passive maintenance of high angle of attack and its lift...

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INTRODUCTIONThe unsteady aerodynamic effects that generate lift forces in insectflapping flight depend on the kinematic pattern of the wing pitchingmotion (Dickinson and Gotz, 1993; Ellington et al., 1996; Dickinsonet al., 1999; Birch and Dickinson, 2003). The fundamental featuresof the kinematic patterns are the maintenance of a high angle ofattack of the wing during the flapping translation and the wingrotation during the stroke reversal (see Fig.1). The high angle ofattack generates a vortex on the leading edge of the wing (leadingedge vortex), which generates a large and instantaneous lift forceon the wing. Since the leading edge vortex is reproduced during thenext half stroke before the previous vortex separates from the wing,sufficient lift is provided continuously in insect flight. The lift duringthe stroke reversal is enhanced by circulation effects resulting fromthe wing rotation. It is therefore important to understand how thesefundamental features of the pitching motion are produced.

Although insects regulate the timing of the wing rotation(Dickinson et al., 1993), it seems that the inertial force can causewing rotation. Ennos (Ennos, 1988b) has suggested using the rigidpendulum model for a dipteran wing that the inertial force of thewing mass is sufficient to account for much of the rotation. Bergouet al. (Bergou et al., 2007) also studied the inertial cause of the wingrotation in some different insects (dragonflies, fruit fly andhawkmoth) using the flapping wing section model andcomputational fluid dynamics and found that the inertial force ofthe wing mass and the added mass from air is sufficient to causethe wing rotation in all tested insects.

Morphological studies on the dipteran wings have shown thatthere exists high torsional flexibility concentrated on the wing basalregion (Ennos, 1987; Ennos, 1988a). This flexibility might prevent

insects from transmitting the active torsional force applied by theirown muscle to the outer wing. It has been suggested that theaerodynamic pressure is sufficient to produce the observed torsionusing the static linear relation between the assumed aerodynamicpressure and the torsional stiffness of the wing (Ennos, 1988a). Inour previous study (Ishihara et al., 2009), we used the flapping wingsection model with a spring to model the wing torsional flexibility,and the finite element method to analyze the motion of the modelwing interacting with the surrounding fluid. Under the dynamicsimilarity between the crane fly flight and our model flight, ourmodel wing passively maintained a high angle of attack during theflapping translation and rotated quickly upon the stroke reversalwithout any prescribed pitching motion. The lift force generated bysuch passive pitching was comparable with but smaller than theweight of the crane fly. This could be attributed to the looselyattached leading edge vortex on the wing that resulted from the longwing chord travel of the crane fly for the two-dimensionalsimulation. In addition, it was not clear what forces are importantfor the production of the features of pitching motion. Under theassumption of the passivity of the wing pitching motion during theflapping translation, which was suggested by our previous study,the equilibrium between the elastic reaction force due to the wingtorsion and the aerodynamic pressure would be a possiblemechanism for the maintenance of the high angle of attack, sincethe acceleration of the stroke during the flapping translation is small.Our purpose in the present study is to provide more evidence forthe passivity of the maintenance of a high angle of attack duringthe flapping translation and its sufficient lift generation.

The elastic wing and surrounding air motions are unsteady andcoupled. Some studies such as those of Combes and Daniel (Combes

The Journal of Experimental Biology 212, 3882-3891Published by The Company of Biologists 2009doi:10.1242/jeb.030684

Passive maintenance of high angle of attack and its lift generation during flappingtranslation in crane fly wing

D. Ishihara*, Y. Yamashita, T. Horie, S. Yoshida and T. NihoKyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka, Japan

*Author for correspondence ([email protected])

Accepted 26 August 2009

SUMMARYWe have studied the passive maintenance of high angle of attack and its lift generation during the crane fly’s flapping translationusing a dynamically scaled model. Since the wing and the surrounding fluid interact with each other, the dynamic similaritybetween the model flight and actual insect flight was measured using not only the non-dimensional numbers for the fluid (theReynolds and Strouhal numbers) but also those for the fluid–structure interaction (the mass and Cauchy numbers). A differencewas observed between the mass number of the model and that of the actual insect because of the limitation of available solidmaterials. However, the dynamic similarity during the flapping translation was not much affected by the mass number since theinertial force during the flapping translation is not dominant because of the small acceleration. In our model flight, a high angleof attack of the wing was maintained passively during the flapping translation and the wing generated sufficient lift force tosupport the insect weight. The mechanism of the maintenance is the equilibrium between the elastic reaction force resulting fromthe wing torsion and the fluid dynamic pressure. Our model wing rotated quickly at the stroke reversal in spite of the reducedinertial effect of the wing mass compared with that of the actual insect. This result could be explained by the added mass fromthe surrounding fluid. Our results suggest that the pitching motion can be passive in the crane fly’s flapping flight.

Key words: insect flight, lift, fluid–structure interaction, passive pitching, dynamic similarity law, Reynolds number, Strouhal number, mass number,Cauchy number.

