flight atRe=O(103). The comparison between the experimental resultsand the numerical solutions, computed using a 3-D NS code,showed good agreement quantitatively in forces and qualitativelyin flow structures. For the forward flight the relevance of thedelayed stall mechanism depended on the advance ratio. Theyobserved that the LEV hardly appeared during upstroke at highadvance ratios (over 0.5).

Comparisons of 2-D computational simulations of an ellipticairfoil in hover against 3-D experimental data of the fruit flymodel [50] were performed by Wang et al. [162]. They concludedthat 2-D computed aerodynamic forces were good approxima-tions of 3-D experiments for the advanced and symmetricalrotation cases considered in their study. Lua et al. [163]experimentally investigated the aerodynamics of a plunging 2-Delliptic airfoil at Re between 6.6�102 and 2.7�103. The resultsshowed that the fluid inertia and the LEVs played dominant rolesin the aerodynamic force generation, and time-resolved forcecoefficients during plunging motion were found to be moresensitive to changes in pitching angular amplitude than toReynolds number. Wang [164–166] carried out 2-D numericalinvestigations on the vortex dynamics associated with a plunging/pitching elliptic airfoil at Re=O(102)–O(104). The result showed adownward jet of counter rotating vortices which were formedfrom LEVs and TEVs. Bos et al. [167] performed 2-D computationalstudies examining different hovering kinematics: simple harmo-nic, experimental model [50], realistic fruit fly [158], and modifiedfruit fly. The results showed that the realistic fruit fly kinematicslead to the optimal mean lift-to-drag ratio compared to otherkinematics. Also they concluded that in the case of realistic fruitfly wing kinematics, the angle of attack variation increases the


aerodynamic performance, whereas the deviation levels the forcesover the flapping cycle. Kurtulus et al. [168] obtained a flow fieldover a pitching/plunging NACA0012 airfoil in hover atRe=1.0�103 experimentally and numerically. A 2-D computationwas compared to a pitching and plunging airfoil in water tank.They found that the more energetic vortices which were the mostinfluential flow features on the resulting forces were visible.

Hong and Altman [169,170] investigated experimentally thelift generation from spanwise flow associated with a simpleflapping wing at Re between 6.0�103 and 1.5�104. The resultssuggested that the presence of streamwise vorticity in the vicinityof the wing tip contributed to the lift on a thin flat plate flappingwith zero pitching angle in quiescent air. Isaac et al. [171] usedboth experimental and numerical methods to investigate theunsteady flow features of a flapping wing at Re between 5.1�102

and 5.1�103. They showed a feasible application of the watertreading kinematics for hovering using insect/bird-like camberedwings.

4.3. Tandem wing in forward/hovering flight condition

Aerodynamics associated with dragonflies differs from othertwo-winged insects because forewing and hindwing interactionsgenerate distinct flow features [12]. Sun and Lan [172] studied thelift requirements for a hovering dragonfly using a 3-D NS solverwith overset grid methods. They showed that the interactionbetween the two wings was not strong and reduced the liftcompared to single wing configuration, however, large enough tostay aloft. Yamamoto and Isogai [173] conducted a study on theaerodynamics of a hovering dragonfly using a mechanical flappingapparatus with a tandem wing configuration and compared thetime history of forces obtained from a 3-D NS solver. The forcecomparison showed a good agreement and the results suggestedthat the phase difference between the flapping motions of thefore- and hind-wings only had small influence on the timeaveraged forces.

Lehmann [44] and Maybury and Lehmann [174] investigatedthe effect of changing the fore- and hindwing stroke-phaserelationship in hover on the aerodynamic performance of eachflapping wing by using a dynamically scaled electromechanicalinsect wing model at Re of approximately 1.0�102. Theymeasured the aerodynamic forces generated by the wings andvisualized flow fields around the wings using PIV. Their resultsshowed that wing phasing determined both mean force produc-tion and power expenditures for flight, in particular, hindwing liftproduction might be varied by a factor of two due to LEVdestruction and changes in the strength and the orientation of thelocal flow vector. Lu et al. [175] showed physical images revealingthe flow structures, their evolution, and their interactions duringdragonfly hovering using an electromechanical model in waterbased on the dye flow visualization. Their results showed adelayed development of the LEV in the translational motion of thewing. Furthermore, in most cases, forewing–hindwing interac-tions were detrimental to the LEVs and were weakened withincrease of the wing–root spacing. For a dragonfly in forwardflight, the conclusions from Wang and Sun [176] were similar inthat the forewing–hindwing interaction was detrimental for thelift, but sufficient to support its weight. They suggested thatthe downward-induced velocity from each wing would decreasethe lift on other wings.

Dong and Liang [177] modeled dragonfly in slow flight byvarying the phase difference between the forewing and hindwingand investigated the changes of aerodynamic performance ofhindwings. They found that the performance of forewings is notaffected by the existence of hindwings; however, hindwings have

obvious thrust enhancement and lift reduction due to theexistence of forewings. For slow flight, by decreasing the phaseangle difference, hindwings will have larger thrust production,slight reduction of lift production, and larger oscillation of forceproduction. Independently, Warkentin and DeLaurier [178] did aseries of wind-tunnel tests on an ornithopter configurationconsisting of two sets of symmetrically flapping wings ofbatten-stiffened membrane structures, located one behind theother in tandem. It was discovered that the tandem arrangementcan give thrust and efficiency increases over a single set offlapping wings for certain relative phase angles and longitudinalspacing between the wing sets. In particular, close spacing on theorder of 1 chord length is generally best, and phase angles ofapproximately 07501 give the highest thrusts and propulsiveefficiencies. Again, referring to other reported studies involvingrigid and flexible wings, the aerodynamics and aeroelasticityassociated with wing–wing interactions need to be furtherstudied. In another study, Broering et al. [179] numericallystudied the aerodynamics of two flapping airfoils in tandemconfiguration in forward flight at a Reynolds number of 104. Therelationship between the phase angle and force production wasstudied over a range of Strouhal numbers and three differentphase angles, 01, 901 and 1801. In general, they found that the lift,thrust and resultant force of the forewing increased compared tothose of the single wing. The lift and resultant force of thehindwing was decreased, while the thrust was increased for the 0phase hindwing and decreased for the 90 and 180 phasehindwings. The lift, thrust and resultant force of the combinedfore and hindwings was also compared to the case of two isolatedsingle wings. In general, the 0 phase case did not noticeablychange the magnitude of the resultant force, but it inclined theresultant forward due to the decreased lift and increased thrust.The 90 and 180 phase cases significantly decreased the resultantforce as well as the lift and thrust. Clearly, more work is needed tohelp unify our understanding of the wing–wing interactions asfunction of the individual and relative kinematics and dimension-less flow and structure parameters.

Wang and Russell [180] investigated the role of phase lagbetween the forewing and the hindwing further by filming atethered dragonfly and computing the aerodynamic forces andpower. They found that the out-of-phase motion in hovering usesalmost minimal power to generate sufficient lift to stay aloft andthe in-phase motion produce additional force to accelerate intakeoffs. Young et al. [181] investigated aerodynamics of theflapping hindwing of Aeschana juncea dragonfly using 3-Dcomputations at Re between 1.0�102 and 5.0�104. The flappingamplitude observed, 34.51, for the dragonfly maximized the ratioof mean vertical force produced to power required. Zhang and Lu[182] studied a dragonfly gliding and asserted that the forewing–hindwing interaction improved the aerodynamic performance forRe=O(102)–O(103).

4.4. Implications of wing geometry

Lentink and Gerritsma [183] considered different airfoil shapesnumerically to investigate the role of shapes in forward flight onthe aerodynamic performance. They computed flow aroundplunging airfoils at Re of O(102) and concluded that the thinairfoil with aft camber outperformed other airfoils including themore conventional airfoil shapes with thick and blunt leadingedges. One exception was the plunging N0010 which due to itshighest frontal area had good performance. Usherwood andEllington [184] examined experimentally the effect of detailedshapes of a revolving wing with planform based on hawkmothwings at Re=5.0�103. The results showed that detailed leading


edge shapes, twist, and camber did not have substantial influenceon the aerodynamic performance. In a companion paper Usher-wood and Ellington [185] examined experimentally the effect ofaspect ratio of a revolving wing with the same hawkmothplanform [184] adjusted to aspect ratios ranging from 4.53 to15.84 with corresponding Re of 1.1�103–2.6�104. The resultsshowed the influence of the aspect ratio was relatively minor,especially at angle of attack below 501. Luo and Sun [186]investigated numerically the effects of corrugation and wingplanform (shape and aspect ratio) on the aerodynamic forceproduction of model insect wings in sweeping motion atRe=2.0�102 and 3.5�103 at angle of attack of 401. The resultsshowed that the variation of the wing shape almost unaffectedthe force generation and the effect of aspect ratio was alsoremarkably small. Moreover, the effects of corrugated wingsections in forward flight were studied in numerical simulations[187,188] and a numerical–experimental approach [189]. Theresults demonstrated that the pleated airfoil produced compar-able and at times higher lift than the profiled airfoil, with a dragcomparable to that of its profiled counterpart [187].

Altshuler et al. [190] tested experimentally the effect ofwing shape (i.e. with sharpened leading edges and withsubstantial camber) of a revolving wing at Re between 5.0�103

and 2.0�104. Their results demonstrated that lift tended toincrease as wing models become more realistic as did the lift-to-drag ratios Ansari et al. [191] used an inviscid model for hoveringflapping wings to show that increasing the aspect ratio, winglength, and wing area enhances lift. Furthermore, they suggestedthat for a flapping wing MAV, the best design configuration wouldhave high aspect ratio, straight leading edge, and large wing areaoutboard. The pitching axis would be then located near the centerof the area in the chordwise direction to provide the bestcompromise for shedding vortices from the leading and trailingedges during the stroke reversal. Green and Smith [192] examinedexperimentally the effect of aspect ratio and pitching amplitude ofa pitching flat plate in forward flight at Re between 3.5�103 and4.3�104 and aspect ratios of 0.54 and 2.25. They measuredunsteady pressure distributions on the wing, and compared to thePIV measurements of the same setup [193]. They concluded thatthe 3-D effects increased with decreasing aspect ratio, or whenthe pitching amplitude increased.

Kang et al. [147] investigated the airfoil shape effect atRe=O(104) by comparing the flow field around pitching andplunging SD7003 airfoil and flat plate using PIV [148], and CFD inforward flight. It was observed that for the flat plate the flow wasnot able to make turn around the sharper leading edge of the flatplate and eventually separated at all phases of motion. The flowseparation led to larger vortical structures on the suction side ofthe flat plate hence increasing the area of lower pressuredistribution on the flat plate surface. From the time history oflift, available for the CFD, it was seen that for the flat plate thesevortical structures increased the lift generation compared to theSD7003, which had a blunter leading edge.

4.5. Implications of wing kinematics

A primary driver of the unsteady aerodynamics in flappingwing flight is the wing motions. Yates [194] suggested that thechoice of the position of the pitching axis may enhanceperformance and control of rapid maneuvers and thus enablethe organism to more adeptly cope with turbulent environmentalconditions, to avoid danger, or to more easily capture food. Theinstantaneous fluid forces, torques, and rate of work done by thepropulsive appendages were computed using 2-D unsteadyaerodynamic theories. For a prescribed motion in forward flight,

the moment and power were further analyzed to find the axes, forwhich the mean square moment was minimal, the mean powerto maintain the moment was zero, the mean square power tomaintain moment was a minimum, and the mean square power tomaintain lift equaled the mean square power. Sane and Dickinson[51] investigated the effect of location of pitching axis on forcegeneration of the flapping fruit fly-like wing in the first strokeusing a scaled-up robotic wing model. They estimated therotational forces based on the quasi-steady treatment and bladeelement theory. The results showed that rotational forcesdecrease uniformly as the axis of rotation moves from the leadingedge towards the trailing edge and change sign at approximatelythree-fourths of a chord length from the leading edge of thewing [50].

Ansari et al. [195] studied the effects of wing kinematics on theaerodynamic performances of insect-like flapping wings in hoverbased on non-linear unsteady aerodynamic models. They foundthat the lift and the drag increased with increasing flappingfrequency, stroke amplitude, and advanced wing rotation. How-ever, such increases were limited by practical considerations.Furthermore, the authors mentioned that variations in wingkinematics were more difficult to implement mechanically thanvariations in wing planform. Hsieh et al. [196] investigated theaerodynamics associated with the advanced, delayed, and sym-metric rotation and decomposed the lift coefficients in terms ofthe lift caused by vorticity, wing velocity, and wing acceleration.The results suggested that while the symmetric rotation had themost lift due to vorticity on the surface of the wing and in theflow, the maximum total lift is found with advanced rotation.Oyama et al. [197] optimized for the mean lift, mean drag, andmean required power generated by a pitching/plungingNACA0012 airfoil in forward flight using a 2-D NS solver atRe=103. The multi-objective evolutionary algorithm was used tofind the pareto front of the objective functions (i.e. by consideringthe reduced frequency, plunge amplitude, pitch amplitude, pitchoffset, and phase shift as the design variables). They found thatthe pitching angle amplitude (between 351 and 451) was optimumfor high-performance flapping motion and the phase anglebetween pitch and plunge of about 901. In addition, the reducedfrequency was a tradeoff parameter between minimization ofrequired power and maximization of lift or thrust where smallerfrequency leads to smaller required power.

4.6. Surrogate modeling for hovering wing aerodynamics

Numerous studies, including Wang et al. [162] and referencescited above, reported similarities as well as differences inaerodynamics and flow structures between 2-D and 3-D flappingwings. There is a need for establishing a comprehensive frame-work to address these matters. From a vehicle developmentperspective, since 3-D Navier–Stokes simulations are expensive torun, if 2-D simplifications can adequately reproduce the mainaerodynamic features of 3-D flapping wing, naturally, the neededdata can be generated much more economically. Trizila et al.[198,199] used surrogate modeling techniques [200–202] toinvestigate a large number of hovering wing cases to developsurrogate models for time averaged lift and thrust. They identifiedregions where 2-D and 3-D results (time averaged lift and thrust)were comparable, as well as those that were substantiallydifferent. Furthermore, based on the guidance from the surrogatemodels, they probed the fluid physics associated with 2-D and3-D cases, and were able to highlight the roles played by the LEVand TiVs in unsteady aerodynamics. The impact on lift fromthe 2-D/3-D unsteady mechanisms is detailed further in thesubsequent sections.


The flapping wing scenario is simplified to better identify andunderstand the competing interactions and influences. The wingis modeled as a flat plate with 2% thickness with respect to thechord; for the finite wing, the aspect ratio is 4. The mid-spancross-section of the 3-D model is used for the 2-D simulation.Simplified wing kinematics are employed (see Section 2.1) andthe mid-chord of the rigid airfoil is selected as the pitching axis. Itwas shown the position of the axis about which the wing pitcheswill influence the aerodynamics [50, 51, 194, 195, 197] andSection 4.5. The results shown will not be independent of thischoice of rotation (which is beyond the scope of the studypresented), however, the understanding of the most prominentfeatures will be relevant to flapping flight in general. The threekinematic parameters (or the design variables in surrogatemodeling), ha (plunge amplitude), aa (pitch amplitude), and f(phase lag between pitching and plunging motion) can be variedindependently.

The range of the design variables is as follows. The normalizedstroke amplitude is representative of a range of flapping wingflyers: 2:0o2ha=cmrg o4:0. Details on pitch amplitude and phaselag are not as plentiful in the literature, so cases are chosen withlow angles of attack (AoAmin=101; high pitching amplitude) andhigh angles of attack (AoAmin=451; low pitching amplitude):451oaa.o801. The bounds on phase lag are chosen symmetricabout the synchronized hovering: 601ofo1201 Although de-layed rotation is not a focus of many studies found in literature,due to its reduced ability to generate lift compared to its normal

Fig. 10. Effect of TiVs on lift generation associated with a pitching/plunging flat plate

(2ha=cmrg ¼ 2:0, aa=451, and f=601); (B) Synchronized rotation case (2ha=cmrg ¼ 3:0, aa

results from the three- and two-dimensional wings, respectively.

hovering and advanced rotation brethren in 2-D flow, it will beshown that the interaction of a TiV with a LEV for a delayedrotation case can exhibit prominent 3-D effects beneficial to thewing’s lift performance.

A Navier–Stokes solver [203] is used to obtain the time-dependent flow solution. Due to the kinematic constraints thereare only two relevant non-dimensional groups in the incompres-sible case. The plunging amplitude to chord ratio, 2ha=cmrg , andthe Reynolds number: since Re is being held constant (for a fixedmedium) the plunging amplitude and flapping frequency are notindependent. The Reynolds number, based on the average tipvelocity, is equal to 65, which is similar to the fruit fly, Drosophila

melanogaster. The reduced frequency, k, contains the sameinformation as the normalized stroke amplitude (see Section 2.1).

The surrogate modeling tools [200–202], trained with CFD data(order of days to obtain the results for one training point), areused to efficiently evaluate off design points (in a fraction of asecond), global trends across the design space, kinematicvariables’ sensitivity, and tradeoffs between lift and power. Thequantities of interest, the objective functions in the surrogatemodeling nomenclature, are mean lift coefficient, CL, and anapproximation of the power required over the stroke cycle. Formore details on the implementation and error quantification thereader is referred to Refs. [198,199] (Fig. 10).

Fig. 11 shows iso-surfaces of the mean lift coefficient whereeach axis corresponds to one of the kinematic parameters, i.e. ha,aa, or f. Fig. 11 (A), (B), and (C) correspond to the 2-D lift, 3-D lift,

(AR=4) at the middle of stroke adopted from Ref. [54]. (A) Delayed rotation case

=801, and f=901). Note that solid and dashed lines in (A-2) and (B-2) indicate the

Fig. 11. Iso-surfaces of the time averaged lift achieved over the entire design space of kinematic combinations evaluated: (A) 2-D model; (B) 3-D model; and (C) the

difference between the two models. Note the magnitude of the discrepancy being a noticeable fraction of the values of lift. The blue region in (C) denotes kinematics for

which the 2-D lift is higher than the equivalent 3-D kinematics. Likewise the yellow/red region denotes higher 3-D mean lift. (For interpretation of the references to color in

this figure legend, the reader is referred to the web version of this article.)


and difference between the two, respectively. Observations thatare immediately apparent are that kinematic combinations withlow aa (high AoA) and advanced rotation (high f) have thehighest mean lift in 2-D and 3-D. This result qualitatively agreeswith the results of Wang et al. [162] and Sane and Dickinson [51].Due to the non-monotonic response in phase lag found in the 2-Dlift response at high pitching amplitudes (see Fig. 11 (A)), thereare two regions where there is low, and possibly negative, meanlift generation. The first region is defined by low plungeamplitudes (low 2ha=cmrg ), low AoA (high aa), and advancedrotation (high f). The second region is defined by low AoA (highaa), and delayed rotation (low f) and is where the 3-D kinematicsalso generate low mean lift values.

