Aspect ratio studies on insect wings

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Aspect ratio studies on insect wings

Cite as: Phys. Fluids 31, 121301 (2019); doi: 10.1063/1.5129191Submitted: 26 September 2019 • Accepted: 21 November 2019 •Published Online: 19 December 2019

S. S. Bhat, J. Zhao (赵季生), J. Sheridan, K. Hourigan, and M. C. Thompsona)

AFFILIATIONSFluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering,Monash University, Melbourne, VIC 3800, Australia

a)Electronic mail: [email protected]

ABSTRACTThe wing aspect ratio (A), that is, the ratio of the wingspan to the mean wing chord, is the most important geometrical parameter describingan insect wing. While studies have shown that a change inA affects the flow structure as well as the aerodynamic force components on wings,the reasons behind the wide variety of aspect ratios observed in nature remain underexplored. Further to this, motivated by the developmentsin micro-air vehicles (MAVs), determining an optimumA is important for their efficient operation. While the effects on flow structureappear to be, at least superficially, broadly consistent across different studies, the effects on aerodynamic forces have been more stronglydebated. Indeed, the considerable variation of force coefficients withA in different studies suggests different optimalAs. To help explainthis, recent studies have pointed out the coupled effects ofA with other parameters. Specifically, the use of Reynolds and Rossby numbersbased on alternative scalings helps to at least partially decouple the effects ofA and also to reconcile previous conflicting trends. This briefreview presents an overview of previous studies on aspect-ratio effects of insectlike wings summarizing the main findings. The suggestedalternative scalings of Reynolds and Rossby numbers, using the wingspan as the characteristic length, may be useful in aiding the selection ofthe optimal aspect ratios for MAVs in the future.

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I. INTRODUCTION

Flying is a way of life for the majority of animals, includingbirds, bats, and insects. The survival and continuing evolution ofthese animals depend on their flight performance. Motivated byresearch into aircraft, the aerodynamics of large flying animals hasbeen studied in detail over the past several decades. However, theaerodynamics of their smaller counterparts is, perhaps, less wellunderstood. Recent developments in the design of micro-air vehi-cles (MAVs) have resulted in a growing interest in the study of theinsect wing aerodynamics. Various morphological as well as kine-matic parameters of flapping wings have been studied by researchersin the past few years.1–10

In general, insect wings are morphologically different fromthose of birds. Bird wings are observed to have sharp tips, whereasinsect wings have a smooth transition of the wing’s leading edge intothe wing tip along a curve. Birds mostly generate lift in the down-stroke of the wing, whereas insects generate lift in both upstroke anddownstroke. Of course, as has been noted by Azevedo et al.11 andWootton,12 not all morphological parameters are related to insectlocomotion. Some are related to wing-folding and thermoregulation

as well. However, wing morphology is important in determining theaerodynamic performance of wings of insects and of insect-inspiredMAVs. The key morphological feature of a wing is its aspect ratio,given by

A =bc

, (1)

where b is the wingspan and c is the mean wing-chord, as shown inthe schematic in Fig. 1. The mean chord is calculated as the ratio ofthe wing area (S) to the wingspan (b).

When scaled with the mean chord, wings with higher val-ues ofA result in a larger wing area (S) than that from small-Awings. This affects the mean lift generated by the wing in a flap-ping cycle as indicated by blade element theory, L = CLρU2

g S/2,where CL is the lift coefficient of the wing, ρ is the density ofthe surrounding medium, and Ug is the mean flapping velocityat the radius of gyration. However, the mean lift is also affectedby wing flapping speeds, depending on the wing Reynolds num-ber Re = Ugc/ν,13–15 where ν is the viscosity of the surroundingmedium. Moreover, a high-A wing has a larger radius of gyra-tion (Rg) scaled with the mean chord. This ratio, in the context

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FIG. 1. Schematic of (a) a fruit fly and its wing characteristics and (b) a wing model used in experimental studies.

of insectlike wings, is known as the Rossby number (Ro = Rg/c= ARg/b).16–19 As a result, the lift acting on the wing can beexpressed as