THE JOURNAL OF EXPERIMENTAL BIOLOGY

3883Passive attack angle maintenance and lift

and Daniel, 2006), Ishihara et al. (Ishihara et al., 2009) and Vanellaet al. (Vanella et al., 2009) addressed such problems directly.Combes and Daniel used actual insect wings. Ishihara et al. andVanella et al. used computer models of wings. By contrast, in thepresent study, we developed a dynamically scaled model of cranefly flight. Since the wing and the surrounding fluid interact witheach other, the dynamic similarity should be measured in terms ofnot only the Reynolds and Strouhal numbers but also the mass andCauchy numbers. A difference was observed between the massnumber of the proposed model and that of the actual insect becauseof the limitation of available solid materials. However, the dynamicsimilarity during the flapping translation was not much affected bythe mass number since the inertial force during the flappingtranslation is not dominant because of the small acceleration.

Our model wing maintained a high angle of attack during theflapping translation, which was similar to that for the actual insectflight. The maintenance was achieved by the equilibrium betweenthe elastic reaction due to the wing torsion and the dynamic fluidpressure. Our model wing also rotated quickly during the strokereversal. This result was surprising since the inertial effect of ourmodel wing mass is very small compared with that of the actualinsect wing mass. This result might be explained by the added massfrom the surrounding fluid. In the flight of insects such as dragonfliesand fruitfly, which have relatively light wings, the added mass forthe wing is comparable to the wing mass during the stroke reversal(Bergou et al., 2007). The crane fly used here also has light wingsthat are a few percent of the body weight. The added mass effecton our model wing was equivalent to that on the actual insect sincethe Reynolds and Strouhal numbers of our model wing flight equalthose of the actual insect flight. Therefore the reduced inertia of themodel wing mass would not change the order of the rotational force.The mean lift coefficient of our model wing flight was close to thatof the previous studies (Dickinson et al., 1999; Usherwood andEllington, 2002b) and it was sufficient for the actual insect to hover.

MATERIALS AND METHODSLumped torsional flexibility model

Our model wing is based on the lumped torsional flexibility modelas a simplified dipteran wing (Ishihara et al., 2009). The lumpedtorsional flexibility model is shown in Fig.2.

One of typical features of the insect wing flexibility is the wingplane twist. The wing plane twist provides the modulation of thepitch angle of the wing plane along the wing length. Although theflexibility of the dipteran wing concentrates on the wing basal region(Ennos, 1987; Ennos, 1988a), the dipteran wing also shows the wingplane twist during its flapping flight. The actual angle of the wingplane twist is typically 10–30degrees (Ellington, 1984b; Ennos,1989; Walker et al., 2009). The angle per unit length is very small

compared with the pitch angle in the wing basal region. Therefore,the fluid force on the wing plane might not be much affected bythe wing plane twist. Indeed Du and Sun (Du and Sun, 2008) haveshown using computational fluid dynamics that the aerodynamicforces are not much affected by the considerable wing plane twist.Therefore the flat-plate wing in the lumped torsional flexibilitymodel is appropriate for our purpose described in the Introduction.

Dynamically scaled modelModeling of the wing flexibility

First, we describe the implementation of the spring in the lumpedtorsional flexibility model. The stiff leading edge and the wingsurface reinforced by the network of veins (see Fig.3A) arerepresented, respectively, by a rigid beam and a rigid plate (seeFig.3B). The rigid beam and the rigid plate are connected by anarrow flexible plate. Note that the rigid leading edge and the rigidwing surface (flat-plate wing) are commonly used in dynamicallyscaled models (Birch and Dickinson, 2003; Usherwood andEllington, 2002a) and computer simulation models (Liu et al., 1998;Sun and Tang, 2002; Miller and Peskin, 2005; Ramamurti andSandberg, 2002; Wang et al., 2004). The narrow flexible plate worksas the plate spring, which is an implementation of the spring in thelumped torsional flexibility model. The reason we employ the platespring is that its torsional stiffness is easy to control by changingthe plate thickness, length and width as described below.

Next, we describe the definition of the wing torsional stiffness. Letus consider the wing with the moment Mq applied around thelongitudinal axis. In the model wing, the narrow flexible plate withthe upper end fixed and moment Mq applied at the lower end generatesslope angle q at the lower end (see Fig.3B). Note that the angulardisplacement of the pitching motion (the pitch angle) for the modelwing surface is equal to q because the model wing surface iscontinuously connected to the narrow flexible part. Under theEuler–Bernoulli beam assumption, q is related to Mq by the relationMqGMq, where GM is the torsional stiffness and is given as:

GM EFP IFP / cFP (1a)

and

IFP lFP tFP3 / 12, (1b)

where EFP is the Young’s modulus, cFP is the length in the chorddirection, IFP is the second moment of the sectional area, lFP is the

Upstroke

DownstrokeInsect body

Fig.1. Typical kinematic patterns of the wing pitching motion. The graylines indicate the wing chord, the black circles indicate the leading edgeand the arrows indicate the direction of chord travel.