A variable’s sensitivity is directly related to the gradients alongthe respective design variables, while a more quantitativemeasure is the global sensitivity analysis; these measures areexamined in Trizila et al. [198,199]. It is seen that the gradientsalong the aa and f axes are much more significant than that alongthe normalized stroke amplitude (2ha=cmrg ). (Note: Lua et al. [163]

find the effect of Re is noticeably smaller than that of aa on themean lift.) With these observations, the trends seen as a variableis varied are more clearly illuminated, but also the associatedlimitations are just as apparent. For example, advancing the phaselag is beneficial in 2-D except when at high aa; in 3-D there is nosuch exception within the bounds studied. Comparisons withSane and Dickinson’s experiments [51] show qualitative agree-ment in the trends in mean lift as a function of aa and f within thecommon ranges, with the noticeable difference in setups beingthe current study using pure translation to represent the plungewhereas the experimental study flaps about a pivot point.

The difference between 2-D and 3-D lift may raise the questionabout areas of the design space for which 2-D computations maysufficiently approximate their analogous 3-D counterparts orwhere there are substantial 3-D effects which would precludesuch a comparison. In order to illustrate comparable 2-D and 3-Dcases, a training point for which the full CFD solution wassimulated was chosen from the region showing similar mean lift.Fig. 12 presents the time history of lift for a synchronized rotation

Fig. 12. The 2-D (solid: red) and 3-D (dashed: black) lift coefficients for the

kinematic parameters, 2ha=cmrg ¼ 3:0;aa ¼ 801;f¼ 901. This case is synchronized

rotation and representative for the cases that show very good agreement between

the 2-D and the 3-D (AR=4) calculations in the time averaged as well as the

instantaneous sense. (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article.)


case with generally low AoA (2ha=cmrg ¼ 3:0;aa ¼ 801;f¼ 901).The figure shows not only good agreement for this case in themean or time averaged sense, which can be deduced from thesurrogate models, but also instantaneously. Fig. 13 shows littlevariation in the spanwise vortex dynamics encountered by eachwing cross-section. This is accomplished by way of an iso-surfaceof the quantity Q, a measure of rotation which separates out theshear and is defined in [204], colored by the spanwise vorticity. Inthis configuration the LEV will be red/yellow, the TEV blue, andthe TiV green. Next to the flow field shots are plots of thespanwise lift at the selected time instances which also showdecent agreement between 2-D and 3-D. The variation that ispresent is generally confined locally to the tips, though themagnitude is small. The TiVs are most prominent at the end oftranslation, but as seen from the spanwise distribution of lift, arelimited in influence. Note that other 2-D and 3-D cases agreed inthe mean sense, but instantaneously differed and this situation isexplored further in Trizila et al. [198,199].

In contrast to the case illustrated above, another set ofkinematics chosen for which the 2-D and 3-D surrogate modelsdid not exhibit such agreement. Fig. 14 illustrates a delayedrotation case with high AoA (2ha=cmrg ¼ 2:0;aa ¼ 451;f¼ 601),where the 2-D and 3-D cases show noticeable differences in theinstantaneous lift. The source of these differences is betterillustrated in Fig. 15. The spanwise variation in the vortexdynamics is seen in the plots of the flow field as well as in thelift distribution across the span of the wing. Once again the iso-surface connected to the wing can be distinguished by the LEV(red and yellow), the TEV (blue) and the TiV (green). The TiV hasnoticeable impact on the resulting flow features and aerodynamicloading. First, the distribution of lift due to pressure shows a localpeak at the tips which is a force enhancement over the 2-Dcounterpart. There also appears to be an LEV anchoringmechanism, which keeps the LEV attached over a portion of thespan near the wing tip thereby increasing the lift farther inboard.The net effect of the tip vortices is a substantial increase inperformance for this set of kinematics.

In Trizila et al. [198,199] the design space was morethoroughly explored. A conclusion that is not apparent from thesteady-state aerodynamics is that TiVs can be utilized to increaseperformance in the context of low Re flapping wing flight. Due tothe competing effects the TiV introduce, the role the TiV plays canvary from case to case (i.e. its mere presence does not guarantee aperformance increase). The TiV can introduce a low-pressureregion on the upper wing surface and anchor the LEV, whichwould have otherwise been shed. On the other hand, it can alsoinduce downwash thereby reducing the effective angle of attackand consequent lift.

4.7. Unsteady flow structures around hawkmoth-like model in hover

There are a number of computational studies of realisticwing configurations of a hornet [205], a bumblebee [206,207], ahawkmoth [42,61–64,79,208], a honeybee [42], a drone fly [209],a hover fly [210], a fruit fly [42,52,80,81,211,212], and a thrips[42] accompanied by appropriate wing kinematics. Moreover,references cited throughout the literature review have elucidatedaspects of the unsteady vortical flow phenomena by way ofdynamically scaled robotic wings and biological flyers. In order toexpand our knowledge about the LEV mechanisms and unsteady3-D flow features associated with a biological flyer-like flappingmodel, a numerical study on the aerodynamics and vortexdynamics around a realistic hawkmoth model with complexflapping wing motions is discussed and highlighted in thissubsection.

Figs. 16 and 17 show the morphological and wing kinematicsmodels of a realistic hawkmoth model. Computations areperformed using ‘‘a biology-inspired dynamic flight simulator[81,213]’’. This framework is capable of simulating an insect withrealistic wing–body morphologies and flapping-wing/bodykinematics. Detailed descriptions of the computational modelingcan be referred to in Refs. [64,79,208,213].

4.7.1. Vortex dynamics of hovering hawkmoth

Iso-vorticity-magnitude surfaces around a hovering hawkmothat nine-time instants in a flapping cycle are shown in Fig. 18. Thecontour color on the iso-vorticity-magnitude surfaces in Fig. 18indicates the magnitude of the normalized helicity density, whichis computed by projecting the spin vector of a fluid element ontoits momentum vector, being positive (red) if these two vectorspoint in the same direction and negative (blue) if they point to theopposite direction. This quantity can be useful for illustratinghelical vortex structures. One aim of the study is to correlate thetime histories of the aerodynamic loadings with the dominantflow features [214]. As such, the time histories of the aerodynamicforces on the wings and body are plotted in Fig. 19: (A) vertical(lift) force, and (B) horizontal (drag or thrust) force, respectively.

4.7.1.1. Flow structures during downstroke. In the first half of thedownstroke (see Fig. 17), Fig. 18 A-1, a horseshoe-shaped vortex isgenerated by the initial wing motion of downstroke. Poelma et al.[72] showed a similar three-dimensional flow structure around animpulsively started dynamically scaled flapping wing using PIV.The horseshoe-shaped vortex is composed of three vortices,namely, a LEV, a TEV, and a TiV, and it grows in size as thetranslational and angular velocity of the wing increases. In par-ticular, the radii of the LEVs and the TEVs expand from the wingroot to the wing tip, resulting in a 3-D vortex structure. Thesevortices produce a low-pressure region in their core and on theupper surfaces of the wing (Fig. 20 (1)) when they are attached.Lift forces shows a peak at the corresponding time instant (Fig. 19(A) and Fig. 17 (b)). Therefore, this would imply that the LEVs, the

Fig. 13. The lift per unit span (right column) and iso-Q surface (Q=0.75) (left column) colored by spanwise vorticity magnitude for flow fields over half the wing associated

with the kinematic parameters, 2ha=cmrg ¼ 3:0;aa ¼ 801;f¼ 901 at t/T=0.05, 0.25, and 0.45. Note that solid and dashed lines indicate the results from the three- and two-

dimensional wings, respectively. The spanwise variation in forces is examined with the 2-D equivalent (redline) marked for reference. The spanwise variation is limited.

(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


TEVs, and the TiVs enhance the lift force generation in hoveringflight of the hawkmoth. It is seen that the TEVs generated by theright and left wings meet at the rear body (Fig. 18 A-2), resulting

in an interaction with each other. These vortex–vortex andvortex–body interactions lead to more complex vortical featuresaround the hovering hawkmoth than the 2-D flow structures seen

Fig. 14. The 2-D (solid: red) and 3-D (dashed: black) lift coefficients for the

delayed rotation and high AoA case governed by the kinematic parameters,

2ha=cmrg ¼ 2:0;aa ¼ 451;f¼ 601. Here the difference between 2-D and 3-D (AR=4)

calculations is significant during translational phase of the wing motion. (For

interpretation of the references to color in this figure legend, the reader is referred

to the web version of this article.)


around pitching/plunging airfoils. During the mid-downstroke(see Fig. 17 (c)), the TEVs are mostly shed from the wings whilethe TEVs stay attached on the body. Moreover, the shed TEVs stayconnected to the TiVs (Fig. 18 A-3). Of the three vortex structures,the LEVs produce the largest and strongest area of low pressure onthe wing surface (Fig. 20 (2)). Immediately after that, the LEVsbegin to break down at a location approximately 70–80% the spanof the wing length. The LEV, the TiV, and the shed TEV togetherform a doughnut-shaped vortex ring around each wing (Fig. 18 A-4). Recent experimental visualization studies showed similarvortex ring structures around a hovering hummingbird [77],during slow forward flights of a bat [99,100], and a free-flyingbumble bee [92]. Looking at the structure of the doughnut-shapedvortex ring close to the wing tip, the vortical structure is twisted(rolled-up) behind each wing, which results from an interactionbetween the broken-down LEV and the shed TV (Fig. 18 A-4). Thishelps to illustrate the rich complexity of 3-D interactionsexperienced in flapping wing flight.

Continuing on, during the second half of the downstroke(see Fig. 17 (d)), the TiVs enlarge gradually. Eventually, when thewings approach the end of the downstroke, the LEVs and the TiVsweaken and begin to detach from the wings. An additional vortexis observed around the wing root and connects with the shed TEV(Fig. 18 A-5). The doughnut-shaped vortex rings of the wing pairbreak up into two circular vortex rings, eventually forming thefar-field wake below the hawkmoth. Two small vortex rings of thewing pair near the wing root are observed to join the two circularvortex rings (Fig. 18 A-6). The weakening of the low-pressure areaon the wing surface due to separation of the vortices leads to asmaller aerodynamic force acting on the wing and the high angleof attack of the wing leads to high thrust but low lift generation(Fig. 20 (3)). During most of the downstroke, the doughnut-shaped vortex ring pair has an intense, downward jet-flowthrough the ‘‘doughnut’’ hole, which forms the downstrokedownwash.

In the first half of the supination (see Fig. 17 (e)), occurringnear the end of the downstroke, as the flapping wing slows down,the attached vortices (the LEVs and the TiVs) are shed from the

wings. At this time instant, a pair of downstroke stopping vorticesis observed wrapping around the two wings (Fig. 18 B-1). Whenthe flapping wings begin to pitch quickly about the spanwise axis,a pair of upstroke starting vortices is detected around the wing tipand the trailing edge (Fig. 18 B-2).

4.7.1.2. Flow structures during upstroke. In the second half of thesupination (Fig. 17 (f)), TEVs and TiVs associated with the be-ginning of the upstroke are generated when the flapping wingsaccelerate rapidly. The downstroke wakes of the circular vortexrings are subsequently captured (Fig. 18 B-3), but a correlationwith the time history of lift shows a relatively minor impact. Theidea of wake capture has been documented many times[1,36,43,44,50,55,56] and Section 3.4. The point here is that therole it plays in the lift generation can vary depending on thecontext. Dickinson et al. [50] experimented with 3-D flappingkinematics at lower Re, i.e. O(102), which showed a significantcontribution to lift from the wake capturing. In contrast, in thepresent study, the hovering hawkmoth-like wing kinematics at aRe of 6.3�103 do not show a prominent wake capturing me-chanism in the lift histories. Another study [81] looking at fruit flywing motions at Re O(102) also demonstrated a situation wherethe wake capturing had little impact on the aerodynamic loading.While still in the first half of the upstroke (Fig. 17 (g)), when theflapping wings begin to accelerate upward, the TEVs and TiVs areshed from the two wings (Fig. 18 C-1). The TiVs are continuallyregenerated and fed while the LEV generation is initiated. To-gether with the TEVs, the LEVs and TiVs form a horseshoe-shapedvortex pair wrapping each wing. Similar to what is seen duringthe downstroke, the horseshoe-shaped vortex grows and evolvesinto a doughnut-shaped vortex ring for each wing. Root vorticesare also detected at this stage and join the vortex ring as illu-strated in Fig. 18 C-2. The difficulty in clearly demarcating andexpressing the relevant phenomena is a direct consequence of the3-D nature of the vortex structures and their subsequent inter-actions. Note that due to asymmetric variation of angle of attackbetween the upstroke and downstroke, the LEVs generated in theupstrokes are smaller than those in the downstrokes. In the sec-ond half of the upstroke (Fig. 17 (h)), the doughnut-shaped vortexrings elongate and deform while maintaining their ring-like shape(Fig. 18 C-3). During most of the upstroke, the downwash inducedthrough the hole of each vortex ring is observed to be similar tothat of the downstroke, despite the asymmetric forward andbackstroke kinematics.

During early pronation (Fig. 17 (i)) at the end of the upstroke,the attachment points of the shed TiV move slightly (approxi-mately 10–20% of the wing length) from the wing tip to the wingroot during the course of the upstroke due to the wing movement(Fig. 18 D-1). Concurrently, the stopping vortices are observedwrapping around each wing (Fig. 18 D-1). As the wings begin torotate quickly, the upstroke stopping vortices are shed from thetrailing edge and the downstroke starting vortices are detected atthe leading edge and wing tip. Thereafter, the upstroke doughnut-shaped vortex rings are shed downward, breaking up into twovortex wake rings; the root vortices are also shed, forming a singlevortex ring under the body (Fig. 18 D-2). During late pronation(Fig. 17 (a)), at the end of the upstroke and beginning of thedownstroke, starting vortices are observed around the leading andtrailing edges and wing tip (Fig. 18 D-3).

4.7.1.3. Flight energetics. As to the aerodynamic force generation,two peaks in the lift force are predicted during each stroke for ahovering hawkmoth (see Fig. 19). Considering the correlationbetween the aerodynamic forces and the key unsteady flow fea-tures associated with flapping wings discussed in Section 3, the

Fig. 15. The lift per unit span (right column) and iso-Q surfaces (Q=0.75) (left column) colored by spanwise vorticity over half of the wing using the kinematic parameters,

2ha=cmrg ¼ 2:0;aa ¼ 451;f¼ 601 at t/T=0.05, 0.25 and 0.45. Note that solid and dashed lines in (A-2) and (B-2) indicate the results from the three- and two-dimensional

wings, respectively. The spanwise variation in forces is examined with the 2-D equivalent (redline) marked for reference. The TiV leads to increased lift in their immediate

region as well as anchor the vortex shed from the leading edge. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of

this article.)


delayed stall of the LEV and contributions from the TEV and TiVare responsible for the first lift peak. The second peak is likely tobe associated with a contribution from the rapid increase invorticity [1,52] as the wing experiences a fast pitching motion.Relevant to estimates of energy consumption in natural flyers andcontrol applications in MAV design/construction, the computedtime histories of the aerodynamic torques are plotted in Fig. 21(A) rolling, (B) pitching and (C) yawing, respectively. Note that theterm ‘total wing’ in Fig. 21 is the sum of the torques of the right

and left wings. Two of the aerodynamic torques are relativelycomparable, the aerodynamic rolling (ART) and yawing (AYT)torques of the two wings (‘total wing’) are much smaller thanthose of the aerodynamic pitching torque (APT). The time-varyingtotal ART and total AYT are mostly zero over a flapping cyclebecause of the symmetrical flapping motion of the left and rightwings and therefore the averaged total ART and AYT becomeapproximately zero (Fig. 21 (A) and (C)). On the other hand, theAPT of the two wings varies over a range, from �0.6�10�3 to

A real hawkmoth A morphological hawkmoth model

Leading-edge

Wing tip

Trailing-

Fig. 16. A morphological model of a hawkmoth, Agrius convolvuli with a

computational model superimposed on the right half. The hawkmoth has a body

length of 5.0 cm, a wing length of 5.05 cm (mean wing chord length cm=1.83 cm),

and an aspect ratio of 5.52.

-90

-45

0

45

90

Ang

les (

degr

ee)

0 0.5 1.0 t/T

(a)

(b)

(c)

(d)(e)

(g)

(h)

(i)

(f)

Positional angle Elevation angle Fethering angle

Fig. 17. Schematic diagrams of the computational system of a hawkmoth, Agrius

convolvuli. Instantaneous positional angle f, feathering angle a, and elevation

angle y of the hawkmoth wing over one complete flapping cycle. Solid, dotted, and

dashed lines represent the positional angle, the feathering angle, and the elevation

angle, respectively. White circles marked: (a) late pronation, (b) early downstroke,

(c) mid downstroke, (d) late downstroke, (e) early supination, (f) late supination,

(g) early upstroke, (h) late upstroke and (i) early pronation. T denotes

dimensionless period of one flapping cycle.


0.4�10�3 Nm in a flapping cycle. Note that a consequence ofnegative APT pulls the body orientation of the insect nose-downwhile a positive APT is associated with nose-up. The transient APTof the two wings has four local maxima, at early/late pronation/supination. In short, the prediction of the APT of the wings showsthat the body may experience a nose-down pitching torqueduring: peak (1) late pronation during the first half of thedownstroke, peak (2) the start of supination during the latedownstroke, and peak (4) the start of pronation during theupstroke. Based on the computed transient aerodynamic forcesand the velocity of the flapping wings, the muscle-mass-specificaerodynamic power Paero can be calculated:

Paero ¼FaeroUVwing

Mmð17Þ

where the mass of flight muscle, Mm is assumed to contribute upto 23% of the total body mass of the hawkmoth (1.6 g), Faero is theaerodynamic force, and Vwing is the velocity vector of the flappingwing. The power necessary to overcome the air resistance isdenoted Paero. The time histories of Paero for the wings are plottedin Fig. 22. The maxima of the Paero coincide with the peaksexperienced in the transient aerodynamic forces. The computed

mean Paero (87.2 W kg�1) is very close to the experimentalestimation by Willmott and Ellington [104].