L = CL12ρν2Re2

A = CL12ρϕ2S2 Ro2

A, (2)

where ϕ is the angular velocity of flapping. Therefore, the effectsof A on the wing aerodynamics are coupled with the effects ofRe and Ro.20,21

The detailed effects of A on the wing aerodynamics havebeen debated over the past few decades. Seemingly inconsistentresults from various experimental studies20,22–31 pose several ques-tions: How does theA affect the flow characteristics over a wing?What is an optimized A? Does the value of the optimized Achange with the flapping speed? Why is there a wide range of Aobserved in nature? This review discusses these issues and, wherepossible, presents some possible answers based on recent researchresults. Motivated by our recent study,32 seemingly contradictoryprevious studies are reconciled using the span-based scaling ofRe and Ro.

In this review, first, a summary of various aspect ratios observedin nature and their relation with Re and Ro is discussed. Fur-thermore, the results from various aspect-ratio studies are cate-gorized into two groups, namely, those showing the effects on

the flow structure and others showing the effects on the aero-dynamic forces. This detailed discussion is necessary to docu-ment important previous studies and to show how they relateto each other. Furthermore, a polynomial model has been devel-oped to predict the lift coefficients for various A-Re-Ro com-binations, which shows that the proposed span-based scalingindeed helps reconcile the apparently conflicting trends in previousstudies.

II. ASPECT RATIOS FOUND IN NATURE AND MAVsSince insect wings have various shapes, the direct measurement

of the mean chord, c, may not be possible in each case. Therefore,Ais also often alternatively defined as a relation between two measur-able quantities, namely, the wingspan b and the wing area S,22,33,34

such that

A =b2

S. (3)

The wingspan and wing area can be measured, for example, by film-ing insect wings.35,36 In some studies, for example, by Usherwoodand Ellington,22

A has been defined as 4b2/S′, where S′ is the totalarea of two wings. For the purpose of comparison, all theA valuesin Appendix A have been scaled according to Eq. (3).

FIG. 2. Data ofA, Re, and r2 of variousinsect wings obtained from the work ofWeis-Fogh,4 Ellington,35 Ennos,41 andZanker.46 Weis-Fogh has assumed mostwings to be semielliptic in shape; thus,having r2 = 0.5. Details can be found inAppendix B.

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Interestingly, birds and bats are found to enhance the lift ontheir wings by dynamically changing the wingspan during their flap-ping motion.37,38 Unlike insects, birds have bones in their wings,which enable them to change their wingspan, thereby changingthe aspect ratio dynamically. This change in the wingspan resultsin changes in the instantaneous flow structure and in the overallflight performance.39 In general, insect wings have aspect ratios inthe range 1.5 ≤ A ≤ 6, whereas bird wings have those in therange 5.5 ≤ A ≤ 19 (see the data by Tennekes40). In comparison

with insects, the longer wingspans of birds and bats help in pro-viding a larger wing area, allowing them to glide while maintaininglow angles of attack. However, insect wings have a lowerA as com-pared to that of birds, owing to the wingspan being limited by theirsmall body lengths (see the work of Ellington35). With a reductioninA, the induced drag increases.41 Simultaneously, the ratio of liftto drag (L/D), also known as the glide ratio, decreases. Thus, lowL/D ratios cause gliding to be not feasible in the case of insects,requiring them to flap their wings to maintain flight. In fact, the

FIG. 3. Various wing planforms, as givenby Combes and Daniel.47 Figure repub-lished with permission from A. Combesand T. L. Daniel, “Flexural stiffness ininsect wings I. Scaling and the influ-ence of wing venation,” J. Exp. Biol. 206,2979 (2003). Copyright 2003 Companyof Biologists Ltd.