Angular displacementof pitching motion

or pitch angle

Torsionalflexibility

Flapping motion

Axis of torsion

Wing chord

Fig.2. Torsional flexibility of the insect wing and the passive pitching wingmodel. The solid line, the spring and the roller support with the arrowrepresent the wing chord, the torsional flexibility concentrated on the wingbasal region, and the active flapping motion, respectively.


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longitudinal length and tFP is the thickness of the flexible plate. Thetorsional flexibilities of actual insect wings were investigated byEnnos (Ennos, 1988a), where the torsional stiffness, GI, was givenby the macroscopic relation MqGIq (see Fig.3A). In the presentstudy, GM is determined such that the Cauchy number Ch given byGM equals that given by GI. Note that Ch describes the ratio betweenthe fluid dynamic pressure and elastic reaction force. Thus, thetorsional flexibility of the present model wing is equivalent to thatof the actual insect wing from the point of view of the dynamicsimilarity under the above definition of the wing torsional stiffness.

Shape of the model wingThe shape of the model wing is geometrically similar to the cranefly wing. As shown in Fig.4 it was made using the plane view ofthe crane fly wing as given in Ennos (Ennos, 1988a). The aspectratio of the model wing is equivalent to that for the crane fly. Theaspect ratio is defined as rA2Lw/c, where Lw is the longitudinallength of the wing (one wing) and c is the average wing chord length.

Flapping motion of the model wingFig.5 is a schematic diagram of the flapping motion of the modelwing in the stroke plane. The flapping motion is similar to thesinusoidal motion, but has higher accelerations and decelerationsduring the stroke reversal and a constant velocity during the middleof each half stroke. This feature of the flapping motion was pointedout by Ellington (Ellington, 1984b) and Ennos (Ennos, 1988b). Thesame feature has also be observed in some other research (Ennos,1989; Liu and Sun, 2008). For the sake of simplicity, the angulardisplacement of the flapping motion or the flapping angle, (t),approximates the sinusoidal motion as (t)�/2 sin2ft, where �is the stroke angle and f is the flapping frequency. Under thisapproximation the maximum speed of the flapping motion of theleading edge center is given as:

Vw,max � f Lw / 2. (2)

Note that, in the experiment described below, (t) given by thestepping motor has the above feature. In the coordinate systemshown in Fig.5, we define the up-stroke as the half stroke when thewing flaps counterclockwise around the stroke axis, whereas thedown-stroke is defined as the half stroke when the wing flapsclockwise around the stroke axis.

Non-dimensional numbers of the actual insect flightWe measured the dynamic similarity between our dynamicallyscaled model and the insect flight using the non-dimensionalnumbers for the fluid–structure interaction system in order to account

D. Ishihara and others

for the interaction between the wing and the surrounding fluidmotions. These non-dimensional numbers include the Reynoldsnumber (Re) and the Strouhal number (St) as well as the massnumber (M) and the Cauchy number (Ch), where M describes theratio between the added mass from the fluid and the structural massand Ch describes the ratio between the fluid dynamic pressure andthe elastic reaction force. The details are described in the Appendix.

Let us define the characteristic length, velocity, and frequencyas the average wing chord length c, the maximum wing speed ofthe flapping motion of the leading edge center Vw,max (see Eqn 2),and the flapping frequency f, respectively. Then, the expressionsof Re, St, M, and Ch are reduced to the following:

St fc / Vw,max c / (Tw), (3a)

Re Vw,max c / , (3b)

Ch f Vw,max c 4 f/ GI, (3c)

M mf / mw f c3 / mw, (3d)

where Tw (� Lw/2) is the travel length of the leading edge centeron the stroke plane, f is the fluid mass density, is the fluid dynamicviscosity, mf (fc3) is the fluid added mass, and mw is the wingmass. Note that, instead of Young’s modulus, the torsional stiffnessGI for the crane fly is used to evaluate the elastic reaction force inCh. The torsional stiffness has the dimension of the Young’smodulus multiplied by the cube of characteristic length. ThereforeCh is reduced to Eqn 3c, which is Eqn A17 multiplied by c3.

The data for the crane fly reported by Ellington (Ellington,1984a; Ellington, 1984b) and Ennos (Ennos, 1988a) used hereinare summarized in Table1. The numbers in Table2 are derivedusing Eqn 3, the average data in Table1, and the materialproperties of air (mass density: f1.205�10–3gcm–3; dynamic

A B

Mθθ θ

Wing initialposition

Leadingedge

Platespring

Wingplane

θ

Mθ

Mθ

Fig.3. Modeling of wing torsional flexibility.(A)Diagram of insect wing. (B)Model of thewing in A. Mq is the applied moment around thewing longitudinal axis, and q is the angulardisplacement of the pitching motion, or the pitchangle. The stiff leading edge in A is modeled bythe rigid black beam (which is also shown incross section in B), and the wing surface that isstrengthened by the network of veins in A ismodeled by the rigid grey plate in B. Thenarrow white part in B is the flexible plate whichworks as the spring in the lumped torsionalflexibility model.