4.8. Effect of the Reynolds number on the LEV structure and

spanwise flow

As discussed previously, the lift enhancement due to thedelayed stall of the LEV is important in flapping wing flight[1,50,57] and Section 3.4. The formation of the LEV depends onthe wing kinematics, the details of wing geometry, and theReynolds number [1,42,65,155,213]. Liu and Aono [42] investi-gated the interaction between LEV, TiV, and vortex ringstructures. Tang et al. [215] investigated numerically the effectsof Reynolds number on 2-D hovering airfoil aerodynamics atRe=7.5�101–1.7�103. They showed that in low Reynoldsnumber regimes, O(102), the viscosity dissipated the vortexstructures quickly and led to essentially symmetric flow struc-tures and aerodynamic forces between the forward and backwardstroke while at higher Reynolds numbers, the history effect wasinfluential, resulting in distinctly asymmetric phenomena be-tween strokes. In order to examine the Re effect on LEV structuresand spanwise flow for realistic wing body configurations withappropriate kinematic motions, a numerical investigation ispresented next: realistic models of hawkmoth (Re=6.3�103,k=0.30), honeybee (Re=1.1�103, k=0.24), fruitfly (Re=1.3�102,k=0.21), and thrips (Re=1.2�101, k=0.25) in hover consideringdifferent representative kinematic parameters (flap amplitude,flap frequency, and type of prescribed actuation) and dimension-less numbers (Reynolds number, reduced frequency) in each case[42]. The morphological and kinematical model of a hawkmoth isshown in Figs. 16 and 17. For other insect models, similarinformation and computational models can be obtained inRefs. [42,213].

Fig. 23 shows the computed velocity vector distributions on anend-view plane at 60% semi-span for these four insects. It may benoted that the LEV structure and the spanwise flow in thehawkmoth and the fruitfly cases (see Fig. 23 (A) and (C)) are ingood qualitative agreement with the corresponding experimentalresults reported in Ref. [155]. For the thrips flight (Re=1.2�101,k=0.25), the LEV forms upstream of the leading edge andspanwise flow is weakest among all cases. For the fruit fly case(Re=1.3�102, k=0.21), the structure of the LEV is smaller than thehawkmoth and the honeybee cases. The fruit fly LEV is tube-likeand ordered, and spanwise flow is observed around the upperregion of trailing edge. The hawkmoth (Re=6.3�103, k=0.30) andhoneybee (Re=1.1�103, k=0.24) cases yield much morepronounced spanwise flow inside the LEV and upper surface ofthe wing, which together with the LEV forms a helical flowstructure near the leading edge (see Figs. 23 (A–1) and (B–1)).Fig. 24 shows the spanwise pressure-gradient contours on thewing of four typical insects during the middle of the downstroke.Compared to the hawkmoth and the honeybee, even though thewing kinematics and the wing–body geometries are different,fruit flies, at a Re of 1.0�102–2.5�102, cannot create as steeppressure-gradients at the vortex core; nevertheless, they seem tobe able to maintain a stable LEV during most of the down andupstroke. While the LEVs on both wings of hawkmoth andhoneybee experience a break-down near the middle of thedownstroke, the LEV on the fruit fly’s wing remains attachedduring the entire each stroke, eventually breaking down duringthe subsequent supination or pronation [81].

To illustrate the Reynolds number effect, ceteris paribus,numerical results were obtained for a single wing–body geometryand kinematics combination [65]. Qualitative similar flowstructures around the flyer were observed. This implied that even


though each flapping flyer had a unique wing and bodykinematics and morphologies, the Reynolds number was thedominant parameter dictating the LEV structure and the spanwiseflow.

The helicopter blade model has been used to help explain theflapping wing aerodynamics; however, spanwise axial flows aregenerally considered playing a minor role in influencing thehelicopter aerodynamics [216,217]. In particular, the helicopterblades operate at a substantially higher Reynolds number andlower angle of attack. The much larger aspect ratio of a blade alsomakes the LEV harder to anchor. These are key differencesbetween the helicopter blades and the typical biological wings.Usherwood and Ellington [184] investigated the aerodynamicperformance associated with ‘propeller-like’ rotation of thehawkmoth wing model (Re=O(103)). Their results reported thatrevolving models produced high lift and drag forces because ofthe presence of the LEV. Knowles et al. [218] performedcomputational studies involving with translating 2-D and rotating3-D flat plate models at Re=5.0�102. Their results presented, thatfor 2-D flows, the LEV was unstable but the rotating 3-D flat plate

Horseshoe-shaped vortex

Joint of the shed TEV and the shedTV

Root vortex

Dougshaped

Fig. 18. Visualization of flow fields around a hovering hawkmoth. Iso-vorticity-magn

supination, (C) the upstroke, and (D) the pronation, respectively. The color of iso-vorticit

the projection of a fluid’s spin vector in the direction of its momentum vector, being po

they point in the opposite direction. (For interpretation of the references to color in th

model at high angle of attack produced a conical LEV, as it wasobserved in the results presented earlier and by others [184,219].Moreover, they mentioned that if the Reynolds number increasedabove a critical value, a Kelvin–Helmholtz instability [220]occurred in the LEV sheet, resulting in the sheet breaking downon outboard sections of the wing. Recently, Lentink and Dickinson[221] revisited existing hypotheses regarding stabilizing the LEV.They systematically investigated the effects of propeller-likemotion of a fruit fly wing model on the aerodynamic performanceand the LEV stability at Re=1.1�102–1.4�104 based on theore-tical and experimental approaches. They stated that the LEV wasstabilized by the ‘‘quasi-steady’’ centripetal and Coriolis accelera-tions that were present at low Rossby numbers (i.e. a half of theaspect ratio) and resulted from the propeller-like sweep of thewing. In addition, they suggested that the force augmentationthrough a stably attached LEV could represent a convergentsolution for the generation of high fluid forces over a range ofReynolds numbers. From the viewpoint of unsteady aerody-namics, the LEV as a lift enhancement mechanism at higher Re

range (O(105–106)) may be questionable because a dynamic-stall

Interaction between the TEVs of each wing

Roll-up

Doughnut-shaped vortex ring

hnut- vortex

Vortex ring (root)

itude surfaces around a hovering hawkmoth during (A) the downstroke, (B) the

y-magnitude surfaces indicates the normalized helicity density which is defined as

sitive (red) if these two vectors point in the same direction and negative (blue) if

is figure legend, the reader is referred to the web version of this article.)

Downstroke stopping vortex

Upstroke starting vortex

Wake capture

Root

Doughnut-shaped vortex ring

Fig. 18. (Continued)


vortex on an oscillating airfoil is often found to break away and toconvect elsewhere as soon as the airfoil translates [222].

5. Flapping wing aeroelasticity

The topic of aeroelasticity in flapping wings has recentlyincreased in activity though a full picture of the basic aeroelasticphenomena is still not clear. Analytical investigations by Danieland Combes [223] suggested that aerodynamic loads wererelatively unimportant in determining the bending patterns inoscillating wings. Subsequently, experimental investigations byCombes [224], and Combes and Daniel [225] found that theoverall bending patterns of a Hawkmoth wing were quite similarwhen flapped (single degree-of-freedom flap rotation) in air and

helium, despite a 85% reduction in fluid density in the latter,suggesting that the contribution of aerodynamic forces wasrelatively small compared to the contribution of inertial–elasticforces during flapping motion. However, they mentioned thatrealistic wing kinematics might include rapid rotation at thestroke reversal that may lead to increased aerodynamic forces dueto unsteady aerodynamic mechanisms (see Section 3). Further-more, static bending tests by Combes and Daniel [10,11] showedanisotropy of wing structures in a variety of insect species. Morerecently, Mountcastle and Daniel [226] investigated the influenceof wing compliance on the mean advective flows (indicative ofinduced flow velocity) using PIV techniques. Their resultsdemonstrated that flexible wings yield mean advective flowswith substantially greater magnitudes and orientations morebeneficial to lift than those of stiff wings.

Upstroke stopping vortex

Doughnut-shapedvortex ring

Vortex ring (root)

Downstroke starting vortex



While Section 4 reviewed studies in rigid flapping wingaerodynamics, this section presents a review of some recentefforts in flexible flapping wing aerodynamics/aeroelasticityfocusing on chordwise-only flexible, spanwise-only flexible, andcombined chordwise/spanwise flexible structures in that order.As summary, the recent studies are listed in Tables A1–A3 withkey information.

5.1. Chordwise-flexible wing structures

Zhu [227] performed a fully coupled fluid–structure interac-tion analyses to investigate a chordwise flexible airfoil prescribedwith pure plunge motion. To clarify the role of inertia on thedeformation, the wing was studied in both water and air. Resultsshowed that when the wing was immersed in air, the chordwiseflexibility reduces both the thrust and the propulsion efficiency.However, when the wing was immersed in water, it increased theefficiency. Refs. [32,228–231] presented studies on both rigid andpartially chordwise flexible airfoils prescribed with both pureplunge and/or combined plunge/pitch motions in water andshowed that flexible wings may be more efficient than the rigidones. In particular, Pederzani and Haj-Hariri [232] suggested thatlighter plunging airfoils were capable of generating more thrustthan heavier ones and were more efficient. They performedcomputational analyses on a rigid wing from which a portion wascut out and covered with a very thin and flexible material (latex)and showed that due to a snapping motion (i.e. non zero velocityin the direction opposite to that of the following stroke) of thelatex at the beginning of each stroke, the strength of the vorticesthat were shed was higher in lighter wing structures, leading tothe generation of more thrust. Furthermore, such structuresrequired less input power in order to be snapped than heavierones. Chaithanya and Venkatraman [233,234] investigated theinfluence of inertial effects due to prescribed motion on the thrustcoefficient and propulsive efficiency of a plunging/pitching thin

plate using inviscid flow theory and beam equations. Their resultsdemonstrated that flexible airfoils with inertial effects yieldedmore thrust than those without inertial effects. This was due tothe increase in the fluid loading in the former which subsequentlyled to an increase in the deformation. Due to their shape,deformed airfoils produced a force component along the forwardvelocity direction [234].

Gopalakrishnan [235] analyzed the effects of elastic camberingof a rectangular membrane flapping wing on aerodynamics inthe forward flight using a linear elastic membrane solvercoupled with an unsteady LES method. Different membranepre-stresses were investigated to give a desired camber inresponse to the aerodynamic loading. The results showed thatthe camber introduced by the wing flexibility increased the thrustand lift production considerably. Analysis of flow structuresrevealed that the LEV stayed attached on the top surface of thewing, followed the camber, and covered a major part of the wing,which resulted in high force production. On the other hand, forrigid wings (which were also considered) the leading edge vortexlifted off from the surface resulting in low force production. Toevaluate the role played by the LEV for a flexibly cambered airfoil,Gulcat [236] investigated (1) a thin rigid plate in a plungingmotion; (2) a flexibly cambered airfoil whose camber waschanged periodically; and (3) the plunging motion of a flexiblycambered airfoil. The leading edge suction force for all cases waspredicted by means of the Blasius theorem and the time-dependent surface velocity distribution of the airfoil is deter-mined by unsteady aerodynamic considerations. Gulcat [236]reported that the viscous effects obtained by the unsteadyboundary-layer solution show very little alteration to theoscillatory behavior of the net propulsive force. They only reducedthe amplitude of the leading edge suction force obtained by theunsteady aerodynamic theory. It was found that the majorcontribution to the thrust is due to heaving plunging; therefore,it is possible to get high propulsion efficiency with limited camberflexibility.

Stroke cycle

Body Right wing Left wing


Down Up

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

Ver

tical

forc

e (N

)H

oriz

onta

l for

ce (N

)Si

desl

ip fo

rce

(N)


Down Up

Stroke cycle

Fig. 19. Time courses of aerodynamic forces over a flapping cycle. (A) Vertical

force (lift). (B) Horizontal force (drag/thrust). (C) Sideslip force. Dashed, dotted,

and solid lines represent the aerodynamic forces acting on right wing, left wing,

and body, respectively.


Miao and Ho [237] prescribed a time-dependent flexibledeformation profile for an airfoil in pure plunge and investigatedthe effect of flexure amplitude on the unsteady aerodynamiccharacteristics for various combinations of Reynolds number andreduced frequency. For a specific combination of Reynoldsnumber, reduced frequency, and plunge amplitude, the resultsshowed that thrust-indicative wake structures were observed

behind the trailing edge of the airfoil for airfoils with flexureamplitudes of 0.0–0.5 of the chord length. It was shown that thiswake structure evolved into a drag-indicative form as the flexureamplitude of the airfoil was increased to 0.6 and 0.7 of the chordlength. Studies conducted under various combinations of Rey-nolds number and reduced frequency showed that the propulsiveefficiency of a chordwise flexible airfoil in pure plunge wasinfluenced primarily by the value of the reduced frequency ratherthan by the Reynolds number. Toomey and Eldredge [238]performed numerical and experimental investigations to under-stand the role of flexibility in flapping wing flight using two rigidelliptical sections connected by a hinge with torsion spring. Thesection at the leading edge was prescribed with fruit fly-likehovering wing kinematics [50], while the trailing edge sectionresponded passively due to the fluid dynamic and inertial/elasticforces. It was found that the lift force and wing deflection areprimarily controlled by the nature of the wing rotation. Fasterwing rotation, for example, led to larger peak deflection and liftgeneration. Advanced rotation also led to a shift in the instant ofpeak wing deflection which increased the mean lift. In contrast tothe rotational kinematics, the translational kinematics wereshown to have very little impact on spring deflection or force.And, while the former was shown to be nearly independent ofReynolds number, the latter was shown to increase withincreasing Reynolds number. Poirel et al. [239] conducted a windtunnel experimental investigation of self-sustained oscillations ofan aeroelastic NACA0012 airfoil occurring in the transitional Reregime, in particular, aeroelastic limit cycle oscillations for theairfoil constrained to rotate in pure pitch. The structural stiffnessas well as the position of the elastic axis were varied. Theirinvestigation suggested that laminar separation plays a role in theoscillations, either in the form of trailing edge separation or due tothe presence of a laminar separation bubble.

Vanella et al. [89] conducted numerical investigations on asimilar structure and found that the best performance (up toapproximately 30% increase in lift) was realized when the wingwas excited by a non-linear resonance at 1/3rd of its naturalfrequency. For all Reynolds numbers considered, the wake capturemechanism was enhanced due to a stronger flow around the wingat stroke reversal, resulting from a stronger vortex at the trailingedge. Heathcote et al. [240] investigated the effect of chordwiseflexibility on aerodynamic performance of an airfoil in pureplunge under hovering conditions. Because the trailing edge was amajor source of shedding of vorticity at zero freestream velocity,they showed that the amplitude and phase angle of the motion ofthe trailing edge affected the strength and spacing of the vortices,and the time averaged velocity of the induced jet. Direct forcemeasurements confirmed that at high plunge frequencies, thethrust coefficient of the airfoil with intermediate stiffness washighest, although the least stiff airfoil can generate larger thrust atlow frequencies. It was suggested that there was an optimumairfoil stiffness for a given plunge frequency and amplitude.Similar conclusions were made in another study [241] wherein,the influence of resonance on the performance of a chordwiseflexible airfoil prescribed with pure plunge motion at its leadingedge was studied. It was shown that while the mean thrust couldincrease with an increase in flexibility, below a certain thresholdthe wing is too flexible to communicate momentum to the flow.On the other hand, too much flexibility led to a net drag andhence, only a suitable amount of flexibility was desirable forthrust generation. Du and Sun [242] numerically investigated theeffect of prescribed time-varying twist and chordwise deforma-tion on the aerodynamic force production of a fruitfly in hover.The results showed that aerodynamic forces on the flappingwing were not affected much by the twist, but by thecamber deformation. The effect of combined camber and twist

(1)

=0.125 (b) in Fig. 17

(2)

=0.25 (c) in Fig. 17

(3)

=0.375 (d) in Fig. 17

Wing surface Section 1 (mid semi-span) Section 2 (75% semi-span)

-1.0 1.0 0

LEV

LEV

LEV

TEV

L. E.

T. E.

Tip

L. E.

Root

T. E.

Tip

L. E.

Root

T. E.

Tip

LEV

LEV

LEV

L. E.

L. E.

L. E.

T. E.

T. E.

T. E.

L. E.

T. E.

L. E.

T. E.

L. E.

T. E.

t /T

t /T

t /T

Fig. 20. Pressure distributions at selected time instants corresponding to Fig. 17 (b), (c), and (d) during the downstroke. Left: Upper wing surface. Middle: Mid semi-span

cross-section. Right: 75% semi-span cross-section from the wing root. Note that LEV, TEV, L.E., and T.E. indicate leading edge vortex, trailing edge vortex, leading edge, and

trailing edge, respectively. Note that the pressure coefficient in the LEV is negative in all sub-figures.


deformation was found to be similar to that of camber deforma-tion alone. With a deformation of 6% camber and 201 twist(typical values observed for wings of many insects), the liftincreased by 10–20% compared to the case of a rigid flat platewing. It was therefore shown that chordwise deformation couldincrease the maximum lift coefficient of a fruit fly wing model andreduce its power requirement for flight.

While most of the recent computational and experimentalstudies explored the role of wing flexibility in augmentingaerodynamic performance while focusing on single wings atrelatively higher Reynolds numbers, Miller and Peskin [243]numerically investigated the effect of wing flexibility on theforces produced during clap and fling/peel motion [244] of a smallinsect (ReE10) focusing on wing–wing interactions. Theyprescribed both clap and fling kinematics separately to a rigidand a chordwise flexible wing and showed that while liftcoefficients produced during the rigid and flexible clap strokeswere comparable, the peak lift forces in the flexible cases werehigher than in the corresponding rigid cases. This was due to thepeel motion which delayed the formation of the trailing edgevortices, thereby maintaining vortical asymmetry and augment-ing lift for longer periods [41].

Zhao et al. [245] investigated the chordwise flexibility effectsusing 16 different dynamically scaled mechanical models of afruit fly wing. Based on the prescribed kinematics, the wing wasstarted impulsively from rest and moved over a 1801 arc in 1.5 s.Each of the 16 wing models (each with a different chordwisebending stiffness) was tested at 23 different static angles of attack

at the leading edge from �9 to 901 in 4.51 increments. Resultsfrom the experiments showed that the overall aerodynamicperformance (lift generation) of flapping wings deteriorated asthey became more flexible. However, it was shown that flexiblewings could generate more lift at higher static angles of attackthan rigid wings due to bending of the wing surface causing aslight enhancement in the lift to drag ratios for such wings overtheir rigid counterparts.