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wing-beat rate in animals is observed to increase nearly exponen-tially with a decrease in wingspan.42 Consequently, smaller insectsare observed to have low-A wings with a high wing-beat frequency.Some large insects, such as butterflies, locusts, and dragonflies, canuse both gliding and flapping motions intermittently.43,44 How-ever, their gliding speeds are limited by their inability to alter thewingspan.45

The wing Reynolds number, Re, depends on the wing size andthe wing-beat frequency. In this case, Re is generally defined by

Re =Ugcν

. (4)

For the wing flapping with a sweep amplitude ϕA at the rate of nHz, the mean velocity Ug is calculated as 4ϕAnRg . The term sweep,in this case, refers to the rotation of the wing about the axis nor-mal to the stroke plane, shown as ϕ in Fig. 1(b). This motion is alsocalled the translation in many studies.13,16,21 Even though the rangeof Re for various insects is large (102

< Re < 104), the range ofA isobserved to be typically within 2 <A < 7.5, as can be seen in Fig. 2.Many insect wings can be reasonably approximated by semiellipticshapes;4 however, some Hymenoptera and Lepidoptera species arebetter approximated by triangular shapes. The range of wing shapescan be seen in Fig. 3.

Ellington35 has derived laws of wing shapes to which most wingsadhere. According to these laws, the normalized radii of the sec-ond and third moments of area of a wing, i.e., r2(S) and r3(S),are related to the centroid of the wing area r1(S) such that r2(S)= 0.929[r1(S)]0.732 and r3(S) = 0.9[r1(S)]0.581. Here, the normalizedradius of a wing’s kth moment of area is given by

rkk(S) = ∫

1

0crkdr, (5)

where c is the local wing chord normalized by c and r is the span-wise distance normalized by the wingspan (b). Broadly, either r1 orr2, in addition toA, might be sufficient to reconstruct an approxi-mate wing planform using a Beta function approximation, as shownby Ellington.35 Data for various insects obtained from the literatureshow that r2 varies in a very narrow range, 0.5 < r2 < 0.6, as can beseen in Fig. 2.

III. EFFECTS OFA ON THE WING AERODYNAMICSInsight into the influence ofA on the wing aerodynamics can

be gained by noting the relation between A and various accel-eration terms in the Navier-Stokes equations. The nondimension-alized Navier-Stokes equations in a noninertial rotating frame ofreference for a flapping wing have been derived by Lentink andDickinson48 as

DuDt

+1

J2 + 1[

1A∗

Ω × r +1A

Ω × (Ω × r) +1A

2Ω × u]

= −∇p +1

Re∇

2u, (6)

where r, u, Ω, and Ω are the normalized displacement, velocity,angular velocity, and angular acceleration vectors, respectively; tand p represent the normalized time and pressure, respectively; andthe nondimensional numbers J and A∗ represent the advance ratio

and the chord-lengths swept by the wing tip, respectively. In thecase of a hovering insect wing, as is considered in most studies dis-cussed in this review, J = 0. It should be noted that the velocity andlength terms in this equation are normalized by the wing-tip velocityand the wing chord, respectively, following the proposed scalings ofLentink and Dickinson. Moreover,A in Eq. (6) has been definedby Lentink and Dickinson as R/c, which did not account for thewing-root offset.

It has been established that the centripetal and Coriolis forcesare important for the stability of the leading edge vortex (LEV)responsible for a high lift force over an insect wing.16,25,49,50 Equa-tion (6) shows that the centripetal and Coriolis accelerationsscale with A. Hence, this will affect the formation and stabil-ity of vortical structures over the wing. In view of this, sev-eral experimental and computational studies have investigated theeffects of A on both the vortical flow structures and the aero-dynamic forces. Observations with respect to the influence on theflow structures are broadly similar; however, inconsistencies existin the reported effects on aerodynamic forces, as described inSecs. III A–III E.