Model wingCrane fly wing

Fig.4. Shape of the model wing, which is geometrically similar to the cranefly wing. The crane fly wing is drawn after Ennos (Ennos, 1988a). Theblack beam and the gray plate are rigid and model the leading edge andthe wing surface, respectively. The narrow white area is a flexible plate(plate spring), which is the implementation of the spring in the lumpedtorsional flexibility model.



viscosity: 1.502�10–1cm2s–1 at 20°C). The Cauchy number(Ch) for supination (Chsp) is approximately seven times largerthan that for pronation (Chpr). We assumed that the realistic valueof Ch exists between Chpr and Chsp. Under this assumption weused the following five values: Chpr4.49�10–3, ChA1.15�10–2

(average value of Chpr and C—h), C—h1.85�10–2 (the average valueof Chpr and Chsp), ChB2.54�10–2 (average value of C—h and Chsp)and Chsp3.24�10–2.

Experimental apparatusFig.6 is a schematic diagram of the experimental apparatus of ourdynamically scaled model. The computer-operated stepping motorrotates the drive shaft via the timing belt and two pulleys to flapthe model wing. The rotational angle of the drive shaft in one stepis 0.35deg. The fluid forces acting on the model wing aremeasured by a force sensor, which is located in the drive shaft.We used a six-axis force and torque sensor (BL Autotec, Ltd, Kobe,Japan), which detects six components of forces and torques withsix pairs of strain gauges affixed to a Y-shaped beam. From thecalibration test using the loads of 10, 20, 30, …, 300gf, the forcesensor revealed the force in the z-direction with an error of lessthan 1.3%. Note that the error is defined as the absolute errordivided by the maximum load 300gram-force, whichapproximately corresponds to the maximum lift force in ourexperiment. The z-axis of the force sensor was set to be coaxialwith the axis of the drive shaft. Thus, the z-axis of the force sensorwas used to detect the lift FL generated by the wing flapping. The

precise flapping angle () was measured by a rotary encoderconnected to the drive shaft via a timing belt and two pulleys. Theresolution of is 0.12deg. We used a high speed video camera(Citius Imaging, Ltd, Finland), which had a resolution of 640�480pixels and a sampling speed of 99framess–1, to record the wholewing motion (camera viewpoints A and B in Fig.5). The pitchangle (q) was calculated using the chord length and its projectionon the stroke plane given by the recording from viewpoint B, i.e.sine of the pitch angle q equals the projection divided by the chordlength. We used a data acquisition system with a resolution of 16bits and a sampling speed of 50,000sampless–1 to collect the datafor and FL or and q at the same time. During data collection,we used a low-pass three-pole Butterworth filter with a cut-offfrequency of 10Hz (implemented via NI LabVIEW), roughly 20times the flapping frequency, f. Each flight of the model wingconsisted of eight continuous strokes. Five such flights wereaveraged for the same experimental condition. We used a siliconoil as the fluid. The x-, y- and z-dimensions of the oil filling the

Stroke angle Φ

Upstroke

Downstroke

Model wing

Cameraview point

Cameraview point

B

A

z

x

ϕ

y

Fig.5. Flapping motion of the model wing. The axes of the camera viewpoints A and B are approximately coaxial with the x- and z-axes,respectively.

Table 1. Summary of the data for the crane fly reported by Ellington and Ennos

Individual mb (g) LW (cm) c (cm) rA mw (g) (deg.) f (Hz)1/GI

pr

[106 deg./(N m)]1/GI

sp

[106 deg./(N m)]

CF01 CF02

1.9 10–2

1.14 10–21.371.27

2.62 10–1

2.32 10–110.510.9

1.48 10–4

1.13 10–4123*123

45.5*45.5

62.3±24.9†

62.3±24.9†449±244†

449±244†

Average 1.52 10–2 1.32 2.47 10–1 10.7 1.31 10–4 123 45.5 62.3 449

*The values are taken from CF02.†The values are taken from Ennos (Ennos, 1988a). The other data is taken from Ellington (Ellington, 1988a; Ellington, 1988b).mb, body mass; LW, longitudinal length of the wing (one wing); c, average wing chord length; rA, aspect ratio of the wing, 2Lw /c; mw, mass of wing; , stroke

angle; f , flapping frequency; GIpr, torsional stiffness of the actual insect wing for pronation; , torsional sitffness of the actual insect wing for supination.GI

sp

Table 2. Values of the non-dimensional numbers for the crane fliesderived using Eqn 3 and the averaged data in Table 1

Re St Chpr Chsp M

333 5.55�10–2 4.49�10–3 3.24�10–2 6.42�10–2

Re, Reynolds number; St, Strouhal number; Chpr, Cauchy number forpronation; Chsp, Cauchy number for supination; M, mass number.

Forcesensor

Rotaryencoder

Steppingmotor

Silicon oil

Model wing

Driveshaft

Strokeaxis

z

x

y

Air

Fig.6. Diagram of the experimental apparatus of the dynamically scaledmodel.