In another study, Heathcote and Gursul [32] performed watertunnel studies (see Fig. 25) to examine the thrust and efficiency ofa flexible 2-D airfoil plunging in forward flight. Subsequently,computations were done by Aono et al. [246] using a nonlinearplanar beam solver [247,248] and a NS solver [122] under theassumption of laminar flow. The airfoil consisted of a rigidteardrop element made out of aluminum and a thin flexible flatplate made out of steel (see Fig. 25). Experiments were conductedon several wing models by considering variations in the platethickness. The key parameters related to all the cases consideredin the experiment are included in Table 3. The computations weredone on some of the experimental wing models and for a specificReynolds number. The flexible flat plate was modeled as acantilevered beam fixed at the trailing edge of the teardrop. Inall the cases considered in both experiments and computations,the leading edge of the teardrop was actuated by a sinusoidalplunge displacement profile. Three different variations areconsidered for the plate thickness in the computations:3.81�10�3 m (‘‘Rigid’’), 1.27�10�4 m (‘‘Flexible’’), and5.04�10�5 m (‘‘Very Flexible’’). The results presented here will

Right wing Left wing Total wing

Right wing Left wing Total_wing

Down Up

Stroke cycle

-1.0E-03

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

-9.0E-04

-6.0E-04

-3.0E-04

0.0E+00

3.0E-04

6.0E-04

-8.0E-04

-4.0E-04

0.0E+00

4.0E-04

8.0E-04

Rol

ling

torq

ue (N

m)

Pitc

hing

torq

ue (N

m)

Yaw

ing

torq

ue (N

m)

Right wing Left wing Total wing

Down Up

Stroke cycle

Fig. 21. Time courses of aerodynamic torques of wings over a flapping cycle.

(A) Rolling torque. (B) Pitching torque. (C) Yawing torque. Dashed, dotted, and

solid lines represent aerodynamic torques acting on right wing, left wing, and two

wings (‘total wing’), respectively.

-50

0

50

100

150

200

250

Pow

er (W

/Kg)

Down Up

Stroke cycle

Right wing Left wing

Fig. 22. Time courses of the muscle-mass-specific aerodynamic power of wings

over a flapping cycle. The dashed and dotted lines represent the muscle-mass-

specific aerodynamic power are produced by the right wing and by the left wing,

respectively.


focus on computations while still including the correspondingexperimental data.

Fig. 26(A) shows the thrust coefficient as a function ofnormalized time (with respect to the period of plunge) for the‘‘Rigid’’, ‘‘Flexible’’, and ‘‘Very Flexible’’ cases. As seen in the figure,the amplitude of instantaneous thrust increases with increasingflexibility. Fig. 26(B) shows the time history of total aerodynamic

force coefficient (total force including both lift and thrust) for allthree cases. Unlike in the case of the thrust coefficient, theinstantaneous amplitude of the resultant aerodynamic forceproduced by the ‘‘Rigid’’ and ‘‘Flexible’’ wings is very close toeach other at all time instants, while being very different fromthat of the ‘‘Very Flexible’’ wing. This behavior is akin to that ofthe lift coefficient (not included in the figure) whose amplitude isthe largest in the ‘‘Rigid’’ case and smallest in the ‘‘Very Flexible’’one. This is due to the redistribution of the total aerodynamicforce between the lift and the thrust directions. Fig. 27 shows themean thrust coefficient plotted as a function of the effectivestiffness (P1) for all variations of chordwise flexibility includingboth experimental and computational data. In the figure, eachdata point corresponds to a particular value of plate thickness(which in turn corresponds to a particular P1). In general, withinthe range of flexibility considered in the computations, the meanthrust coefficient increases with increasing flexibility. However,for the cases considered in the experiments wherein morevariations in flexibility were considered, the mean thrustinitially increases with increasing flexibility but decreases in themost flexible case. Fig. 29 shows the time history of the effectiveangle of attack (illustrated in Fig. 28) for the three variations offlexibility considered in the computations. Fig. 29(A) correspondsto the effective angle of attack considering the whole airfoil(i.e. with reference to the leading edge point of the teardrop) andFig. 29(B) to the angle considering only the flexible flat plate(i.e. with reference to the trailing edge point of the teardrop). And,Fig. 29(C) shows the instantaneous airfoil shape for all threeflexible wings at selected time instants as illustrated under eachsnapshot. As seen in the figures, in general, with increasingchordwise flexibility, the effective angle of attack decreases.Considering the time instant corresponding to the middle of thedownstroke (i.e. at t/T=0.25), it is seen in the figure that theeffective angle of attack in the ‘‘Rigid’’ and ‘‘Flexible’’ cases is morethan that of the ‘‘Very Flexible’’ case. This is because, withincreasing chordwise deformation (as seen in Fig. 29(C), it is thelargest in the ‘‘Very Flexible’’ case), the angle ‘‘a’’ increaseswhich subsequently results in a decrease of the effective angle ofattack ‘‘ae’’.

In order to estimate the individual contribution of the teardropand the flexible plate to force generation, the time histories ofthrust coefficient are shown in Fig. 30 separately for each case. Itis seen in the figure that the thrust response with variation inflexibility is different in each of the two cases: with increasing


chordwise flexibility of the plate, the instantaneous thrust actingon the teardrop reduces because of a decrease in the leading edgesuction peak (Fig. 31(C)). However, with increasing flexibilityfrom ‘‘Rigid’’ to the ‘‘Very Flexible’’ case, the instantaneous thrustcontributed by the flexible plate increases. This is because, thechordwise deformation of the rear flexible plate in both ‘‘Flexible’’and ‘‘Very Flexible’’ cases result in an effective projected area forthe thrust forces to develop. Streamlines and pressure contours

(A-1) Streamlines

(B-1) Streamlines

(C-1) Streamlines

Hawkmoth model (Re

Honeybee model (Re

Fruit fly model (Re=

Fig. 23. Comparison of near-field flow fields between a hawkmoth, a honeybee, a fruit

models of (A) a hawkmoth model, (B) a honeybee model, (C) a fruit fly model, and (D

corresponding velocity vectors in a plane cutting through the left wing at 60% of the s

around the three wing configurations are shown in Fig. 31 at thetime instant t/T=0.25 that corresponds to the middle of downstroke. Furthermore, in Fig. 31 (A–2), (B–2), and (C–2), thepressure coefficient on the airfoil is plotted as a function of thechordwise position. The grey vertical line in the plots representsthe boundary between the tear drop and the flexible thin plate. Asseen in these figures, the leading edge suction is more dominantin the ‘‘Rigid’’ and ‘‘Flexible’’ cases than in the ‘‘Very Flexible’’ case.

(A-2) Spanwise flow

(B-2) Spanwise flow

(C-2) Spanwise flow

= 6.3×103 , k= 0.30)

= 1.1×103 , k= 0.24)

1.3×102 , k= 0.21)

fly, and a thrips around the middle of the downstroke. Wing-body computational

) a thrips model, with the LEVs visualized by instantaneous streamlines and the

emi-span.

Fig. 24. Spanwise pressure gradient contours on flapping wings around the

middle of the downstroke: (A) a hawkmoth model (Re=6.3�103, k=0.30); (B) a

honeybee model (Re=1.1�103, k=0.24); (C) a fruit fly model (Re=1.3�102,

k=0.21), and (D) a thrips model (Re=1.2�101, k=0.25), respectively.

(D-2) Spanwise flow(D-1) StreamlinesThrips model (Re= 1.2×101 , k= 0.25)



Considering the outcome from the studies conducted onchordwise flexible plunging/pitching structures discussed so far,three factors may be identified that play a vital role inaerodynamic force generation in hover/forward flight: (i) airfoilplunge motion modifies the effective angle of attack andaerodynamic forces; (ii) relative moment of leading edge andtrailing edge creates the pitch angle which dictates the directionof the net force; and (iii) airfoil shape deformation modifieseffective geometry such as camber.

5.2. Spanwise-flexible wing structures

Numerical simulations were performed by Liu and Bose [248] for a3-D pitching and plunging wing in forward flight. Their results

showed that the phase of the flexing motion of the wing relative tothe prescribed heave motion plays a key role in determining thrustand efficiency characteristics of the fin. Zhu [227] performednumerical investigations on a thin flapping foil prescribed with pureplunge motion in forward flight in different both air and water. Hisresults showed that when the wing was immersed in air, thespanwise flexibility (through equivalent plunge and pitch flexibility)increased the thrust without an efficiency reduction and when thesame wing was immersed in water, the spanwise flexibility reducedboth the thrust and efficiency.

Heathcote et al. [250] conducted water-tunnel studies to studythe effect of spanwise flexibility on the thrust, lift, and propulsiveefficiency of a plunging flexible wing configuration in forwardflight. A schematic of the experimental setup is shown in Fig. 25.Three wings of 0.3 m span and 0.1 m chord with varying levels offlexibility were constructed (‘‘Inflexible’’, ‘‘Flexible’’, and ‘‘HighlyFlexible’’, see Fig. 25(C)). The leading edge at the wing root wasactuated with a prescribed sinusoidal plunge displacementprofile. Wing shape was recorded with a 50-frames-per-second,high-shutter-speed, digital video camera. Overall wing thrustcoefficient and tip displacement response were measured. Therepresentations of the cross-section constructions are reproducedin Fig. 25(C). While the ‘‘Inflexible’’ wing cross-section was built-up from nylon and reinforced by steel rods, the ‘‘Flexible’’ and‘‘Highly Flexible’’ cross-sections were built-up from PDMS(rubber) and stiffened with a thin metallic sheet made out ofsteel and aluminum respectively. Table 4 shows the keyparameters related to the test cases. In a subsequent effort,computations were conducted on these wing configurations byChimakurthi et al. [251] and Aono et al. [252] only at the chord-based Reynolds number of 3�104 using an aeroelastic frameworkdeveloped by coupling a 3-D NS solver [122] with a nonlinearbeam solver [253]. The flow is assumed to be laminar. It might benoted that in the computations, the PDMS around the metallicstiffeners was neglected. While the contribution of it to theoverall stiffness of the wings will be negligible (due to a relativelyvery low Young’s modulus of the material compared to eithersteel or aluminum), the effect of it on the mass properties mightbe significant. Furthermore, it might also be noted that, thebending stiffness of the Highly Flexible wing was adjusted due tosome discrepancies found by the experimentalists who conductedstatic loading tests and revealed the actual stiffness data viaprivate communication.

Fig. 32 shows the time history of the instantaneous thrustcoefficient and that of the total aerodynamic force coefficient for

Fig. 25. Experimental set-up of for the chordwise and spanwise flexible wing studies: (A) water tunnel setup; (B) airfoil configuration for chordwise flexible wing studies

[extracted from Ref. [32]]; (C) wing cross-sections used in the experiment for spanwise flexible wing studies [extracted from Ref. [240]].

Table 3Geometric, mechanical, flow, and dimensionless properties associated with the chordwise flexible wing cases in Ref. [32,236].

Reference velocity [m/s]: Uref=UN 1.0�10�1

Water density [kg/m3]: qt 1.0�103

Mean chord length [m]: cm 9.0�10�2

Young’s modulus of steel [Pa]: E 205�109

Density of steel; [kg/m3]: qs 7.8�103

Plunge amplitude [m]: ha 1.75�10�2

Plunge frequency [Hz]: f 9.62�10�1

Relative thickness (Experiment) 0.56�10�3, 0.84�10�3, 1.12�10�3, 1.41�10�3, 1.69�10�3, 2.25�10�3, 4.23�10�3

Relative thickness (Computation) 0.56�10�3 (Very Flexible), 1.41�10�3 (Flexible), 4.23�10�3 (Rigid)

Re 9.0�103

St 0.34

q 7.8

P1 (Experiment) 3.30�100, 1.11�101, 2.64�101, 5.26�101, 9.06�101, 2.14�102, 1.42�103

P1 (Computation) 3.30�100, 5.26�101, 1.42�103


all three wing configurations for the case of Re=3.0�104. Inthe figure, the experimental data is indicated as ‘‘EXP’’ and thecomputational data as ‘‘COMP’’. From Fig. 33(A), it is seen that the‘‘Flexible’’ wing case generates the highest thrust peaks, while inthe ‘‘Very Flexible’’ wing case, the smallest. Similarly, as seen inFig. 32(B), the amplitude of the total aerodynamic force coefficientis the largest in the ‘‘Flexible’’ wing and the smallest in the ‘‘VeryFlexible’’ one. Furthermore, there is also a difference in phaseamong the three wing cases. Fig. 33 shows the mean thrust

coefficient for all three wings plotted as a function of the non-dimensional parameter ‘‘effective stiffness (dimensionlessparameter P1)’’. As shown in the figure, the mean thrustresponse is non-monotonic. In particular, it is observed that thelargest and smallest mean thrust coefficients are produced by the‘‘Flexible’’ and the ‘‘Very Flexible’’ wing cases, respectively. Itmight be noted that in all these cases, while there is an overallagreement between the experiments and the computations, thereare noticeable discrepancies in the ‘‘Very Flexible’’ case due to

Fig. 26. Time response of thrust (CT) and total aerodynamic force coefficients (Cf) of the chordwise flexible wing structures at Re=9.0�103 and St=0.34: (A) Thrust

coefficient; (B) Total aerodynamic force coefficient.


uncertainties in both modeling (mass density of PDMS was notaccounted for in the computations) and experiments (materialproperties). Interested readers might refer to Chimakurthi et al.[251] and Aono et al. [252] for additional details aboutcomputational results and their correlations with experimentaldata.

The reasons for the diminishment of the lift/thrust forces inthe ‘‘Very Flexible’’ wing case and the enhancement of the same inthe ‘‘Flexible’’ one are as follows. Fig. 34 shows the plot of theinstantaneous angle of attack along the span position at the timeinstant t/T=0.25 (this is the time instant corresponding to themiddle of downstroke during which the maximum lift/thrustforces are produced in the ‘‘Rigid’’ and ‘‘Flexible’’ cases) for allthree plunging wing configurations. As seen from the plot, thearea under the ‘‘Flexible’’ wing curve is the largest and the oneunder the ‘‘Very Flexible’’ one, the smallest. Since it is the

cumulative effect of the angle of attack seen by the flow at allsections through the span that is important in total forcegeneration on the wing, the forces in the ‘‘Flexible’’ case are thelargest and those on the ‘‘Very Flexible’’ wing, the smallest. Thedecrease in the angle of attack in the case of the ‘‘Very Flexible’’wing (at several sections through the wing) over that of the othersled to a decrease in the leading edge suction which in turn causeda loss of total lift/thrust on the wing.

Sample pressure contours for all four wing configurations atthe mid-span section at time instant t/T=0.25 are shown inFig. 35. The dominance of leading edge suction in the ‘‘Flexible’’case and the reduction of it in the ‘‘Very Flexible’’ case are visiblein that figure. The leading edge suction is obtained due to thenormal projected area contributed by the leading edge curvatureof the wing (see Fig. 35) that supports the generation of thehorizontal force. The phase lag between the prescribed motion

Fig. 27. Plot of mean thrust coefficient as a function of the effective stiffness P1 for the chordwise flexible wing structures at Re=9.0�103 and St=0.34. Experimental data

was extracted from Ref. [32].

Fig. 28. Schematics of the effective angle of attack for the chordwise flexible wing

structures: (A) effective angle of attack of the wing at the leading edge point of the

tear drop (aeL.E.) [32]; (B) effective angle of attack of the wing at the trailing edge

point of the tear drop (aeT.E.). aL.E. and aT.E. indicate the angles subtended by the

line joining the leading edge and trailing edge of the teardrop with the horizontal,

respectively.


and the deformation of the wing is another quantity that could beused to explain the aerodynamic force generation in flexibleflapping wings [251,252]. The phase lag at the wing tip withrespect to the prescribed motion for the ‘‘Rigid’’, ‘‘Flexible’’, and‘‘Very Flexible’’ cases are, 01, �25.71, and �126.31, where the inphase percentages are 100%, 86%, and 24%, respectively. As seen inthe table, the ‘‘Very Flexible’’ wing has the largest phase lag andthe shortest time of in-phase motion at the wing tip in a stroke. Incontrast, the ‘‘Flexible’’ wing has the smallest phase lag and thelongest in-phase motion in a stroke. As a result of the substantialphase lag in the ‘‘Very Flexible’’ case, the wing tip and root movein opposite directions during most of the stroke resulting in lowereffective angles of attack and consequently lesser aerodynamicforce generation. This is seen by correlating the phase lag

information with the mean thrust response of the ‘‘Flexible’’ andthe ‘‘Very Flexible’’ wing cases in Fig. 32(A).

The results presented so far focused on a single case of theReynolds number 3�104. Results related to parametric studiesare available in Heathcote et al. [250]. To summarize the same, itwas found that the mean thrust coefficient is a function of theStrouhal number (based on the amplitude of the mid-span of thewing), and is very weakly dependent on the Reynolds numberp[250].

5.3. Combined chordwise-and-spanwise flexible wing structures

Watman and Furukawa [254] investigated the effects ofpassive pitching motions of flapping wings on the aerodynamicperformance using robotic wing models. They considered twotypes of passive flapping wing models: first model used a rigidconnection between all parts of the structure. This design wasutilized in several MAVs [1,5,6] and served as a common designused for analysis of flapping wings. Second model was designed ina way to allow free rotation of ribs and membrane over a limitedangle. This design was used recently in a small MAV prototype[255]. They showed that the former passive flapping wing(constrained) was found to have better performances comparedto latter one due to favorable variation of passive pitching angle ofthe wing. Hui et al. [256] examined various flexible wingstructures to evaluate their implications on flapping wingaerodynamics. They showed that the flexible membrane wingswere found to have better overall aerodynamic performance(i.e., lift-to-drag ratio) over the rigid wing for soaring flight,especially for high-speed soaring flight or at relatively high angleof attack. The rigid wing was found to have better lift productionperformance for flapping flight in general. The latex wing, whichis the most flexible among the three tested wings, was found tohave the best thrust generation performance for flapping flight.The less flexible nylon wing, which has the best overallaerodynamic performance for soaring flight, was found to be theworst for flapping flight applications. Shkarayev et al. [257]investigated the aerodynamics of cambered membrane flappingwings. Specifically, a cambered airfoil was introduced into thewing by shaping metal ribs attached to the membrane skin of the25 cm-wing-span model. The thrust force generated by a 9%camber wing was found to be 30% higher than that of the samesize flat wing. Adding a dihedral angle to the wings and keeping

Fig. 29. Time histories of the effective angle of attack for the chordwise flexible

wing structures at Re=9.0�103 and St=0.34: (A) effective angle of attack at the

leading edge of the teardrop (aeL.E.); (B) effective angle of attack at the trailing edge

of the teardrop (aeT.E.); (C) deformed airfoil shapes at selected time instants.