A. Effects on the flow structureA flapping stroke of an insect can be divided into the for-

ward and the backward half strokes. During the majority of eachhalf stroke, the wing rotates about the vertical axis, which is calledthe sweep motion. Toward the end of a half stroke, the wing flipsand changes its orientation, which is called the pitch motion. Insectwings flap at relatively low Reynolds numbers, maintaining highangles of attack (α ∼ 45). A purely translating wing would stall atsuch angles with a loss of lift. However, in flapping insect wings, ahigh lift is maintained, which can be attributed to several unsteadymechanisms. Nevertheless, it has been widely accepted that the highlift over an insect wing is primarily obtained as a result of a stableattachment of the LEV.5,51

The LEV is formed from the shear layer separating from theleading edge during the sweep motion. The shear layer curls up onthe wing suction side to form the LEV, which is conical in shape,growing in size from the wing root to the wing tip. LEVs have alsobeen observed to be important in the hydrodynamics of the flappingmotion of fish tails52 and stingrays.53 In insects, the LEV has beenobserved to be stabilized during the sweep motion by the action ofthe centripetal and Coriolis accelerations.16,54,55 Since these impor-tant terms in Eq. (6) scale with A, the stability of the LEV mustbe dependent onA. This can ultimately affect the lift acting on thewing.

In their experiments on a rotating wing at Re = 13 000, Kruytet al.29 observed that the LEV remains attached up to the spanwiselocation r/c < 4. Further outboard, the LEV lifts away from the wingsurface, as can be seen in Fig. 4. The proximity of the LEV to thesurface for low A results in a higher magnitude of the normal-ized suction pressure beneath the LEV. Kruyt et al. showed that theLEV circulatory lift coefficient (Cl) is higher at lowA and decreasessteeply beyondA > 4. The flow over anA = 10 wing largely resem-bles that over a purely translating wing, except in the inboard region(at r/c < 4), where the 3D effects are prominent. For locations beyondr/c = 4, the size of the LEV at that location is limited by the trailingedge and it will be similar for largerA.

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FIG. 4. The vorticity contours inside the LEV core at various spanwise sections are shown for variousA. Republished with permission from Kruyt et al., “Power reductionand the radial limit of stall delay in revolving wings of different aspect ratio,” J. R. Soc., Interface 12, 20150051 (2015). Copyright 2015 Royal Society.

At a specific spanwise location normalized by wingspan, thesize of the LEV is observed to be different for differentA, as hasbeen observed by Luo and Sun.23 For example, Fig. 5 shows that atlocation ra = r/b = 0.4, the LEV size for A = 2.8 is smaller thanthat for a wing ofA = 5.5. However, at location rb = r/b = 0.6, theLEV forA = 2.8 is slightly larger, whereas that forA = 5.5 is splitinto two corotating vortices called the dual-LEVs. Here, the primaryLEV at the leading edge is lifted away from the wing surface. It can benoted that the split may be caused by induced opposite-signed sec-ondary vorticity that has developed between the two vortex cores, as

has been discussed by Lu et al.56 This secondary vorticity is devel-oped as a result of the interaction of the two corotating LEVs withthe wing surface. The secondary vortex (SV) is lifted away from thesurface by the action of the larger LEV core, ultimately causing theLEV to split.

Han et al.26 also observed that the LEV over a flapping wing islifted away from the surface at a higherA. Figure 6 shows that theLEV over a wing ofA = 1.5 is close to the wing surface, allowing thereattachment of separated flow beyond the midchord. However, forA = 3 and 6, the LEV is lifted off the wing surface and the separated

FIG. 5. Vorticity contours over a wing are shown at two different spanwise locations forA = 2.8 andA = 5.5. ra and rb represent the locations r /b = 0.4 and r /b = 0.6,respectively. Figure adapted with permission from G. Luo and M. Sun, “The effects of corrugation and wing planform on the aerodynamic force production of sweeping modelinsect wings,” Acta Mech. Sin. 21, 531–541 (2005). Copyright 2005 Springer Nature.

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FIG. 6. The contours of the normalized vorticity (ω∗ = ωc/Ut ) are superimposed with the streamlines of flow over wings ofA = 1.5, 3, and 6 at r /b = 0.6. Figure adaptedwith permission from J.-S. Han, J. W. Chang, and H.-K. Cho, “Vortices behavior depending on the aspect ratio of an insect-like flapping wing in hover,” Exp. Fluids 56, 181(2015). Copyright 2015 Springer Nature.

flow does not reattach. The detachment of the LEV at higherAmaybe due to reduced Coriolis effects, according to Eq. (6). It shouldbe noted that, in this figure, the difference in the LEV structure isshown on the plane at r/b = 0.6. Also, note that the local Rossbynumbers for wings ofA = 1.5, 3, and 6 are r/c = 0.9, 1.8, and 3.6,respectively.