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tank were 45, 75, and 33cm, respectively. The silicon oil had adensity of f0.96gcm–3 and a dynamic viscosity of 0.5cm2s–1

(25°C). The wing longitudinal length (Lw), and the average chordlength (c) were 22.5cm and 4.2cm, respectively, which satisfythe aspect ratio (rA) for the crane fly (see Table1). The strokeangle (�) was set to 123deg., which is equivalent to that for thecrane fly (see Table1). The rigid beam used for the leading edgewas made of stainless steel and had a cross section of0.6cm�0.6cm and length of 17.5cm. The rigid plate for the wingsurface was made of polyethylene terephthalate (PET), with athickness of 0.12cm. The flexible plate (plate spring) was madeof polyoxymethylene [POM; Young’s modulus: EFP2.59�1010g/(cms2)], which had a thickness of tFP0.03cm or 0.05cmand a length in the chord direction of cFP1.0cm. The mass of themodel wing, excluding the rigid beam, was mw10.7g.


Non-dimensional numbers of the proposed modelWe determined the flapping frequency (f) and the wing longitudinallength of the flexible plate (lFP), such that the non-dimensionalnumbers for the proposed model were equivalent to those for theactual insect flight.

First, the Strouhal number (St) is considered. Eqn 3a can bereduced to St4/(�rA). Thus, St is equivalent to that for the cranefly (St5.55�10–2; see Table2) because � and rA are equivalent tothose for the crane fly.

Next, the Reynolds number (Re) is considered. Eqn 3b can bereduced to the following equation:

f 2 Re / ( � Lw c). (4)

Using Eqn 4 and Re333 in Table2, the flapping frequency, f,is given as 0.52Hz.

7 cycles 7.5 cycles

7.125 cycles 7.625 cycles



Leadingedge Stroke axis

Plate spring

Wing surface

Fig.7. Sequence of snapshotsfor the motion of our model wingin the case of ChC—h during theseventh stroke captured using ahigh-speed video camera (thecamera view point A in Fig.5).The model wing consists of theleading edge (rigid beam), theplate spring (flexible plate) andthe wing surface (rigid plate).The model wing flaps from leftto right (upstroke) in the leftcolumn. The model wing flapsfrom right to left (downstroke) inthe right column. The rod in thecenter of the images is the driveshaft or the stroke axis.



Next, the Cauchy number (Ch) is considered. Eqn 3c can bereduced to the following equation:

GM f Vw,max c 4 f/ Ch. (5)

Using Eqn 5, ChChpr, ChA, C—h, ChB and Chsp, and Eqn 1, thelongitudinal length of the flexible plate lFP is given as 5.0cm forChpr, 9.3cm for ChA, 6.0cm for C—h, 4.2cm for ChB and 3.3cm forChsp. Note that we used tFP0.05cm for the first lFP and tFP0.03cmfor the rest.

Finally, the mass number (M) is considered. Using Eqn 3d, themass number M for the proposed model is equal to 6.65, which isroughly 100 times larger than that for the crane fly (M6.42�10–2).It is difficult to satisfy the mass number condition since a solidmaterial having a mass density 100 times larger than the presentone is required to satisfy the mass number condition. A mass numberroughly 100 times larger than that for the crane fly means that theinertial effect of the present model wing is roughly 1% of that ofthe actual insect wing (the reduced inertia of the model wing).However the dynamic similarity during the flapping translation wasnot much affected by the mass number since the inertial force duringthe flapping translation is not dominant because of the smallacceleration. By contrast, the reduced inertia of our model wingwould affect the wing rotation upon the stroke reversal where theacceleration is very large. In spite of this reduced inertia, however,the order of the rotational force might not be changed. The addedmass during the stroke reversal is comparable to the wing mass inthe insect flapping flight with the relative light wing (Bergou et al.,2007) and the added mass effect on our model wing is equivalentto that on the actual insect.

RESULTS AND DISCUSSIONInitially, the model wing was set to the position with a flappingangle of –�/2 (see Fig.5, the upstroke is first). Then, afterstabilizing at the static state, the up-stroke was started.

Passive pitching motionFig.7 shows sequences of snapshots of the motion of the proposedmodel wing in the case of ChC—h during the seventh stroke usinga high-speed video camera. Fig.8 shows the time history of thepitch angle (q) as well as that of the flapping angle (), theflapping angular velocity (d/dt), and the lift force (FL) in the caseof ChC—h. The wing chord motion is shown in Fig.9, where thepitching motion was derived using the stroke and pitch angles fromFig.8A and C. These figures illustrate the typical flapping motionof our model wing during the whole of one stroke. Our model wingmaintained a high angle of attack during the flapping translationand rotated quickly upon stroke reversal in all cases of Ch.

The model mid-stroke pitch angles were close to those of actualcrane flies. The mid-stroke pitch angles are 27deg. for Chpr, 40deg.for ChA, 49deg. for C—h, 54deg. for ChB and 61deg. for Chsp, whereasEllington (Ellington, 1984b) reported the mid-stroke pitch angle forthe actual crane fly are 45 or 55deg. during the downstroke and 55or 65deg. during the upstroke at 70% of the wing length.