Fig. 30. Time histories of thrust coefficient contribution due to the teardrop and

the flexible plate separately at Re=9.0�103 and St=0.34: (A) response of the

teardrop; (B) response of the flexible plate.


the flapping amplitude constant improved the cambered wing’sperformance even further. The aerodynamic coefficients aredefined using a reference velocity as a sum of two components:a free stream velocity and a stroke-averaged wing tip flappingvelocity. The lift, drag, and pitching moment coefficients obtainedusing this procedure collapse well for studied advance ratios,especially at lower angles of attack. Kim et al. [258] developed abiomimetic flexible flapping wing using micro-fiber compositeactuators and experimentally investigated the aerodynamicperformance of the wing under flapping and non-flapping motionin a wind tunnel. Results showed that the camber due to wingflexibility could produce positive effects (i.e. stall delay, dragreduction, and stabilization of the LEV) on flapping wingaerodynamics in quasi-steady and unsteady region. Muelleret al. [259] presented a new versatile experimental test for

measuring the thrust and lift of a flapping wing MAV. Theyshowed increase in average thrust due to increased wingcompliance and detrimental influence of excessive complianceon drag forces during high-frequency operation. Also theyobserved the useful effect of compliance on the generation ofextra trust at the beginning and end of flapping motions.

Hamamoto et al. [260] conducted a fluid–structure interactionanalysis on a deformable dragonfly wing in hover and examinedthe advantages and disadvantages of flexibility. They tested threetypes of flapping flight: a flexible wing driven by dragonflyflapping motion, a rigid wing (stiffened version of the originalflexible dragonfly wing) driven by dragonfly flapping motion, anda rigid wing driven by modified flapping based on tip motion ofthe flexible wing. They found that the flexible wing with nearlythe same average energy consumption generated almost the sameamount of lift force as the rigid wing with modified flappingmotion, which realized the same angle of attack at theaerodynamically dominant sections of the wing. However, therigid wing required 19% more peak torque and 34% more peakpower, indicating the usefulness of wing flexibility. Luo et al.[261] have simulated the interaction between a viscous, unsteadyflow and deformable thin structures to simulate a dragonfly wing.

Fig. 31. Vorticity contours, streamlines, and surface pressure distribution for the chordwise flexible wing structures at the time instant t/T=0.25 at Re=9.0�103 and

St=0.34. Vorticity contours are in the range between �5.0 and +5.0. The dashed and solid lines in the Cp plot correspond to the upper and lower surfaces of the airfoil,

respectively. The grey vertical line in same plot indicates the junction between the teardrop and the flexible plate. Note that CWV and CCWV in the vorticity contours

indicate clock-wise and counter clock-wise vortex.

Table 4Flow, geometrical, and structural properties along with dimensionless parameters associated to the spanwise flexible wing cases in Refs. [240–242].

Rigid Flexible Very flexible

Reference velocity [m/s]: Uref=UN 3.0�10�1 3.0�10�1 3.0�10�1

Water density [kg/m3]: qf 1.0�103 1.0�103 1.0�103

Mean chord length [m]: cm 1.0�10�1 1.0�10�1 1.0�10�1

Semi-span [m]: R 3.0�10�3 3.0�10�3 3.0�10�3

Thickness of wing [m]: hs 1.0�10�3 1.0�10�3 1.0�10�3

Young’s modulus [Pa]: E 2.1�1011 4.0�1010

Density of structure [kg/m3]: qs 7.8�103 2.7�103

Plunge amplitude [m]: ha 1.75�10�2 1.75�10�2 1.75�10�2

Plunge frequency [Hz]: f 1.74�100 1.74�100 1.74�100

Re 3.0�104 3.0�104 3.0�104

St 0.20 0.20 0.20

q 7.8 2.7

P1 Infinite 2.14�102 4.07�101


They focused on the situations where the wing inertia is largecompared to the fluid forces, and therefore, the flow–structureinteraction is essentially unidirectional (from the structure to theflow, and not the other way around.) However, the two-wayinteraction is built in the code, and it only takes longer time toperform the simulations. Aono et al. [262] reported a combinedcomputational and experimental study of a well-characterizedflapping wing structure was conducted. An aluminum wing wasprescribed with single degree-of-freedom flapping at 10 Hzfrequency and 7211 amplitude. Flow velocities and deformationwere measured using digital image correlation and digital PIVtechniques, respectively. In the most flexible flapping wing case,

the elastic twisting of the wing was shown to producesubstantially larger mean and instantaneous thrust due to shapedeformation-induced changes in effective angle of attack. Rele-vant fluid physics were documented including the counter-rotating vortices at the leading and the trailing edge, whichinteracted with the tip vortex during the wing motion.

Singh and Chopra [263] developed an experimental apparatuswith a bio-inspired flapping mechanism to measure the thrustgenerated for a number of wing designs. The key conclusions thatstemmed from this study were that the inertial loads constitutedthe major portion of the total loads acting on the flapping wingstested on the mechanism and that for all the wings tested, the

Fig. 32. Effect of spanwise flexibility on instantaneous aerodynamic force generation at Re=3.0�104 and St=0.2: (A) Thrust (CT). (B) Total aerodynamic force (Cf).


thrust dropped at higher frequencies. Further, the author foundthat at such frequencies, the light-weight and highly flexiblewings used in the study exhibited significant aeroelastic effects.Young et al. [121] conducted numerical investigations on atethered desert locust called Schistocerca gregaria [118]. Resultsdemonstrated that time-varying wing twist and camber wereessential for the maintenance of attached flow. The authorsemphasized that while high-lift aerodynamics was typicallyassociated with massive flow separation and large LEVs, whenhigh lift was not required, attached flow aerodynamics couldoffer greater efficiency. Their results further showed that, indesigning robust lightweight wings that could support efficientattached flow, it was important to build a wing that undergoesappropriate aeroelastic wing deformation through the course of awing beat.

Agrawal and Agrawal [264] investigated the benefits of insectwing flexibility on flapping wing aerodynamics based on experi-ments and numerical simulations. They compared the performance

of two synthetic wings: (i) a flexible wing based on a bio-inspireddesign of the hawkmoth (Manduca Sexta) wing and (ii) a rigid wingof similar geometry. The results demonstrated that more thrustwas generated by the bio-inspired flexible wing compared to therigid wing in all wing kinematic patterns considered. Theyemphasized that the results provided motivation for exploringthe advantages of passive deformation through wing flexibility andthat coupled fluid–structure simulations of flexible flapping wingswere required to gain a fundamental understanding of the physicsand to guide optimal flapping wing MAV designs.

As a flight vehicle exploring the merit of aeroelasticity effect,Jones and Platzer [265] and Jones et al. [266] devised a flappingconcept which allowed the wing to flap with a constant amplitudeto produce thrust from root to tip. In a follow-up study conductedby Kaya et al. [267], the flapping motion was optimized formaximum thrust. Specifically, the flapping airfoils were attachedto swing arms by an elastic joint, which could be treated as aspring-mass system. The stiffness coefficient and the mass

Fig. 33. Plot of mean thrust coefficient as a function of the non-dimensional parameter P1 for the spanwise flexible wing structures at Re=3.0�104 and St=0.2. Note that

the experimental data point for the Rigid airfoil overlaps with the computational data point. Experimental data were extracted from Ref. [240].

Fig. 34. Instantaneous angle of attack along the wing span at the time instant

t/T=0.25 corresponding to all three spanwise flexible plunging wing configurations.

Rigid

Very Flexible

XX

X

Z Z

0 0.05 0.1

0.05 0.10

0 0.05 0.1

0

0.02

-0.02

Flexible

0.02

0

-0.02

-0.04

Z

0.02

0

-0.02

-0.04

Pressure

Fig. 35. Pressure contours at mid-span station at t/T=0.25 for all three spanwise

flexible plunging wing configurations. Leading edge suction is the most dominant

in the Flexible case and the least in the Very Flexible one.


moment of inertia of the airfoil were then optimized using thegradient-based method for maximum thrust. The optimizationresults showed that the aeroelastic pitching with optimum elastic

properties produces significantly higher thrust than that of anoptimum sinusoidal pitching. The aeroelastic pitching providedlower effective incidence angles and delayed the leading edgevortex formation, which in turn augmented the thrust generation.Wu et al. [268] and Sallstrom et al. [269] presented a multi-disciplinary experimental endeavor correlating flapping wingMAVs aeroelasticity and thrust production by quantifying andcomparing elasticity, dynamic responses, and air flow patterns ofsix different pairs of MAV wings (in each one, the membrane skinwas reinforced with different leading edge and batten configura-tions) of the Zimmerman planform (two ellipses meeting at thequarter chord) with varying elastic properties. In their experi-ment, single degree-of-freedom flapping motion was prescribedto the wings in both air and vacuum. Amongst many conclusions,they found that, within the range of flexibility considered, moreflexible wings are more thrust-effective at lower frequencieswhereas stiffer wings are more effective at higher frequencies.They hypothesized that flexible wings may have a certainactuation frequency for peak thrust production and the perfor-mance would degrade once that frequency is passed.

These studies as well as the investigations of Heathcote et al.[32,250], Chimakurthi et al. [251], Aono et al. [252], and Tang et al.[230] offer consistent findings. It seems like the spanwiseflexibility increases aerodynamic forces by creating highereffective angles of attack via spanwise deformation. However,apart from affecting the overall aerodynamic force generation, thechordwise flexibility can redistribute lift versus thrust bychanging the projection angle of the wing with respect to thefreestream by changing airfoil via camber deformation, forexample. Of course, similar behavior may be observed in spanwiseflexible with passive twist deformation along the wing span.Overall, both spanwise and chordwise flexibility need to beconsidered together in order to optimize the aerodynamicperformance under different flight speed and environmentuncertainty such as wind gust.

6. Conclusions

In this article, a number of previous efforts focusingon flapping wing aerodynamics and aeroelasticity have beenhighlighted. Furthermore, important features related to theaerodynamics associated with rigid and flexible flapping wingshave been reviewed.


For Re=O(102), corresponding to the small insect flight regime,the aerodynamic performances associated with various winggeometry and flapping kinematics have been with extensivecomparison between 2-D and 3-D studies, it has been observed that:

(i)
There is significant variance in the spanwise distribution offorces in the 3-D cases. Cases which suppress the TiVgeneration, those with the highest pitching amplitudes andthus low angles of attack, and small translational velocitywhen vertical (e.g. near synchronized rotation), appear tohave a relatively constant response along the span. Incontrast, 3-D cases with a prominent TiV exhibit significantvariations along the span which do not have a strongcorrelation to the 2-D lift values experienced.
(ii)
At Reynolds number at 65, the 3-D fluid physics and TiVeffects are able to augment the lift by: (a) the presence of alow pressure region near the tip and, (b) the anchoring of anotherwise shed vortex (in 2-D) near the tip.
(iii)
The surrogate modeling techniques reveals that the hierarchyof variable sensitivity in the mean lift changes between 2-Dand 3-D in space. In 2-D the importance is pitching angularamplitude, phase lag and normalized stroke amplitude. In3-D the hierarchy switches to phase lag, pitching angularamplitude, and normalized stroke amplitude. Regions forwhich 2-D kinematics outperformed 3-D and vice versa areidentified in the design space considered.
(iv)
That a 3-D low aspect ratio wing can produce higher lift thana 2-D airfoil is not supported by the classical stationary wingtheory [78], which suggests that TiVs decrease performance.In unsteady flow around a hovering wing, however, the TiVscan contribute to lift generation rather than just draggeneration on the wing.
The LEV is common and important to the flapping wingaerodynamics at the Reynolds number of O(104) or lower, whichcorresponds to the hummingbird and insect flight regimes.However, the LEV structures and distribution of spanwise flowinside the LEV change with the variation of Reynolds number(wing sizing, flapping frequency, etc.) and with the interactionsbetween LEVs and TiVs and hence influence the aerodynamicforce generation. In the hawkmoth and honeybee studies, athigher Reynolds numbers (O(103) and O(104)), it is seen that theLEV has a spiral structure and breaks down at the middle of thedownstroke. In the fruit fly studies, at lower Reynolds number(O(102)), it is observed that the LEV has an ordered structure andit connects with the TiV. A dependency on the Reynolds number isobserved in both the spanwise flow inside the LEV and thespanwise variation of the pressure-gradient on the wing surfaces.

The LEV as a lift enhancement mechanism for flapping wing atReynolds number of O(105–106) seem less certain because, aspointed out in Ref. [222], a dynamic-stall vortex on an oscillatingairfoil is often found to break away and to convect elsewhere assoon as the airfoil translates.

The impact of structural flexibility on aerodynamics undergoingplunging motion in forward flight is also highlighted as follows:

(i)
The thrust generation consists of contributions due to bothleading edge suction and the pressure projection of thechordwise deformed rear foil. When the rear foil’s flexibilityincreases, the thrust of the teardrop element decreases, aswell as the effective angle of attack.
(ii)
Within a certain range as chordwise flexibility increases,even though the effective angle of attack and the netaerodynamic force are reduced due to chordwise shapedeformation, both mean and instantaneous thrust are
enhanced due to the increase of projected area normal to theflight trajectory.

(iii)
For the spanwise flexible case, correlations of the motionfrom the root to the tip play a role. Within a suitably selectedrange of spanwise flexibility, the effective angle of attack andthrust forces of a plunging wing are enhanced due to thewing deformations.
While airfoil shape and spanwise geometry obviously affect theaerodynamics of a flapping wing, it seems from the growing bodyof the studies that the implications of structural flexibility areconsistent with respect to different wing geometries.

Since the various scaling parameters vary with the length and timescales in different proportionality, the scaling invariance of both fluiddynamics and structural dynamics as the size changes is fundamen-tally difficult and challenging. It also seems that there is a desirablelevel of structural flexibility to support desirable aerodynamics.Significant work needs to be done to better understand theinteraction between structural flexibility and aerodynamic perfor-mance under unpredictable wind gust conditions.

In summary, there have been many efforts ongoing to increaseour understanding of the fluid physics of biology-inspiredmechanisms that simultaneously provide lift and thrust, enablehover, and grant high flight control authority, while minimizingpower consumption. Detailed experimental efforts and firstprinciples-based computational modeling and analysis capabil-ities are essential in support of the investigation of issues relatedto fluid–structure interactions, unsteady freestream (wind gust),and unsteady aerodynamics. Other issues for further research are:

(i)
In order to support the broad flight characteristics such astake-off, forward flight of varying speed, wind gust response,hover, perch, threat avoidance, station tracking, and payloadvariations, a variety of wing kinematics and body/legmaneuvers will need to be employed. Considering thevariations in size and flapping patterns of natural flyers,there is significant potential and need for us to further refineour knowledge regarding the interplay between kinematicsand aerodynamics.
(ii)
It is established that local flexibility can significantly affectaerodynamics in both fixed and flapping wings. Furthermore,as already discussed above, insects’ wing properties areanisotropic, with the spanwise bending stiffness about 1–2orders of magnitude larger than the chordwise bending one.As the vehicle size changes, the scaling parameters cannot allmaintain invariance due to different scaling trends associatedwith them. This means that if, for measurement precision,instrumentation preference, etc., one cannot do laboratorytest of a flapping wing design using different sizes or flappingfrequencies. A closely coordinated computational and experi-mental framework is needed in order to facilitate theexploration of the vast design space (which can have O(102)or more design variables including geometry, materialproperties, kinematics, flight conditions, and environmentalparameters) while searching for optimal and robust designs.
(iii)
The implications of the dynamics and stability of a flappingvehicle [270–272], in association with flapping wing aero-dynamics is still inadequately understood. In particular, thevehicle stability via passive shape deformation due to flexiblestructures needs to be addressed.
(iv)
Bio-inspired mechanisms [273,274] need to be developed forthe flapping wing. These mechanisms include, for example,joints and distributed actuation to enable flapping andmorphing. Most importantly, these mechanisms should becapable of mitigating wind gust.


(v)

TableStudi

Au

He

Ish

D

Van

P

a

Zhu

Yam

N

I

Pre

a

Tan

Cha

Ped

Cha

V

Go

Gu

Mia

Too

He

G

Mic

Du

Mil

Zha

S

Vision-based sensing techniques will be very helpful forflight control as well as estimating the aeroelastic states ofthe vehicle. For example, both rigid-body and deformationstates of a vehicle can be extracted by noting frequency-varying properties by optical means.

(vi)
Closed-loop active control can be used to target a desiredaerodynamic performance, to eliminate the influence ofdisturbances, or to achieve both objectives. Adaptive controltechniques have the advantage of tuning the feedback gainsin response to the true plant dynamics and exogenoussignals. These techniques usually require additional referencemodels along with system identification techniques. Closecoupling between aerodynamics and control is a key in MAVdevelopment. Integration of aerodynamics, structural dy-namics, and control holds the key of MAV development. Withsubstantial uncertainty in flight environments due to windgusts, possible movement of targets, intrinsically unsteadyaerodynamics, and the substantial disparity in time scalesbetween flapping motion, fluid physics, structural dynamics,and vehicle response, the interactions between differentcomponents of MAVs are highly complex.
A1es of chordwise-flexible wing structures.

thors Ref no. Model Kinematics

athcote and Gursul [32] Teardrop

NACA0012

Plunge

ihara, Horie and

enda

[88] Shell/Beam Pitch/plunge

ella, Fitzgerald,

reidikman, Balaras

nd Balachandran

[89] Beam Pitch/plunge

[227] Plate Plunge

amoto, Terada,

agamatu and

maizumi

[228] Airfoils Pitch/plunge

mpraneerach, Hover

nd Triantafyllou

[229] NACA0014 Pitch/plunge

g, Viieru and Shyy [230] Teardrop plate Plunge

ndar and Damodaran [231] Teardrop plate Plunge

erzani and Haj-Hariri [232] Airfoils Plunge

ithanya and

enkatraman

[232,234] Plate Pitch/plunge

palakrishnan [235] Rectangular

membrane

Pitch/plunge

lcat [236] Thin airfoil Plunge

o and Ho [237] NACA0012 Plunge

mey and Eldredge [238] Two ellipses Pitch/plunge

athcote, Martin and

ursul

[32,240] Teardrop Plunge

helin and Smith [241] Plate Plunge

and Sun [242] 3% flat plate fruit

fly

Pitch/flap

ler and Peskin [243] Beam Clap and fling

o, Huang, Deng and

ane

[245] Fruitfly-like Impulsive/pitch/

flap

The work in this area has begun recently, and it is clear thatsignificant progress needs to be made before MAVs can performreal missions envisioned by the community.

Acknowledgements

The work has been supported in part by the Air Force Office ofScientific Research’s Multidisciplinary University Research Initia-tive (MURI) and by the Michigan/AFRL (Air Force ResearchLaboratory)/Boeing Collaborative Center in Aeronautical Sciences.Dr. Jian Tang assisted in producing some of the results related tochordwise flexibility.