These observations show that important features of the LEVstructure change with A even when r/c < 4. The changes canbe seen more clearly in three-dimensional (3D) vortical structuresidentified by isosurfaces of total pressure,25 shown in Fig. 7. Here,the LEV over an A = 1 wing is a single coherent vortex. Withan increase in A, the LEV core becomes relatively compact, withincreased suction beneath it. However, at a certain spanwise loca-tion, the LEV expands to a bubblelike structure, which bursts to formsmaller noncoherent structures. The LEV burst is typically char-acterized by the spanwise flow stagnation or reversal, the entrain-ment of the opposite-signed vorticity, and the local expansion ofthe vortex size.57 The spanwise locations of the bubble and the LEV

burst, both normalized by wingspan, are observed to move inboardwith an increase inA. This weakens the wing-surface suction fur-ther outboard, as is clear from the lower magnitude of suction seenoutboard. In the case of A = 4, the LEV is observed to formnarrow dual LEVs that display a helical structure prior to burst-ing. The narrower size is a result of an increased spanwise flux ofvorticity.

Overall, most studies indicate that the spanwise vorticity fluxat inboard locations increases withA. Moreover, the LEV split andbursting occur at a more inboard location for high-A wings due tothe decreased Coriolis and centripetal effects, consistent with Eq. (6).Although the suction magnitude beneath the LEV increases withA, the early bursting of the LEV results in a loss of suction out-board. Therefore, according to these observations, the lift coefficientof wings can be expected to increase withA up to a limit, beyondwhich the loss of the outboard suction will dominate, causing the liftcoefficient to decrease. Hence, it is of interest to further characterizethe effects ofA on the lift coefficient.

FIG. 7. The 3D LEV structure, identified using the isosurface of total pressure, changes significantly withA. Contours of the pressure coefficient are shown on the wingsuction surface. Figure adapted with permission from D. J. Garmann and M. R. Visbal, “Dynamics of revolving wings for various aspect ratios,” J. Fluid Mech. 748, 932–956(2014). Copyright 2014 Cambridge University Press.

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FIG. 8. Relation between lift andA for (a) symmetric and (b) asymmetric wing shapes is shown. Figure reproduced with permission from S. A. Ansari, K. Knowles, andR. Zbikowski, “Insectlike flapping wings in the hover. Part I: Effect of wing kinematics,” J. Aircr. 45, 1945–1954 (2008). Copyright 2008 by S. A. Ansari.

B. Effects on the lift coefficient

As discussed earlier, the changes in the flow structure over awing withA are expected to affect the lift (L) acting on the wing.For an assumed value of the lift coefficient (CL), the expression for Lfor a givenA (for a square wing) is given by Ansari et al.58 as

LA =12ρCLω2c

b3

3, (7)

where ρ is the fluid density and ω is the angular velocity of the wing.It should be noted that the area scaling used in this expression isbc, which is the same as the wing area S used in most studies. Onthe other hand, Dickinson and co-workers have used Rc as the areascaling, where the root cutout is not taken into consideration. IfAis scaled by a factor of x (xA = xb/c), to maintain the same wingarea (S = bc), the chord and wingspan should be changed to c/

√

xand b

√

x, respectively. Therefore, the lift acting of the scaled wingshape will be

LxA =12ρCLω2 c

√

x(

√

xb)3

3= xLA. (8)

The numerical simulations of Ansari et al. have shown that var-ious symmetric and asymmetric wing shapes adhere to the linearrelation between L andA, as can be seen in Fig. 8. This relation isderived under the assumption that CL remains constant across var-ious A. However, it may be noted that some shapes, such as theellipse and semiellipse, do not follow the linear relationship betweenL andA. Ansari et al. have attributed this deviation to the “hinder-ing effects of the wake-induced components” of the lift for the wingsof higherA.