Fig.10 shows the relation between the Cauchy number andthe mid-stroke pitch angle. The mid-stroke pitch angle hasapproximately linear dependency on the torsional stiffness. Thisresult indicates that the present pitch angle during the flappingtranslation was maintained by the equilibrium between the elasticreaction force due to the wing torsion and the fluid dynamic pressure.As a consequence it is suggested that the equilibrium between theelastic reaction force and the aerodynamic pressure maintains a highangle of attack during the crane fly flapping translation.

The quick rotation of our model wing is surprising since the massof our model wing provided only 1% inertial effect compared withthe mass of the actual crane fly wing. This result would be explainedby the added mass from the surrounding fluid. Recent study on thepassive rotation using computational fluid dynamics (Bergou et al.,2007) has shown that the added mass during the stroke reversal iscomparable to the wing mass for insects such as dragonflies andfruitflies, which have relatively light wings. Crane flies also havelight wings, the weight of which is only a few percent of bodyweight. Note that the added mass effect on our model wing wasequivalent to that on the actual crane fly wing because of the fluiddynamic similarity. Thus, the added mass might not change the orderof the inertial force to rotate the wing upon the stroke reversal.

Lift force generated by the flapping wing with passivepitching

As shown in Fig.8C,D, the pitch angle, q, and the lift force, FL,

covaried. It seems that they followed the kinematic characteristicsof the flapping angle, , or the flapping angular velocity d/dt.Fig.11 shows the time histories of the lift force FL for all Ch as

A B

–300

–150

0

150

300

dϕ/d

t (de

g. s

–1)

ϕ (d

eg.)

C

7.0 8.0

Cycles

Pitc

h an

gle

(deg

.)

D

–0.5

0

0.5

1.0

1.5

2.0

7.5 7.0 8.07.5

7.0 8.07.5 7.0 8.07.5

Lift

(N)

–90

–60

–30

0

30

60

90

–90

–60

–30

0

30

60

90

Fig.8. Time histories of flapping angle (; A), flapping angular velocity(d/dt; B), pitch angle (q; C) and lift force (FL, D). We repeated the modelwing flights five times. The FL data for the individual flights (gray lines) andthe average of the five flights (black line) are shown. d/dt was derivedusing the numerical time differential of .

Upstroke

Downstroke

Fig.9. The wing chord motion in the case of C—h. The pitching motion wasderived using the stroke and pitch angles from Fig.8A and C. Solid linesindicate the wing chord, white squares indicate the leading edge andarrows indicate the flapping direction of the wing chord.


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well as the relation between Ch and the mean lift coefficient (cL).The time histories of FL are similar to each other but their averagevalues are different from each other. The average lift forces, FL, are0.557N for Chpr, 0.753N for ChA, 0.772N for C—h, 0.746N for ChB

and 0.672N for Chsp. The corresponding lift coefficients CL are 1.35,1.83, 1.88, 1.81 and 1.63, respectively. These lift coefficients areclose to those of the previous studies (Dickinson et al., 1999;Usherwood and Ellington, 2002b). The lift coefficients werecalculated according to the following equations:

CL FL / [1/2 f Aw (r2Vw)2], (6)

where the fluid mass density f960kgm–3 (silicon oil, 25°C), thearea of the wing surface Aw0.00945 m2 given by c�Lw , the secondmoment of wing area r20.6 (Ellington, 1984a) and the mean wingtip velocity of the flapping motion Vw0.502ms–1 given by 2�fLw.The relation between Ch and the mean lift coefficient CL is


summarized in Fig.11F. It is interesting that the maximum CL wasachieved when C—h, which is the average of Chpr and Chsp, was used.

A mean lift coefficient (CLI) of 1.58 is required for the crane fly

to hover if it is assumed that the insect can hover when the mean liftFL (both wings) equals its body weight. The following parameters ofthe actual crane fly flight were used to calculate CL

I: f1.205kgm–3

(air: 20°C), Aw3.26�10–5 m2, Vw2.58 ms–1 and mb1.52�10–5kg.The present CL for ChA, C—h, ChB and Chsp are larger than CL

I, whilethe present CL for Chpr is slightly smaller than CL

I. The translationallift would be well simulated in our experiment since, during theflapping translation, the dynamic similarity was not much affectedby the inertial force because of the small acceleration. The rotationallift would be partly simulated because of the added mass effect fromthe surrounding fluid, but it would be weakened because of thereduced wing inertia. Evidence of the weakened rotational lift mightbe slight lift peaks observed before and after the stroke reversal (seeFig.11). The lift, which was composed mainly of translational lift,was sufficient to support the weight of the insect. This result isconsistent with conventional studies of the aerodynamics of insectflight, i.e. the translational lift accounts for much of the lift requiredfor the insect to hover while the rotational lift enhances the lift requiredfor forward, upward accelerations or turn.

Our results suggest that the pitching motion in the actual crane flyflapping flight can be passive. Our purpose in this study has beenachieved but the issue concerning the mass number still remains. Wewill address it in future work. We will also examine other dipteransto examine the applicability of our conclusion to their flapping flights.