Appendix

See Tables A1–A3.

Re St or k Structural information

9�103 0.1–0.7 E=205 GPa, rs=7800 kg/m3,

hs=5�10�5–3.8�10�4 m1.8�104

2.7�104

2.0�102 0.054 Torsional stiffness: 0.8–

15.4 g cm2/(s2rad)

75 – Frequency ratio: 1/2, 1/3, 1/4,

and 1/62.5�102

103

2.0�104 0.2 E=20–2000 Gpa, rs=7800 kg/

m3, hs=4�10�3 m

104–105 – Elasticity of flexible part:

3.3 mm/N–40 mm/N

4.0�104 0.1–0.45 Flexible rubber: E=3.18–

50.6 GPa

9.0�103 0.5 E=205 GPa, rs=7850 kg/m3,

hs=5�10�5–3.8�10�4 m

9.0�103 0.5 E=205 GPa, rs=7850 kg/m3,

hs=5�10�5–3.8�10�4 m

5.0�102 5.5 Membrane (latex): rs=0.5-

1.0 kg/m3, Fixed trailing edge

or free trailing edge

Inviscid – –

1.0�104 1.7 E=114 GPa, rs=4500 kg/m3,

hs=15�10�6 m

103–105 0.5–1.5 Prescribed camber

1.0�103 1–6 Prescribed wing deformation

1.0�104

1.0�105

1.2�103–

6.1�103

– liner spring-damper model:

spring stiffness

coefficient=0.0068 kg m2/s2,

damping

coefficient=0.0039 kg m2/s

0.75–2.1�104 0.17–0.40 E=205 GPa, rs=7800 kg/m3,

hs=5�10�5–3.8�10�4 m

Inviscid 0.08–0.8 Euler–Bernoulli equation:

P1=10�2-102

3.3�102 6.0�103 – Prescribed wing deformation

10 – Bending stiffness: 6.9�10�7–

1.1�10�5 Nm2

2.0�103 – EIB=8.6�10�6–5.3�10�4

Table A3Studies of combined spanwise-and-chordwise flexible wing structures.

Authors Ref. no. Model Re St or k Structural information

Zheng, Wang, Khan, Vallance,

Mittal and Hedrick

[120] Hawkmoth-like 5.0�103 0.2–0.4 –

Young, Walker, Bomphrey,

Taylor and Thomas

[121] Realist locust wings 4.0–5.0�103 – –

Daniel and Combes [232] Beam – – EIB=6.0�10�6 Pa m4

Mountcastle and Daniel [226] Real moth wing 4.0–7.0�103 0.2-0.3 EIB=10�6–10�5 Pa m4

Wu, Ifju, Stanford, Sallstrom,

Ukeiley, Love and Lind

[268] Zimmerman 1.0–3.0�104 0.4 Composite wing: capran (E=2.5 GPa,

rs=1600 kg/m3, hs=12�10�6 m) and

carbon fiber robs (E=73.4 GPa, rs=1740 kg/

m3, hs=100�10�6 m)

Sallstrom, Ukeiley, Wu and

Ifju

[269]

Watman and Furukawa [252] Half ellipse 5.0�105 0.83 Mylar sheet (6.25�10�6 m) and carbon

fiber robs (0.5 and 0.25�10�3 m)

Hui, Kumar, Abate and

Albertani

[256] Bird-like 2.0–8.0�104 0.6-3.3 Composite wings: carbon fiber rods and

wood/nylon/latex

Shkarayev, Silin, Abate and

Albertani

[257] Bird-like 1.6–4.8�104 0.46–1.4 Composite wings:

Membrane: 0.015 mm Mylar

Front spar: carbon rods T315-4 of diameter

0.8 mm

Ribs: steel wires of diameter 0.5 mm d

Kim, Han and Kwon [258] Bird-like 2.0—3.0�104 1-7 Composite wing (flexible PVC skin and MFC

actuators)

Muller, Bruck and Gupta [259] Bird-like 103–104 – Mylar sheet and carbon fiber rods

Hamamoto, Ohta, Hara, and

Hisada

[260] Dragonfly-like 103 1 Corrugation, Frame: E=70 GPa,

hs=9�10�6 m

Root of corrugation: : E=70 GPa,

hs=27�10�6 m

Film: : E=3.9 GPa, hs=5�10�6 m

Luo, Yin, Dai and Doyle [261] Plate Dragonfly-like 50 0.94 Reduced bending stiffness:5.63

250 25 Reduced bending stiffness:0.08

Aono, Chimakurthi, Wu,

Sallstrom, Stanford, Cesnik,

Ifju, Ukeiley and Shyy

[262] Zimmerman 2.6�103 0.56 Aluminum (E=70 GPa, rs=2700 kg/m3,

hs=4.0�10�4 m)

1.0�103 2.35 Hypothetical wing (E=0.1–70 GPa,

rs=2400 kg/m3, hs=4.0�10�4 m)

Singh and Chopra [263] Fruit fly-like

mechanical model

1.5�104 0.31 Aluminum (E=60 GPa, rs=2400 kg/m3,

hs=5.0�10�4 m)

Mylar (E=7 GPa, rs=1250 kg/m3,

hs=1.0�10�4 m)

Agrawal and Agrawal [264] Hawkmoth-like 7000 0.2 Carbon rods (E=200 GPa), nylon 6/6 rods

(E=3 GPa), nylon strips (E=3 GPa), rubber

(E=1.5 MPa), latex film (E=1.7 MPa,

hs=0.1�10�3 m)

Table A2Studies of spanwise-flexible wing structures.

Authors Ref. no. Model Kinematics Re St or k Structural information

Zhu [227] Plate Plunge 2.0�104 0.2 E=20–2000 GPa, rs=7800 kg/m3, hs=4�10�3 m

Liu and Bose [249] Immature fin like Pitch/plunge Inviscid – –

Heathcote, Wang and Gursul [250] NACA0012 Plunge 1–3.0�104 0.05–0.9 (1) Inflexible,

(2) Steel (E=210 GPa, rs=7800 kg/m3, hs=1�10�3 m),

(3) Aluminum (E=70 GPa, rs=2700 kg/m3, hs=1�10�3 m)

Chimakurthi, Tang, Palacios,

Cesnik and Shyy

[251] NACA0012 Plunge 3.0�104 0.4–1.82 (1) Rigid

(2) Steel (E=210 GPa, rs=7800 kg/m3, hs=1�10�3 m)

Aono, Chimakurthi,

Cesnik, Liu and Shyy

[252] NACA0012 Plunge 3.0�104 1.82 (1) Inflexible,

(2) Flexible (E=210 GPa, rs=7800 kg/m3, hs=1�10�3 m),

(3) Very flexible (E=70 GPa, rs=2700 kg/m3, hs=1�10�3 m)


References

[1] Shyy W, Lian Y, Tang J, Viieru D, Liu H. Aerodynamics of low Reynolds

number flyers. New York: Cambridge University Press; 2008.

[2] Shyy W, Berg M, Ljungqvist D. Flapping and flexible wings for biological and

micro air vehicles. Prog Aerosp Sci 1999;35(5):455–505.

[3] Muller TJ. Fixed and flapping wing aerodynamics for micro air vehicle

applications. AIAA Prog Astronaut Aeronaut 2001:195.

[4] Shyy W, Ifju P, Viieru D. Membrane wing-based micro air vehicles. Appl

Mech Rev 2005;58:283–301.

[5] Pines DJ, Bohorquez F. Challenges facing future micro-air-vehicle develop-

ment. J Aircraft 2006;43(2):290–305.

[6] Platzer M, Jones K, Young J, Lai J. Flapping wing aerodynamics: progress and

challenges. AIAA J 2008;46(9):2136–49.

[7] Lian Y, Shyy W, Viieru D, Zhang B. Membrane wing aerodynamics for micro

air vehicles. Prog Aerosp Sci 2003;39(6-7):425–65.

[8] Stanford BK, Ifju P, Albertani R, Shyy W. Fixed membrane wings for micro air

vehicles: experimental characterization, numerical modeling, and tailoring.

Prog Aerosp Sci 2008;44(4):258–94.

[9] Dalton S. The miracle of flight. London: Merrell; 2006.


[10] Combes SA, Daniel TL. Flexural stiffness in insect wings I. Scaling and theinfluence of wing venation. J Exp Biol 2003;206:2979–87.

[11] Combes SA, Daniel TL. Flexural stiffness in insect wings II. Spatialdistribution and dynamic wing bending. J Exp Biol 2003;206:2989–97.

[12] Azuma A. The biokinetics of flying and swimming. 2nd edition, Virginia:American Institute of Aeronautics and Austronautics Inc. 2006.

[13] Norberg UM. Vertebrate flight: mechanics, physiology, morphology,ecology, and evolution. New York: Springer; 1990.

[14] Dudley R. The biomechanics of insect flight: form, function, evolution. NewJersey: Princeton University Press; 2002.

[15] Biewener AA. Animal locomotion. New York: Oxford University Press;2003.

[16] Weis-Fogh T. Quick estimates of flight fitness in hovering animals, includingnovel mechanism for lift production. J Exp Biol 1973;59:169–230.

[17] Maxworthy T. The fluid dynamics of insect flight. Ann Rev Fluid Mech1981;13:329–50.

[18] Sane SP. The aerodynamics of insect flight. J Exp Biol 2003;206:4191–208.[19] Lehmann F-O. The mechanisms of lift enhancement in insect flight.

Naturewissenschaften 2004;91:101–22.[20] Lehmann F-O. Aerial locomotion in flies and robots: kinematic control and

aerodynamics of oscillating wings. Arthropod Struct Dev 2004;33:331–45.[21] Dickinson MH. The ignition and control of rapid flight maneuvers in fruit

flies. Integr Comp Biol 2005;45:274–81.[22] Dickinson MH. Insect flight. Curr Biol 2006;16(9):R309–14.[23] Tobalske BW. Biomechanics of bird flight. J Exp Biol 2007;210:3135–46.[24] Ho S, Nassef H, Pornsinsiriak N, Tai Y-C, Ho C-M. Unsteady aerodynamics

and flow control for flapping wing flyers. Prog Aerosp Sci 2003;39:635–81.

[25] Zbikowski R, Ansari SA, Knowles K. On mathematical modelling of insectflight dynamics in the context of micro air vehicles. Bioinsp Biomim2006;1:R26–37.

[26] Shyy W, Lian Y, Tang J, Liu H, Trizila P, Stanford B, et al. Computationalaerodynamics of low Reynolds number plunging, pitching and flexiblewings for MAV applications. Acta Mech Sin 2008;24:351–73.

[27] van Breugel F, Regan W, Lipson H. From insects to machines: demonstrationof a passively stable, untethered flapping-hovering micro-air vehicle. IEEERobot Automat Mag 2008;Dec:68–74.

[28] Ellington CP. The aerodynamics of hovering insect flight III. Kinematics. PhilTrans R Soc Lond B 1984;305:41–78.

[29] Tobalske BW, Warrick DR, Clarck CJ, Powers DR, Hedrick TL, Hyder GA, et al.Three-dimensional kinematics of hummingbird flight. J Exp Biol2007;210:2368–82.

[30] Hodges DH, Pierce GA. Introduction to structural dynamics and aeroelas-ticity. New York: Cambridge University Press; 2002.

[31] Jones KD, Dohring CM, Platzer MF. Experimental and computationalinvestigation of the Knoller–Betz effect. AIAA J 1998;36(7):1240–6.

[32] Heathcote S, Gursul I. Flexible flapping airfoil propulsion at low Reynoldsnumbers. AIAA J 2007;45(5):1066–79.

[33] Christensen RM. Mechanics of composite materials, reprint ed.. New York:Dover Publications Inc.; 2005.

[34] Reddy JN. Mechanics of laminated composite plates and shells: theory andanalysis, 2nd ed.. Florida: CRC Press LLC; 2003.

[35] Triantafyllou MS, Triantafyllou GS, Yue DKP. Hydrodynamics of fishlikeswimming. Annu Rev Fluid Mech 2000;32:33–53.

[36] Greenewalt CH. Dimensional relationships for flying animals. SmithsonMisc Collect 1962;144:1–46.

[37] Lehmann F-O, Sane SP, Dickinson MH. The aerodynamic effects of wing–wing interaction in flapping insect wings. J Exp Biol 2005;208:3075–92.

[38] Sun M, Yu X. Flows around two airfoils performing fling and subsequenttranslation and translation and subsequent clap. Acta Mech Sin2003;19(2):103–17.

[39] Sun M, Yu X. Aerodynamic force generation in hovering flight in a tinyinsect. AIAA J 2006;44(7):1532–40.

[40] Miller LA, Peskin CS. When vortices stick: an aerodynamic transition in tinyinsect flight. J Exp Biol 2004;208:195–212.

[41] Miller LA, Peskin CS. A computational fluid dynamics of ‘clap and fling’ inthe smallest insects. J Exp Biol 2005;208:195–212.

[42] Liu H, Aono H. Size effects on insect hovering aerodynamics: an integratedcomputational study. Bioinsp Biomim 2009;4:1–13.

[43] Lehmann F-O, Pick S. The aerodynamic benefit of wing–wing interactiondepends on stroke trajectory in flapping insect wings. J Exp Biol2007;210:1362–77.

[44] Lehmann F-O. Wing–wake interaction reduces power consumption in insecttandem wings. Exp Fluids 2009;46:765–75.

[45] Cooter RJ, Baker PS. Weis-Fogh clap and fling mechanisms in Locusta.Nature 1977;269:53–4.

[46] Gotz KG. Course-control, metabolism and wing interference during ultra-long tethered flight in Drosophila melangaster. J Exp Biol 1987;128:35–46.

[47] Brodsky AK. Vortex formation in the tethered flight of the peacock butterflyInachis io L and some aspects of insect flight evolution. J Exp Biol1991;161:77–95.

[48] Brachenbury J. Wing movements in the bush cricket Tettigonia viridissimaand the mantis Ameles spallanziana during natural leaping. J Zool Long1990;220:593–602.

[49] Srygley RB, Thomas ALR. Unconventional lift-generating mechanisms infree-flying butterflies. Nature 2002;420:660–4.

[50] Dickinson MH, Lehmann F-O, Sane SP. Wing rotation and the aerodynamicbasis of insect flight. Science 1999;284(5422):1954–60.

[51] Sane SP, Dickinson MH. The aerodynamic effects of wing rotation and arevised quasi-steady model of flapping flight. J Exp Biol 2002;205:1087–96.

[52] Sun M, Tang J. Unsteady aerodynamic force generation by a model fruit flywing in flapping motion. J Exp Biol 2002;205:55–70.

[53] Birch JM, Dickinson MH. The influence of wing–wake interactions on theproduction of aerodynamic forces in flapping flight. J Exp Biol2003;206:2257–72.

[54] Wang ZJ. Dissecting insect flight. Annu Rev Fluid Mech 2005;37:183–210.[55] Shyy W, Trizila P, Kang C, Aono H. Can tip vortices enhance lift of a flapping

wing?AIAA J 2009;47(2):289–93[56] Jardin T, David L, Farcy A. Characterization of vortical structures and loads

based on time-resolved PIV for asymmetric hovering flapping flight. ExpFluids 2009;46:847–57.

[57] Ellington CP, van den Berg C, Willmott AP, Thomas ALR. Leading-edgevortices in insect flight. Nature 1996;384:626–30.

[58] van den Berg C, Ellington CP. The three-dimensional leading-edge vortexof a ‘hovering’ model hawkmoth. Phil Trans R Soc Lond B 1997;352:329–40.

[59] van den Berg C, Ellington CP. The vortex wake of a ‘hovering’ modelhawkmoth. Phil Trans R Soc Lond B 1997;352:317–28.

[60] Willmott AP, Ellington CP, Thomas ALR. Flow visualization and unsteadyaerodymamics in the flight of the hawkmoth, Manduca sexta. Phil Trans RSoc Lond B 1997;352:303–16.

[61] Liu H, Kawachi K. A numerical study of insect flight. J Comput Phys1997;146(1):124–56.

[62] Liu H, Ellington CP, Kawachi K, van den Berg C, Willmott AP. Acomputational fluid dynamic study of hawkmoth hovering. J Exp Biol1998;201:461–77.

[63] Liu H. Computational biological fluid dynamics: digitizing and visualizinganimal swimming and flying 2002;42(5):1050–9.

[64] Liu H. Simulation-based biological fluid dynamics in animal locomotion.Appl Mech Rev 2005;58:269–82.

[65] Shyy W, Liu H. Flapping wings and aerodynamics lift: the role of leading-edge vortices. AIAA J 2007;45(12):2817–9.

[66] Milano M, Gharib M. Uncovering the physics of flapping flat plates withartificial evolution. J Fluid Mech 2005;534:403–9.

[67] Gharib M, Rambod E, Shariff K. A universal time scale for vortex ringformation. J Fluid Mech 1998;360:121–40.

[68] Dabiri JO. Optimal vortex formation as a unifying principle in biologicalpropulsion. Ann Rev Fluid Mech 2009;41(1):17–33.

[69] Rival D, Prangemeier T, Tropea C. The influence of airfoil kinematics on theformation of leading-edge vortices in bio-inspired flight. Exp Fluids2009;46:823–33.

[70] Tarascio M, Ramasamy M, Chopra I, Leishman JG. Flow visualization of MAVscaled insect based flapping wings in hover. J Aircraft 2005;42(2):355–60.

[71] Ramasamy M, Leishmann JG. Phase-locked particle image velocimetrymeasurements of a flapping wing. J Aircraft 2006;43(6):1867–75.

[72] Poelma C, Dickinson WB, Dickinson MH. Time-resolved reconstruction ofthe full velocity field around a dynamically-scaled flapping wing. Exp Fluids2006;41:213–25.

[73] Lu Y, Shen GX. Three-dimensional flow structure and evolution of theleading-edge vortices on a flapping wing. J Exp Biol 2008;211:1221–30.

[74] Pick S, Lehmann F-O. Stereoscopic PIV on multiple color-coded light sheetsand its application to axial flow in flapping robotic insect wings. Exp Fluids2009, doi:, doi:10.1007/s00348-009-0687-5.

[75] Liang Z, Dong H, Wei M. Computational analysis of hovering hummingbirdflight. In: 48th AIAA aerospace sciences meeting including the new horizonsforum and aerospace exposition. 2010, AIAA 2010-555.

[76] Warrick DR, Tobalske BW, Powers DR. Aerodynamics of the hoveringhummingbird. Nature 2005;435:1094–7.