Moreover, the reference velocity used here is√

ω2b2/3. Follow-

ing the observation that r2 of most insect wings is close to 1/√

3 (seeFig. 2), the reference velocity can be expressed as ωr2. Indeed, usingthis reference velocity, Luo and Sun showed that the variation inAhas only a minimal effect on CL. Their numerical simulations withrotating wings of variousA showed a negligible effect on CL at twodifferent Re, as can be seen in Fig. 9, supporting the assumptionsof Ansari et al. Furthermore, Carr et al.27 also observed CL to beonly slightly decreasing withA for rectangular wings rotating witha constant wing tip velocity.

On the other hand, some researchers have observed the liftcoefficient CL and the drag coefficient CD to be significantly affected

FIG. 9. Variation of CL with the sweep angle ϕ shows a negligible effect ofA at (a) Re = 200 and (b) Re = 3500. Figure adapted with permission from G. Luo and M. Sun,“The effects of corrugation and wing planform on the aerodynamic force production of sweeping model insect wings,” Acta Mech. Sin. 21, 531–541 (2005). Copyright 2005Springer Nature.

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FIG. 10. Variations in force coefficients with the sweep angle are shown for variousA. Figure adapted with permission from D. J. Garmann and M. R. Visbal, “Dynam-ics of revolving wings for various aspect ratios,” J. Fluid Mech. 748, 932–956(2014). Copyright 2014 Cambridge University Press.

by A. Garmann and Visbal,25 for example, showed that CL andCD during the sweep motion are lower for A = 1 and higherfor larger A. However, there is no significant change in the CLand CD values between A = 2 and A = 4, as can be seenin Fig. 10.

Han et al.26 extended the range ofA and observed that increas-ing A beyond the value of 3 results in a reduction in CL. Thisvariation is similar to the spanwise variation in the LEV circula-tory lift reported by Kruyt et al.29 (see Fig. 4). According to thesestudies, A = 3 is optimum. Many insects flying at low Re have

wings of A ≈ 3. However, it should be noted that the experi-ments in both of these studies were carried out at relatively highReynolds numbers (∼104), where a wider range of A is observedin insects (see Fig. 2). Hence, the optimization ofA based on CLmay be insufficient to explain the existence of other aspect ratios innature.

Although it has been established by researchers, e.g., Luo andSun,23 that the velocity at the radius of gyration, Ug , is the appropri-ate reference velocity, many other researchers, for example, Carr etal.27 and Phillips et al.,28 have used the wing-tip velocity, U t , as thereference. This choice of reference also affects the trends observedin the lift coefficient as a function ofA. Figure 11 summarizes thetrends in the mean lift coefficient based on Ug [i.e., CL = 2L/(ρU2

g S),where L is the mean lift] and that based on U t [i.e., CLt = 2L/(ρU2

t S)]with changingA extracted from various studies. It is clear from thisfigure that not only the actual values of the lift coefficient but alsotheir trends are inconsistent between various studies.

Broadly, the values of CL in these studies are in the rangebetween 1 and 1.8, as can be seen in Fig. 11(a). The data fromHan et al.26 and Shahzad et al.30 are outside this range, perhapsdue to a difference in scaling, not clear from the available infor-mation. Overall, the observed variations in the trends would resultin differences in the selection of the optimum aspect ratios basedon the maximum CL. For example, for Han et al.26 and Jardinand Colonius,31

A = 3 is optimum, whereas for Harbig et al.,24

A = 5 is optimum. Harbig et al.20 and Bhat et al.32 showeda monotonic decrease in CL after a certain value of A, indi-cating that a low A is optimum. On the other hand, Phillipset al.28 and Shahzad et al.30 showed a monotonic increase in CLwithA.

These variations may be due to the differences in the valuesof parameters, such as Re, the reference scales, the wing shapes,and the angles of attack, which are summarized in Appendix A.It should be noted that Harbig et al.,20 Carr et al.,27 and Bhatet al.32 have used the constant span-based Reynolds number

FIG. 11. Comparison of variations in (a) the lift coefficient based on Ug (CL) withA and of those in (b) the lift coefficient based on Ut (CLt ) withA, extracted from variouspast studies, is shown. The dashed line represents the LEV-circulatory lift coefficient.