APPENDIXNon-dimensional numbers for the dynamic similarity between

fluid–structure interaction systemsThere are many non-dimensional numbers for the fluid–structureinteraction (FSI) system, such as reduced velocity (Blevins, 1990;

0

30

60

90

0.040.020

Cauchy number

Pitc

h an

gle

(deg

.)

Fig.10. The relation between the Cauchy number and the mid-stroke pitchangle.

E

–0.5

0

0.5

1.0

1.5

2.0

8.07.57.06.0 6.5Cycles

Lift

(N)

C

–0.5

0

0.5

1.0

1.5

2.0

8.07.57.06.0 6.5

D

–0.5

0

0.5

1.0

1.5

2.0

8.07.57.06.0 6.5

A

–0.5

0

0.5

1.0

1.5

2.0

8.07.57.06.0 6.5

B

–0.5

0

0.5

1.0

1.5

2.0

8.07.57.06.0 6.5

Cycles

Lift

(N)

F

1.0

1.3

1.5

1.8

2.0

0.040.03Cauchy number

Mea

n lif

t coe

ffici

ent

0.020.010

Fig.11. Time histories of lift force (FL) for the cases ofChChpr0.0049 (A), ChChA0.0115 (B), ChC—h0.0185 (C), ChChB0.0254 (D) andChChsp0.0324 (E). In each case we repeated themodel wing flights for five times. The FL data for theindividual flights (gray lines) and the average of the fiveflights (black line) are shown. The relation between theCauchy numbers and the mean lift coefficient is alsoprovided (F).



Chakrabarti, 2002; Dowell, 1999), Cauchy number (Chakrabarti,2002; Fung, 1956; Sedov, 1959), Stokes number (Paidoussis, 1998),mass number (Blevins, 1990; Dowell, 1999; Fung, 1956; Sedov,1959) and reduced damping (Blevins, 1990), etc. We summarizehere the equations governing the FSI system, the dimensionalanalyses for these equations and a set of the non-dimensionalnumbers used in the present study.

Equations governing fluid–structure interactionThe body forces acting on the wing and the surrounding fluid areassumed to be zero. This assumption is justified by the fact that thegravitational force acting on the wing is only a few percent of thelift. Superscripts f and s denote fluid and solid quantities,respectively.

The fluid motion is described by the following Navier–Stokesequations for the incompressible viscous fluid:

where and vi are the mass density and velocity, respectively, andthe stress tensor, ij, for a Newtonian fluid is:

where p and are the fluid pressure and viscosity, and ij is theKronecker delta.

The wing motion is described by the following equation:

where d/dt in the left-hand side is the Lagrangian time derivative.The second Piola–Kirchhoff stress tensor for the linear isotropicHookean elastic body is:

where and G are the Lame constants and ui is the displacement.The following equilibrium condition is satisfied on the

fluid–structure interface:

fij · nf

j +sij · ns

j 0, (A5)

where nfj and ns

j denote the outward unit normal vectors for the fluidand the structure, respectively.

Dimensional analyses for the governing equationsWe assume that the fluid and elastic body under the FSI share thefollowing reference or characteristic quantities: length (L),displacement (U), velocity (V), pressure (PfV2) and time (T). Interms of these common reference quantities, let us define thefollowing non-dimensional variables:

xi xi / L, (A6a)

u si us

i / U, (A6b)

v si vs

i / V, (A6c)

v fi v f

i / V, (A6d)

p p / P, (A6e)

t t / T. (A6f)

ρf ∂vi

f

∂t+ ρf vj

f ∂vif

∂xj=

∂σ jif

∂xj

∂v if

∂xi= 0 ,, (A1a,b)

σ ijf = − pδij + μ

∂vif

∂xj+

∂vjf

∂xi

⎛

⎝⎜⎞

⎠⎟ , (A2)

ρs dvis

dt=

∂σ jis

∂xj , (A3)

σ ijs = λδij

∂uks

∂xk+ G

∂uis

∂xj+

∂ujs

∂xi

⎛

⎝⎜⎞

⎠⎟ , (A4)

First, we apply dimensional analysis to the equation of the fluidmotion (Eqn A1a) as described by Katz and Plotkin (Katz andPlotkin, 2001). Eqn A1a can be rewritten using the variables in EqnA6 as:

Note that the term:

f V2 / L (A8)

in Eqn A7 represents the fluid inertial force due to the convectiveacceleration. Dividing Eqn A7 by Eqn A8 to obtain the non-dimensional form of Eqn A1a, as follows:

where

St L / (TV), (A10a)

Eu P / (f V2) 1, (A10b)

and

Re f VL / , (A10c)

are the Strouhal, Euler and Reynolds numbers, respectively.Following the procedure similar to that for the fluid, we obtain

the following form of the equation of motion for the elastic body,as follows:

Note that the term:

s V / T (A12)

in the left-hand side of the above equation represents the magnitudeof the structural inertial force due to the Lagrangian time derivativeacceleration and that the coefficients of the two terms in the right-hand side represent the magnitude of the elastic force. Dividing EqnA11 by Eqn A12 and using the relation UVT, we obtain the non-dimensional form of Eqn A3, as follows:

The coefficients of the first and second terms in the right-hand sideof Eqn A13 are summarized using the relations 2G/(1–2) andE2G(1+) ( is Poisson’s ratio, E is Young’s modulus) as:

Rs s L2 / (ET2), (A14)

which represents the ratio between the structural inertial force dueto the Lagrangian time derivative acceleration and the elastic force.