[77] Altshuler DL, Princevac M, Pan H, Lozano J. Wake patterns of the wings andtail of hovering hummingbirds. Exp Fluids 2009;46:835–46.

[78] Anderson JD. Fundamentals of aerodynamics, 4th ed.. New York: McGrawHill Higher Education; 2006.

[79] Aono H, Shyy W, Liu H. Near wake vortex dynamics of a hoveringhawkmoth. Acta Mech Sin 2009;25:23–36.

[80] Ramamurti R, Sandberg WC. A computational investigation of the three-dimensional unsteady aerodynamics of Drophila hovering and maneuver-ing. J Exp Biol 2007;210:881–96.

[81] Aono H, Liang F, Liu H. Near- and far-field aerodynamics in insect hoveringflight: an integrated computational study. J Exp Biol 2008;211:239–57.

[82] Ringuette MJ, Milano M, Gharib M. Role of the tip vortex in the forcegeneration of low-aspect-ratio normal flat plates. J Fluid Mech 2007;581:453–68.

[83] Taira K, Colonius T. Three-dimensional flows around low-aspect-ratio flat-plate wings at low Reynolds numbers. J Fluid Mech 2009;623:187–207.

[84] Ennos AR. A comparative study of the flight mechanism of Diptera. J ExpBiol 1987;127:355–72.

[85] Ennos AR. The importance of torsion in the design of insect wings. J. Exp Biol1988;140:137–60.

[86] Ennos AR. The inertial cause of the wing rotation in Diptera. J Exp Biol1988;140:161–9.

[87] Ennos AR. Inertial and aerodynamic torques on the wings of Diptera inflight. J Exp Biol 1989;142:87–95.

dx.doi.org/10.1007/s00348-009-0687-5.3d


[88] Ishihara D, Horie T, Denda M. A two-dimensional computational study onthe fluid-structure interaction cause of wing pitch changes in dipteranflapping flight. J Exp Biol 2009;212:1–10.

[89] Vanella M, Fitzgerald T, Preidikman S, Balaras E, Balachandran B. Influenceof flexibility on the aerodynamic performance of a hovering wing. J Exp Biol2009;212:95–105.

[90] Willmott AP, Ellington CP, Thomas ALR. Flow visualization and unsteadyaerodymamics in the flight of the hawkmoth, Manduca sexta. Phil Trans RSoc Lond B 1997;352:303–16.

[91] Thomas ALR, Taylor GK, Srygley RB, Nudds RL, Bomphrey RJ. Dragonflyflight: free-flight and tethered flow visualzations reveal a diverse array ofunsteady lift-generating mechanisms, controlled primarily via angle ofattack. J Exp Biol 2004;207:4299–323.

[92] Bomphrey RJ, Taylor GK, Thomas ALR. Smoke visualization of free-flyingbumblebees indicates independent leading-edge vortices on each wing pair.Exp Fluids 2009;46:811–21.

[93] Hubel TY, Hristov NI, Swart SM, Breuer KS. Time-resolved wake structureand kinematics of bat flight. Exp Fluids 2009;46:933–43.

[94] Tobalske BW, Hearn JWD, Warrick DR. Aerodynamics of intermittentbounds in flying birds. Exp Fluids 2009;46:963–73.

[95] Videler JJ, Stamhuis EJ, Povel GDE. Leading-edge vortex lifts swifts. Science2004;306:1960–2.

[96] Bomphrey RJ, Lawson NJ, Harding NJ, Taylor GK, Thomas ALR. Theaerodynamic of Manduca sexta: digital particle image velocimetry analysisof the leading-edge vortex. J Exp Biol 2005;208:1079–94.

[97] Bomphrey RJ. Insects in flight: direct visualization and flow measurements.Bioinsp Biomim 2006;1:1–9.

[98] Warrick DR, Tobalske BW, Powers DR. Lift production in the hoveringhummingbird. Proc R Soc Lond B 2009;276(1674):3747–52.

[99] Hedenstrom A, Johansson LC, Wolf M, von Busse R, Winter Y, Spedding GR.Bat flight generates complex aerodynamic tracks. Science 2007;316:894–7.

[100] Muijres FT, Johansson LC, Barfield R, Wolf M, Spedding GR, Hedenstrom A.Leading-edge vortex improves lift in slow-flying bats. Science 2008;319:1250–3.

[101] Hedenstrom A, Muijres FT, von Busse R, Johansson LC, Winter Y, SpeddingGR. High-speed stereo PIV measurement of wakes of two bat species flyingfreely in a wind tunnel. Exp Fluids 2009;46:923–32.

[102] Wakeling JM, Ellington CP. Dragonfly flight II. Velocities, accelerations andkinematics of flapping flight. J Exp Biol 1997;200:557–82.

[103] Willmott AP, Ellington CP. The mechanics of flight in the hawkmothManduca sexta I. Kinematics of hovering and forward flight. J Exp Biol1997;200:2705–22.

[104] Willmott AP, Ellington CP. The mechanics of flight in the hawkmothManduca sexta II. Aerodynami consequences of kinematic and morphologi-cal variation. J Exp Biol 1997;200:2723–45.

[105] Fry SN, Sayaman R, Dickinson MH. The aerodynamics of free-flightmaneuvers in Drosophila. J. Exp. Biol. 2003;300:495–8.

[106] Tian X, Iriate-Diaz J, Middleton K, Galvao R, Israeli E, Roemer A, et al. Directmeasurements of the kinematics and dynamics of bat flight. Bioinsp Biomim2006;1:S10–8.

[107] Riskin DK, Willis DJ, Iriarte-Diaz J, Hedrick TL, Kostandov M, Chen J, et al.Quantifying the complexity of bat wing kinematics. J Theor Biol2008;254:604–15.

[108] Wallance ID, Lawson NJ, Harvey AR, Jones JDC, Moore AJ. High-speedphotogrammetry system for measuing the kinematics of insect wings. ApplOpt 2006;45(7):4165–73.

[109] Hedrick TL. Software techniques for two- and three-dimensional kinematicmeasurements of biological and biomimetic systems. Bioinsp Biomim2008;3:1–6.

[110] Ristroph L, Berman GJ, Bergou AJ, Wang ZJ, Cohen I. Automated hullreconstruction motion tracking (HRWT) applied to sideways maneuvers offree-flying insects. J Exp Biol 2009;212:1324–35.

[111] Zeng L, Hao Q, Kawachi K. A scanning projected line method for measuring abeating bumblebee wing. Opt Commun 2000;183:37–43.

[112] Zeng L, Hao Q, Kawachi K. Measuring the body vector of a free flightbumblebee by the reflection beam method. Meas Sci Technol2001;12:1886–90.

[113] Zeng L, Matsumoto H, Kawachi K. A fringe shadow method for measuringflapping angle and torsional angle of a dragonfly wing. Meas Sci Technol1996;7:776–81.

[114] Song D, Wang H, Zen L, Yin C. Measuring the camber deformation of adragonfly wing using projected comb fringe. Rev Sci Instrum2001;72(5):2450–4.

[115] Wang H, Zeng L, Liu H, Yin C. Measuring wing kinematics, flight trajectoryand body attitude during forward flight and turning maneuvers indragonflies. J Exp Biol 2003;206:745–57.

[116] Zhang G, Sun J, Chen D, Wang Y. Flapping motion measurement of honeybeebilateral wings using four virtual structured-light sensors. Sens Actuators A2008;148:19–27.

[117] Walker SM, Thomas ALR, Taylor GK. Deformable wing kinematics infree-flying hoverflies. J R Soc Interface 2009, doi:, doi:10.1098/rsif.2009.0120.

[118] Walker SM, Thomas ALR, Taylor GK. Deformable wing kinematics in thedesert locust: how and why do camber, twist and topography vary throughthe stroke? J R Soc Interface 2008;6(38):735–47.

[119] Wu G, Zeng L. Measuring the kinematics of a free-flying hawkmoth(Macroglossum stellatarum) by a comb-fringe projected method. Acta MechSin 2009, doi:, doi:10.1007/s10409-009-0306-y.

[120] Zheng L, Wang X, Khan A, Vallance RR, Mittal R. A combined experimental–numerical study of the role of wing flexibility in insect flight. In: 47th AIAAaerospace sciences meeting including the new horizons forum andaerospace exposition. 2009, AIAA 2009-382.

[121] Young J, Walker SM, Bomphrey RJ, Taylor GK, Thomas ALR. Details of insectwing design and deformation enhance aerodynamic function and flightefficiency. Science 2009;325:1549–52.

[122] Shyy W, Udaykumar HS, Rao MM, Smith RW. Computational fluid dynamicswith moving boundaries. New York: Dover Publications Inc.; 2006.

[123] Tezduyar TE, Behr M, Mittal S, Liou J. A new strategy for finite elementcomputations involving moving boundaries and interfaces – the deforming-spatial-domain/space-time procedure: II. Computation of free-liquid flows,and flows with drifting cylinders. Comput Method Appl M 1992;94:353–71.

[124] Lesoinne M, Farhat C. Geometric conservation laws for flow problems withmoving boundaries and deformable meshes, and their impact on aeroelasticcomputations. Comput Method Appl M 1996;134(1-2):71–90.

[125] Shyy W. Computational modeling for fluid flow and interface transport.Amsterdam: Elsevier Science Publisher; 1994, also New York: DoverPublications Inc.; 2006.

[126] Mittal R, Iaccarino G. Immersed boundary method. Annu Rev Fluid Mech2005;37:239–61.

[127] Katz J, Plotkin A. Low-speed aerodynamics, 2nd ed.. New York: CambridgeUniversity Press; 2001.

[128] Ansari SA, Zbikowski R, Knowles K. Aerodynamic modelling of insect-likeflapping flight for micro air vehicles. Prog Aerosp Sci 2006;42(2):129–72.

[129] von Ellenrieder KD, Parker K, Soria J. Flow structures behind a heaving andpitching finite-span wing. J Fluid Mech 2003;490:129–38.

[130] Godoy-Diana R, Aider J-L, Wesfried JE. Transitions in the wake of a flappingfoil. Phys Rev E 2008;77:016308-1-016308-5.

[131] Godoy-Diana R, Marias C, Aider J-L, Wesfried JE. A model for the symmetrybreaking of the reverse Benard-von Karman vortex street produced by aflapping foil. J Fluid Mech 2009;622:23–32.

[132] Lee J-S, Kim J-H, Kim C. Numerical study on the unsteady-force-generationmechanism of insect flapping motion. AIAA J 2008;46(7):1835–48.

[133] Anderson JM, Streitlin K, Barrett DS, Triantafyllou MS. Oscillating foils ofhigh propulsive efficiency. J Fluid Mech 1998;360:41–72.

[134] Read DA, Hover FS, Triantafyllou MS. Forces on oscillating foils forpropulsion and maneuvering. J Fluid Struct 2003;17:163–83.

[135] Hover FS, Haugsdal Ø, Triantafyllou MS. Effect of angle of attack profiles inflapping foil propulsion. J Fluid Struct 2004;19:37–47.

[136] Schouveiler L, Hover FS, Triantafyllou MS. Performance of flapping foilpropulsion. J Fluid Struct 2005;20:949–59.

[137] Lai JCS, Platzer MF. Jet characteristics of a plunging airfoil. AIAA J1999;37(12):1529–37.

[138] Cleaver DJ, Wang Z, Gursul I. Vortex mode bifurcation and lift force of aplunging airfoil at low Reynolds numbers. In: 48th AIAA aerospace sciencesmeeting including the new horizons forum and aerospace exposition. 2010,AIAA 2010-390.

[139] Lewing GC, Haj-Hariri H. Modeling thrust generation of a two-dimensionalheaving airfoil in a viscous flow. J Fluid Mech 2003;492:339–62.

[140] Lua KB, Lim TT, Yeo KS, Oo GY. Wake-structure formation of a heaving two-dimensional elliptic airfoil. AIAA J 2007;45(7):1571–83.

[141] Bohl DG, Koochesfahani MM. MTV measurements of the vortical field in thewake of an airfoil oscillating at high reduced frequency. J Fluid Mech2009;620:63–88.

[142] von Ellenrieder KD, Pothos SPIV. measurements of the asymmetric wake of atwo dimensional heaving hydrofoil. Exp Fluids 2008;44:733–45.

[143] Jones AR, Babinsky H. Three-dimensional effects of a waving wing. In: 48thAIAA aerospace sciences meeting including the new horizons forum andaerospace exposition. 2010, AIAA 2010-551.

[144] Calderon DE, Wang Z, Gursul I. Lift enhancement of a rectangular wingundergoing a small amplitude plunging motion. In: 48th AIAA aerospacesciences meeting including the new horizons forum and aerospaceexposition. 2010, AIAA 2010-386.

[145] Ol MV et al. Unsteady aerodynamics for micro air vehicles. RTO AVT-149Report, 2009, p. 1–145.

[146] Ol MV, Bernal L, Kang C, Shyy W. Shallow and deep dynamic stall forflapping low Reynolds number airfoils. Exp Fluids 2009;46:883–901.

[147] Kang C, Aono H, Trizila P, Baik YS, Rausch JM, Bernal LP, et al. Modeling ofpitching and plunging airfoils at Reynolds number between 1�104 and6�104. In: Proceeding of 39th AIAA fluid dynamics conference. 2009, AIAA2009-4100.

[148] Baik YS, Rausch JM, Bernal LP, Ol MV. Experimental investigation of pitchingand plunging airfoils at Reynolds number between 1�104 and 6�104. In:39th AIAA fluid dynamics conference 2009, AIAA 2009-4030.

[149] Menter FR. Two equation eddy-viscosity turbulence models for engineeringapplications. AIAA J 1993;32:269–89.

[150] Baik YS, Raush J, Bernal L, Shyy W, Ol M. Experimental Study of governingparameters in pitching and plunging airfoil at low Reynolds number. In:48th AIAA aerospace sciences meeting including the new horizons forumand aerospace exposition. 2010, AIAA 2010-388.

[151] Visbal MR, Gordiner RE, Galbraith MC. High-fidelity simulations of movingand flexible airfoils at low Reynolds numbers. Exp Fluids 2009;46:903–22.

dx.doi.org/10.1098/rsif.2009.0120

dx.doi.org/10.1098/rsif.2009.0120

dx.doi.org/10.1007/s10409-009-0306-y.3d


[152] Radespiel R, Windte J, Scholz U. Numerical and experimental flow analysisof moving airfoils with laminar separation bubbles. AIAA J 2007;45(6):1346–56.

[153] Sane SP, Dickinson MH. The control of flight force by a flapping wing: liftand drag production. J Exp Biol 2001;204:2607–26.

[154] Birch JM, Dickinson MH. Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 2001;412:729–33.

[155] Birch JM, Dickson WB, Dickinson MH. Force production and flow structureof the leading edge vortex on flapping wings. J Exp Biol 2004;207:1063–72.

[156] Dickinson MH. The effects of wing rotation on unsteady aerodynamicperformance at low Reynolds numbers. JEB 1994;192:179–206.

[157] Dickson WB, Dickinson MH. The effect of advance ratio on the aerodynamicsof revolving wings. J Exp Biol 2004;207:4269–81.

[158] Fry Sayaman, Dickinson MH. The aerodynamics of hovering flight inDrosophila. J Exp Biol 2005;208:2303–18.

[159] Altshuler DL, Dickson WB, Vance JT, Roberts SP, Dickinson MH. Short-amplitude high-frequency wing strokes determine the aerodynamics ofhoneybee flight. Proc Natl Acad Sci 2005;102(50):18213–8.

[160] Sunada S, Takahashi H, Hattori T, Yasuda K, Kawachi K. Fluid-dynamiccharacteristics of a bristled wing. J Exp Biol 2002;205:2737–44.

[161] Nagai H, Isogai K, Fujimoto T, Hayase T. Experimental and numerical studyof forward flight aerodynamics of insect flapping wing. AIAA J2009;47(3):730–42.

[162] Wang ZJ, Birch MB, Dickinson MH. Unsteady forces and flows in lowReynolds number hovering flight: two-dimensional computations vs roboticwing experiments. J Exp Biol 2004;207:461–74.

[163] Lua KB, Lim TT, Yeo KS. Aerodynamic forces and flow fields of a two-dimensional hovering wing. Exp Fluids 2008;45:1047–65.

[164] Wang ZJ. Two dimensional mechanism for insect hovering. Phys Rev Lett2000;85(10):2216–9.

[165] Wang ZJ. Vortex shedding and frequency selection in flapping flight. J FluidMech 2000;410:323–41.

[166] Wang ZJ. The role of drag in insect hovering. J Exp Biol 2004;207:4147–55.

[167] Bos FM, Lentik D, van Oudheusden BW, Bijl H. Influence of wing kinematicson aerodynamic performance in hovering insect flight. J Fluid Mech2008;594:341–68.

[168] Kurtulus DF, David L, Farcy A, Alemdaroglu N. Aerodynamic characteristicsof flapping motion in hover. Exp Fluids 2008;44:23–36.

[169] Hong YS, Altman A. Streamwise vorticity in simple mechanical flappingwings. J Aircraft 2007;44(5):1588–97.

[170] Hong YS, Altman A. Lift from spanwise flow in simple flapping wings. JAircraft 2008;45(4):1206–16.

[171] Isaac KM, Rolwes J, Colozza A. Aerodynamic of a flapping and pitching wingusing simulations and experiments. AIAA J 2008;46(6):1505–15.

[172] Sun M, Lan SL. A computational study of the aerodynamic forces and powerrequirements of dragonfly hovering. J Exp Biol 2004;207:1887–901.

[173] Yamamoto M, Isogai K. Measurement of unsteady fluid dynamic forces for amechanical dragonfly model. AIAA J 2005;43(12):2475–80.

[174] Maybury WJ, Lehmann F-O. The fluid dynamics of flight control bykinematic phase lag variation between two robotic insect wings. J Exp Biol2004;207:4707–26.

[175] Lu Y, Shen GX, Su WH. Flow visualization of dragonfly hovering via anelectromechanical model. AIAA J 2007;45(3):615–23.

[176] Wang JK, Sun M. A computational study of the aerodynamic and forewing-hindwing interaction of a model dragonfly in forward flight. J Exp Biol2005;208:3785–804.

[177] Dong H, Liang Z. Effects of ipsilateral wing-wing interactions on aero-dynamic performance of flapping wings. In: Proceedings of 48th AIAAaerospace sciences meeting including the new horizons forum andaerospace exposition. 2010, AIAA 2010-71.

[178] Warkentin J, DeLaurier J. Experimental aerodynamic study of tandemflapping membrane wings. J Aircraft 2007;44(5):1653–61.

[179] Broering TM, Lian Y. Numerical study of two flapping airfoils in tandemconfiguration. In: 48th AIAA aerospace sciences meeting including the newhorizons forum and aerospace exposition. 2010, AIAA 2010-865.