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(Reb = Re × A), which will be discussed later. Moreover, ourrecent study32 indicates that CL derived from the LEV-circulatorylift, such as that by Phillips et al.,28 may not represent the actual CLand its trends accurately. Furthermore, the values of CL by Kruytet al.29 are obtained at relatively high Reynolds numbers (Re ∼ 104)inspired by hummingbird wings. At such Re, the variations inCL may not be comparable to those for insect-scale Re sincebird wings at higher Re maintain a lower α and typically havehigher A, which makes it easier for them to glide, unlike forinsects.

When scaled with U t , the trends in CLt from most studies showa decrease with A, as can be seen in Fig. 11(b). The differencein the trends in CLt from some studies with respect to the trendsin CL is on account of the change in the values of Rg/b withA inthose studies. The relation between CLt and CL can be given byCLt = CL × R2

g/b2. For a given wing planform without a root off-set, the ratio Rg/b should remain constant even after changingA.However, this ratio may also be affected by the wing offset ratiob0, as shown later in Eq. (10). In some studies, the values of Rg/bhave changed withA. This has resulted in trends in CLt different tothose in CL.

C. Coupled effects ofA and other parametersAerodynamics of a rotating or a flapping wing is determined

by parameters from a large parameter space. There might be manyfactors, including Re, Ro, α, and wing shape, that can affect the wingaerodynamics.13,18,31,59 Kruyt and co-workers, for example, showedthat low-Awings outperform high-Awings in terms of the aerody-namic forces and power economy if the angle of attack is in the rangeα > 20. However, at lower angles, the high-A wings outperformthe low-A wings. Therefore, the values of CL predicted by various

researchers may not compare well unless all the affecting variablesare maintained to be the same.

For the comparison in Fig. 11, we have tried to extract mostresults from the studies on wings in pure rotation with α ∼ 45.For the flapping wing experiments of Han et al., the values shownwere extracted during the wing sweep motion, during which thewing was rotated with a near-constant angular velocity and a con-stant α. Nevertheless, the values of Re from various studies varyover a very large range of [102–104]. Moreover, most computa-tional studies rotate the wings about their roots, whereas mostexperimental studies require a central body to hold and rotatethe wing. The presence of a central body causes the wing to beoffset from the rotation axis, changing its Rossby number. Vari-ous wing shapes can be described by the characteristic length Rg ,which also affects Ro. The variations in Ro may in turn affect theLEV as well as the forces, depending on the size of the centralbody.32,60

The coupled effects of A and Re were studied systemati-cally by Harbig et al.20 They observed that the large-scale vorti-cal structures over a rotating wing change with both A and Re.As can be seen in Figs. 12(a)–12(c), a single LEV is observed ata low Re, which is observed to split into the dual LEVs, namely,LEV1 and LEV2, at higher Re. The reason of the split has beenattributed to the growth of the secondary vortex (SV) beneath theLEV between the two corotating vortex cores. As Re increases, thestrength of the SV also increases, causing the LEV to split at a moreinboard location. This split is followed by vortex bursting at a cer-tain spanwise location. It should be noted that the LEV structuresin these figures have been shown as the isosurfaces of the constantQ value.20

Interestingly, Harbig and co-workers observed a similar phe-nomenon even with an increase in A and a constant Re,

FIG. 12. The LEV structures identified by the isosurfaces of Q over a rotating wing ofA = 2.91 are shown at (a) Re = 300, (b) Re = 750, and (c) Re = 1500. Furthermore,the LEV structures over wings of (d)A = 2.91, (e)A = 5.1, and (f)A = 7.28 are shown for Re = 300. Figure adapted with permission from R. R. Harbig, J. Sheridan, andM. C. Thompson, “Reynolds number and aspect ratio effects on the leading-edge vortex for rotating insect wing planforms,” J. Fluid Mech. 717, 166–192 (2013). Copyright2013 Cambridge University Press.