Finally, the non-dimensional numbers for the FSI are reduced asfollows. Using the variables in Eqn A6, the fluid force f

i on thefluid–structure interface can be rewritten as:

where P and V/L represent the fluid pressure and viscous force onthe fluid–structure interface. Similarly, the elastic force s

i on thefluid–structure interface can be rewritten as:

dvis

dt=

(λ + G )T 2

ρs L2

∂2uks

∂xk∂xi+

GT 2

ρs L2

∂2uis

∂x j∂x j . (A13)

ρsVT

dvis

dt=

(λ + G )UL2

∂2uks

∂xk∂xi+

GUL2

∂2uis

∂x j∂x j . (A11)

St∂vi

f

∂t+ v j

f ∂vif

∂x j= − Eu

∂p∂xi

+1

Re∂2vi

f

∂x j2

, (A9)

ρfVT

∂vif

∂t+

ρfV 2

Lv j

f ∂vif

∂x j= −

PL

∂p∂xi

+μVL2

∂2vif

∂x j2

. (A7)

τ if = σ ij

f ⋅ njf = − Pp +

μVL

∂vif

∂x j+

∂v jf

∂xi

⎛

⎝⎜⎞

⎠⎟nj

f , (A15)

τ is = σ ij

s ⋅ njs =

λUL

∂uks

∂xk+

GUL

∂uis

∂x j+

∂u js

∂xi

⎛

⎝⎜⎞

⎠⎟nj

s . (A16)


3890 D. Ishihara and others

Using the relations 2G/(1–2) and E2G(1+), U/L andGU/L appearing in the right-hand side of Eqn A16 can be reducedto EU/L, which represents the elastic force on the fluid–structureinterface. Dividing Eqn A15 by this term EU/L, two non-dimensional numbers for the non-dimensional form of Eqn A5can be obtained. One is the ratio between the fluid dynamicpressure (PfV2) and the structural elastic force on thefluid–structure interface (EU/L):

RPE P / (EU/L) fVL / (TE). (A17)

The other is the ratio between the fluid viscous (V/L) and elastic(EU/L) forces on the fluid–structure interface:

RVE / (EU/L) / (TE), (A18)

where the relation UVT is used. The ratio RPE has the physicalmeaning equivalent to the Cauchy number ChfV2/E (Chakrabarti,2002; Fung, 1956; Sedov, 1959) but it is the product of Chmultiplied by St. Instead of the usual expression, we use Eqn A17as Ch since the FSI systems considered in the present study havea periodic input.

The other number for the FSI is the mass number (Blevins, 1990;Dowell, 1999; Fung, 1956; Sedov, 1959), which represents the ratiobetween the representative structural mass mf and the fluid addedmass ms:

M mf / ms f L3 / ms f / s, (A19)

where the other expression of the representative mass, L3, is usedin the third and fourth expressions. Note that the fourth expressionin Eqn A19 is equivalent to the ratio between the fluid inertial forcedue to the Eulerian time derivative acceleration (fV/T in Eqn A7)and the structural inertial force due to the Lagrangian time derivativeacceleration (sV/T, Eqn A12).

Non-dimensional numbers for the FSI systemThe complete set of the non-dimensional numbers of the FSI systemwhose motion is described by the present governing equations areSt, Re, Rs, M, RVE and Ch (RPE), which have the following tworelations:

M St � Re � Rs � RVE (A20a)

and

Ch Re � RVE. (A20b)

We used the well-known numbers St, Re, M and Ch in the presentstudy since the two relations (Eqn A20) make only four of St, Re,Rs, M, RVE and Ch independent.

LIST OF ABBREVIATIONSq angular displacement around the wing longitudinal axis or

the pitch angleAw the area of the wing surfacec average wing chord lengthcFP flexible plate length in the chord directionCh Cauchy numberCL lift coefficientEFP Young’s modulus of the flexible platef flapping frequencyFL total lift force acting on the wingGI torsional stiffness of the actual insect wingGM torsional stiffness of the model winglFP flexible plate length in the longitudinal direction (one wing)Lw longitudinal length of the wing (one wing)mb body massmf added fluid mass

mw mass of wingM mass numberMq moment around the wing longitudinal axispr (superscript) pronationrA aspect ratio of the wing, 2Lw /cRe Reynolds numbersp (superscript) supinationSt Strouhal numbertFP thickness of the flexible plateTw travel length of the leading edge center in the stroke planeV_

w mean velocity of the flapping motionVw,max maximum speed of the flapping motion of the leading

edge centerf mass density of fluid angular displacement of the flapping motion or flapping

angle� stroke angle

This research was supported by a Grant-in-Aid from Japan Ministry of Education,Culture, Sports, Science and Technology. We would like to thank Professor M.Denda for the helpful discussion on the dynamic similarity law.

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