[180] Wang ZJ, Russell D. Effect of forewing and hindwing interactions onaerodynamic forces and power in hovering dragonfly flight. Phys Rev Lett2007;99:14801.1–4.

[181] Young J, Lai JCS, Germain C. Simulation and parameter variation offlapping-wing motion based on dragonfly hovering. AIAA J 2008;46(4):918–24.

[182] Zhang J, Lu X-Y. Aerodynamic performance due to forewing and hindwinginteraction in gliding dragonfly flight. Phys Rev E 2009;80:017302.1–4.

[183] Lentink D, Gerritsma M. Influence of airfoil shape of performance in insectflight. In: Proceeding of 33rd AIAA fluid dynamics conference and exhibit.2003, AIAA 2003-3447.

[184] Usherwood JR, Ellington CP. The aerodynamics of revolving wings I. Modelhawkmoth wings. J Exp Biol 2002;205:1547–64.

[185] Usherwood JR, Ellington CP. The aerodynamics of revolving wings II.Propeller force coefficients from mayfly to quail. J Exp Biol 2002;205:1565–76.

[186] Luo G, Sun M. The effects of corrugation and wing planform on theaerodynamic force production of sweeping model insect wings. Acta MechSin 2005;21:531–41.

[187] Vargas A, Mittal R, Don H. A computational study of the aerodynamicperformance of a dragonfly wing section in gliding flight. Bioinsp Biomim2008;3(026004):1–13.

[188] Kim W-K, Ko JH, Park HC, Byun D. Effects of corrugation of the dragonflywing on gliding performance. J Theor Biol 2009;260:523–30.

[189] Levy D-E, Seifert A. Simplified dragonfly airfoil aerodynamics at Reynoldsnumbers below 8000. Phys Fluids 2009;21:071901-1–17.

[190] Althsuler DL, Dudley R, Ellington CP. Aerodynamic forces of revolvinghummingbird wings and wing models. J Zool Lond 2004;264:327–32.

[191] Ansari SA, Knowles K, Zbikowski R. Insectlike flapping wings in the hover.Part 2: Effect of wing geometry. J Aircraft 2008;45(6):1976–90.

[192] Green MA, Smith AJ. Effects of three-dimensionality on thrust production bya pitching panel. J Fluid Mech 2008;615:211–20.

[193] Buchholz JH, Smits AJ. The wake structure and thrust performance of a rigidlow-aspect-ratio pitching panel. J Fluid Mech 2008;603:331–65.

[194] Yates GT. Optimum pitching axes in flapping wing propulsion. J Theor Biol1985;120:255–76.

[195] Ansari SA, Knowles K, Zbikowski R. Insectlike flapping wings in the hover.Part 1: Effect of wing kinematics. J Aircraft 2008;45(6):1945–54.

[196] Hsieh CT, Chang CC, Chu CC. Revisiting the aerodynamics of hovering flightusing simple models. J Fluid Mech 2009;623:121–48.

[197] Oyama A, Okabe Y, Shimoyama K, Fujii K. Aerodynamic multi-objectivedesign exploration of a flapping airfoil using a Navier–Stokes solver.J Aerospace Comput Inf Commun 2009;6:256–70.

[198] Trizila P, Kang C, Visbal M, Shyy W. Unsteady fluid physics and surrogatemodeling of low Reynolds number, flapping airfoils. In: 38th AIAA fluiddynamics conference and exhibit. 2008, AIAA-2008-3821.

[199] Trizila P, Kang C-K, Visbal M, Shyy W. A surrogated model approach in 2-Dversus 3-D flapping wing aerodynamic analysis. In: 12th AIAA/ISSMOmultidisciplinary analysis and optimization conference. 2008. AIAA 2008-5914.

[200] Madsen JI, Shyy W, Haftka RT. Response surface techniques for diffusershape optimization. AIAA J 2000;38:1512–8.

[201] Shyy W, Papila N, Vaidyanathan R, Tucker PK. Global design optimization foraerodynamics and rocket propulsion components. Prog Aerosp Sci2001;37:59–118.

[202] Quiepo N, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK. Surrogate-based analysis and optimization. Prog Aerosp Sci 2005;41:1–25.

[203] Kamatoki R, Thakur S, Wright J, Shyy W. Validation of a new parallel all-speed CFD code in a rule-based framework for multidisciplinary applica-tions. In: 36th AIAA fluid dynamics conference an exhibit. 2006, AIAA2006–3063.

[204] Chong MS, Perry AE, Cantwell BJ. A general classification of three-dimensional flow fields. Phys Fluids A 1990;2:765–77.

[205] Togashi F, Ito Y, Murayama M, Nakahashi K, Kato T. Flow simulation offlapping wings of an insect using overset unstructured grid. In: 15th AIAAcomputational fluid dynamics conference. 2001, AIAA Paper 2001–2619.

[206] Zuo D, Chen W, Peng S, Zhang W. Modeling and simulation study of aninsect-like flapping-wing micro air vehicle. Adv Robotics 2006;20(7):807–24.

[207] Zuo D, Chen W, Peng S, Zhang W. Numerical simulation of flapping-winginsect hovering flight at unsteady flow. Int J Numer Meth Fluids2007;53:1801–17.

[208] Aono H, Liu H. Vortical structure and aerodynamics of hawkmoth hovering.J Biomech Sci Eng 2006;1(1):234–45.

[209] Liu YP, Sun M. Wing kinematics measurement and aerodynamics ofhovering droneflies. J Exp Biol 2008;211:2014–25.

[210] Liu YP, Sun M. Wing kinematics measurement and aerodynamic force andmoments computation of hovering hoverfly. In: Proceedings of internationalconference on bioinformatics and biomedical engineering. 2007.p.452–5.

[211] Gopalakrishnan P, Tafti DK. A parallel boundary fitted dynamic mesh solverfor applications to flapping flight. Comput Fluids 2009;38:1592–607.

[212] Yu X, Sun M. A computational study of the wing–wing and wing-bodyinteractions of a model insect. Acta Mech Sin 2009;25(4):421–31.

[213] Liu H. Integrated modeling of insect flight: from morphology, kinematics toaerodynamics. J. Comput Phys 2009;228(2):439–59.

[214] Moffatt HK, Tsinober A. Helicity in laminar and turbulent flow. Annu RevFluid Mech 1992;24:281–312.

[215] Tang J, Viieru D, Shyy W. Effect of Reynolds number and flapping kinematicson hovering aerodynamics. AIAA J 2008;46(4):967–76.

[216] De Vries O. On the theory of the horizontal-axis wind turbine. Annu RevFluid Mech 1983;15:77–96, doi:10.1146/annurev.fl.15.010183.000453.

[217] McCroskey WJ, Carr LW, McAlister KW. Dynamic stall experiments onoscillating airfoils. AIAA J 1976;14(1):57–63.

[218] Knowles K, Wilkins PC, Ansari SA, Zbikowski RW. Integrated computationaland experimental studies of flapping-wing micro air vehicle aerodynamics.In: Proceedings of 3rd international symposium on integrating CFD andexperiments in aerodynamics. 2007, p. 1–15.

[219] Liu H, Kawachi K. Leading-edge vortices of flapping and rotary wings at lowReynolds number. In: Mueller Thomas J, editor. Fixed and flapping wingaerodynamics for micro air vehicle applications. Vol. 195, AIAA ProgAstronaut Aeronaut 2001. p. 275–285.

[220] Drazin PG, Reid WH. Hydrodynamic Stability. New York: CambridgeUniversity Press; 1981.

dx.doi.org/10.1146/annurev.fl.15.010183.000453


[221] Lentink D, Dickinson MH. Rotational accelerations stabilize leading edgevortices on revolving fly wings. J Exp Biol 2009;212:2705–19.

[222] McCroskey WJ, McAlister KW, Carr LW, Pucci. An experimental study ofdynamic stall on advanced airfoil section. NASA TM-84245 1982.

[223] Daniel TL, Combes SA. Flexible wings and fins: bending by inertial or fluid-dynamic forces?Integr Comp Biol 2002;42:1044–9

[224] Combes SA. Wing flexibility and design for animal flight. Ph.D Dissertation,University of Washington, 2002.

[225] Combes SA, Daniel TL. Into thin air: contributions of aerodynamic andinertial–elastic forces to wing bending in the hawkmoth Manduca sexta.J Exp Biol 2003;23:2999–3006.

[226] Mountcastle AM, Daniel TL. Aerodynamic and functional consequences ofwing compliance. Exp Fluids 2009;46:873–82.

[227] Zhu Q. Numerical simulation of a flapping foil with chordwise or spanwiseflexibility. AIAA J 2007;45(10):2448–57.

[228] Yamamoto I, Terada Y, Nagamatu T, Imaizumi Y. Propulsion system withflexible/rigid oscillating fin. IEEE J Ocean Eng 1995;20(1):23–30.

[229] Prempraneerach P, Hover FS, Triantafyllou MS. The effect of chordwiseflexibility on the thrust and efficiency of a flapping foil. In: Proceedings of13th international symposium unmanned untethered submersible technol-ogy, 2003. p. 1–10.

[230] Tang J, Viieru D, Shyy W. A study of aerodynamics of low Reynolds numberflexible airfoils. In: AIAA 37th fluid dynamics conference and exhibit. 2007,AIAA 2007–4212.

[231] Chandar DDJ, Damodaran M. Computational fluid-structure interaction of aflapping wing in free flight using overlapping grids. In: 27th AIAA appliedaerodynamics conference, 2009. AIAA 2009–3849.

[232] Pederzani J, Haj-Hariri H. Numerical analysis of heaving flexible airfoils in aviscous flow. AIAA J 2006;44(11):2773–9.

[233] Chaithanya MR, Venkatraman K. Hydrodynamic propulsion of a flexible foil.In: 12th Asian congress of fluid mechanics, 2008.

[234] Chaithanya MR, Venkatraman K. Hydrodynamic propulsion of a flexible foilundergoing pitching motion. In: Proceedings of the 10th annual CFDsymposium, 2008.

[235] Gopalakrishnan P. Unsteady aerodynamic and aeroelastic analysis offlapping flight. Ph.D. Dissertation. Blacksburg, VA: Department of Mechan-ical Engineering, Virginia Polytechnic Institute and State University; 2008.

[236] Gulcat U. Propulsive force of a flexible flapping thin airfoil. J Aircraft2009;46(2):465–73.

[237] Miao J-M, Ho M-H. Effect of flexure on aerodynamic propulsive efficiency offlapping flexible airfoil. J Fluid Struct 2006;22:401–9.

[238] Toomey J, Eldredge JD. Numerical and experimental study of the fluiddynamics of a flapping wing with low order flexibility. Phys Fluids2008;20(073603):1–10.

[239] Poirel D, Harris Y, Benaissa A. Self-sustained aeroelastic oscillations of aNACA0012 airfoil at low-to-moderate Reynolds numbers. J Fluid Struct2008;24:700–19.

[240] Heathcote S, Martin D, Gursul I. Flexible flapping airfoil propulsion at zerofreestream velocity. AIAA J 2004;42(11):2196–204.

[241] Michelin S, Smith SGL. Resonance and propulsion performance of a heavingflexible wing. Phys Fluids 2009;21:071902-1–15.

[242] Du G, Sun M. Effects of unsteady deformation of flapping wing on itsaerodynamic forces. Appl Math Mech 2008;29(6):731–43.

[243] Miller LA, Peskin CS. Flexible clap and fling in tiny insect flight. J Exp Biol2009;212:3076–90.

[244] Ellington CP. The aerodynamics of hovering insect flight. IV. Aerodynamicmechanics. Phil Trans R Soc Lond B 1984;305:79–113.

[245] Zhao L, Huang Q, Deng X, Sane S. The effect of chord-wise flexibility on theaerodynamic force generation of flapping wings: experimental studies.In: Proceedings of IEEE international conference on robotics and automation,2009. p. 4207–12; also J R Soc Interface, 2009, doi:10.1098/rsif.2009.0200.

[246] Aono H, Tang J, Chimakurthi SK, Cesnik CES, Liu H, Shyy W. Spanwise andchordwise flexibility effects on aerodynamics of plunging wings. under review.

[247] Belytschko T, Hsieh BJ. Nonlinear transient finite element analysis withconvected coordinates. Int J Numer Methods Eng 1973;7(3):255–71.

[248] Belytschko T, Schwer L, Kelin MJ. Large displacement, transient analysis ofspace frame. Int J Numer Methods Eng 1977;11(1):65–84.

[249] Liu P, Bose N. Propulsive performance from oscillating propulsors withspanwise flexibility. Philos R Soc A - Math Phys 1997;453(1963):1763–70.

[250] Heathcote S, Wang Z, Gursul I. Effect of spanwise flexibility on flapping wingpropulsion. J Fluid Struct 2008;24:183–99.

[251] Chimakurthi SK, Tang J, Palacios R, Cesnik CES, Shyy W. Computationalaeroelasticity framework for analyzing flapping wing micro air vehicles.AIAA J 2009;47(8):1865–78.

[252] Aono H, Chimakurthi SK, Cesnik CES, Liu H, Shyy W. Computationalmodeling of spanwise flexibility effects on flapping wing aerodynamics.In: 47th AIAA aerospace sciences meeting including the new horizons forumand aerospace exposition, 2009. AIAA 2009–1270.

[253] Palacios R, Cesnik CES. A geometrically nonlinear theory of active compositebeams with deformable cross-sections. AIAA J 2008;46(2):439–50.

[254] Watman D, Furukawa T. A parametric study of flapping wing performanceusing a robotic flapping wing. In: Proceedings of IEEE internationalconference on robotics and automation. 2009. p. 3638–43.

[255] Wood R. Design fabrication and analysis of a 3 dof, 3 cm flapping-wing mav.In: IEEE/RSJ international conference on proceeding in intelligent robots andsystems, 2007. p. 1576–81.

[256] Hui H, Kumar AG, Abate G, Albertani R. An experimental study of flexiblemembrane wings in flapping flight. In: 47th AIAA aerospace sciencesmeeting including the new horizons forum and aerospace exposition, 2009.AIAA 2009-876.

[257] Shkarayev S, Silin D, Abate G, Albertani R. Aerodynamics of camberedmembrane flapping wing. In: 48th AIAA aerospace sciences meetingincluding the new horizons forum and aerospace exposition, 2010, AIAA2010-58.

[258] Kim D-K, Han J-H, Kwon K-J. Wind tunnel tests for a flapping wing modelwith a changeable camber using macro-fiber composite actuators. SmartMater Struct 2009;18:024008-1–8.

[259] Mueller D, Bruck HA, Gupta SK. Measurement of thrust and lift forcesassociated with drag of compliant flapping wing for micro air vehicles usinga new test stand design. Exp Mech 2009:1–11, doi:10.1007/s11340-009-9270-5.

[260] Hamamoto M, Ohta Y, Hara K, Hisada T. Application of fluid-structureinteraction analysis to flapping flight of insects with deformable wings. AdvRobot 2007;21(1-2):1–21.

[261] Luo H, Yin B, Dai H, Doyle J. A 3D computational study of the flow-structureinteraction in flapping flight. In: 48th AIAA aerospace sciences meetingincluding the new horizons forum and aerospace exposition, 2010, AIAA2010-556.

[262] Aono H, Chimakurthi SK, Wu P, Sallstrom E, Stanford BK, Cesnik CES, et al.A computational and experimental study of flexible flapping wingaerodynamics. In: 48th AIAA aerospace sciences meeting including thenew horizons forum and aerospace exposition, 2010, AIAA 2010-554.

[263] Singh B, Chopra I. Insect-based hover-capable flapping wings for micro airvehicles: experiments and analysis. AIAA J 2008;46(9):2115–35.

[264] Agrawal A, Agrawal SK. Design of bio-inspired flexible wings for flapping-wing micro-sized air vehicle applications. Adv Robot 2009;23(7-8):979–1002.

[265] Jones KD, Platzer MF. Experimental investigation of the aerodynamiccharacteristics of flapping-wing micro air vehicles. In: 41st aerospacesciences meeting & exhibit, 2003, AIAA 2003-0418.

[266] Jones KD, Bradshaw CJ, Papadopoulos J, Platzer MF. Improved performanceand control of flapping-wing propelled micro air vehicles. In: 42ndaerospace sciences meeting, 2004, AIAA 2004-0399.

[267] Kaya M, Tuncer IH, Jones KD, Platzer MF. Optimization of flapping motion ofairfoils in biplane configuration for maximum thrust and/or efficient. In:45th aerospace sciences meeting and exhibit, 2007, AIAA 2007-484.

[268] Wu P, Ifju P, Standford BK, Sallstrom E, Ukeiley L, Love R, et al.A multidisplinary experimental study of flapping wing aeroelasticity inthrust production. In: 50th AIAA/ASME/ASCE/AHS/ASC structures, structuraldynamics, and materials conference, 2009, AIAA 2009–2413.

[269] Sallstrom E, Ukeiley L, Pin W, Ifju P. Three-dimensional averaged flow andwing deformation around flexible flapping wings. In: 39th AIAA fluiddynamics conference, 2009, AIAA 2009–3813.

[270] Orlowski C, Girard A, Shyy W. Derivation and simulation of the nonlineardynamics of a flapping wing micro-air vehicle. In: The European micro aerialvehicle conference and flight competition 2009. p. 1–9.

[271] Oppenheimer MW, Doman DB, Sigthorsson DO. Dynamics and control of abiomimetic vehicle using biased wingbeat forcing functions: part 1aerodynamic model. In: 48th AIAA aerospace sciences meeting includingthe new horizons forum and aerospace exposition, 2010, AIAA 2010–1023.

[272] Doman DB, Oppenheimer MW, Sigthorsson DO. Dynamics and control of abiomimetic vehicle using biased wingbeat forcing functions: part 2Controller. In: 48th AIAA aerospace sciences meeting including the newhorizons forum and aerospace exposition, 2010, AIAA 2010–1024.

[273] Katz J, Weihs D. Hydrodynamic propulsion by large amplitude oscillation ofan airfoil with chordwise flexibility. J Fluid Mech 1978;88(3):485–97.

[274] Weihs D. Stability versus maneuverability in aquatic locomotion. IntegComp Biol 2002;42:127–34.

doi:10.1098/rsif.2009.0200

dx.doi.org/10.1007/s11340-009-9270-5.3d

dx.doi.org/10.1007/s11340-009-9270-5.3d

Other - Recent Progress in Flapping Wing Aerodynamics and Aeroelasticity

Documents

Transcript of Other - Recent Progress in Flapping Wing Aerodynamics and Aeroelasticity