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similar to that observed by Luo and Sun23 and Garmann andVisbal.25 As can be seen in Figs. 12(d)–12(f), at Re = 300, theLEV splits at higher A and the split location shifts inboardwith an increase in A. Harbig and co-workers tracked the LEVsplit location by systematically varying both Re and A overa wide range. From their study, they successfully attempted todecouple the effects of the two parameters by suggesting thewingspan as an alternate length scale for the Reynolds number.According to their suggestion, the span-based Reynolds number isdefined as

Reb = Ugb/ν. (9)

Note that the length scale c from Eq. (4) has been replaced withb. With this modified definition, Harbig and co-workers showedthat the LEV split occurs at approximately the same spanwiselocation across a range of A at constant Reb. This, in turn,shows that the large-scale vortical structure remains indepen-dent of A if Reb is maintained to be constant. Carr et al.27

also followed the span-based definition of Re and showed thatthe spanwise circulation around the LEV at a high rotationangle (ϕ = 120) was minimally affected between A = 2andA = 4.

Furthermore, Lee et al.21 found the effects of Ro andA also tobe coupled. This is because, for the wings with a root offset, the rela-tion between Ro andA is affected by the wing-root offset distance.For the wing models with the normalized root offset of b0 = b0/bfrom the rotation axis, the radius of gyration can be approximated

to Rg ≈ b√

r′22 + b0 + b20, where r′2 is the normalized radius of the

second moment of area of the wing without any offset. Therefore,the Rossby number in this case will be

Ro ≈A√

r′22 + b0 + b20. (10)

This shows that, for a given wing planform (i.e., a constant r′2) and aconstant wing offset ratio (i.e., a constant b0), Ro changes linearlywith A. However, in some experimental studies, b0 also changeswith a change inA due to the extension of the wingspan. Hence,the variations in Ro in these studies were different from those with aconstant b0. This could have resulted in differences in the perceivedeffects ofA in various studies.

D. Reconciling the past studiesThe first attempt to reconcile the past studies was under-

taken in a remarkable study by Lee et al.21 They classified previ-ous studies into two groups, depending on their offsets. The firstgroup involved the “constant r2 configurations” of Usherwood andEllington,22 Luo and Sun,23 and Han et al.26 The second groupinvolved the “constant ΔR configurations” or, consistent with thepresent notation, the “constant b0 configurations” of Garmann andVisbal,25 Carr et al.,27 Phillips et al.,28 and Kruyt et al.29 Thevariations in Ro with respect to A in all of these studies weredifferent.

In order to reveal the coupling between Ro and A, Lee andco-workers simulated four series of configurations. Series-1 wasthe root-flapping series with b0 = 0, in which A was varied byextending the wingspan. Series-2 involved a constant b0, whileAwas varied by extending the wing chord. Series-3 also involved aconstant b0; however, A was varied by changing the wingspan,thereby simultaneously changing b0. Series-4 was the constant Roseries, in which b0 was changed for eachA such that Ro remainedconstant across different A. By varying both A and Ro overlarge ranges, Lee and co-workers obtained a map of the predic-tions of CL on the plane of A and Ro, as shown in Fig. 13(a).These contours are superimposed with the lines showing thevariations in A and Ro in Series-1 to 4. Depending on thesevariations, the relation between CL andA may vary, as can be seen

FIG. 13. In (a), contours of CL are mapped on the plane ofA and Ro. In (b), the variations in CL withA are shown for various series of configurations. Figures adaptedwith permission from Y. J. Lee, K. B. Lua, and T. T. Lim, “Aspect ratio effects on revolving wings with Rossby number consideration,” Bioinspiration Biomimetics 11, 056013(2016). Copyright 2016 IOP Publishing.

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in Fig. 13(b). Therefore, the choice of optimumAmay change withthe series.

It should be noted that all the configurations in this study weresimulated at Re = 500, which is close to Re of only small insects.At this Re, the phenomenon of the LEV-splitting and LEV-burstingcan be observed only for relatively very highA. However, for largerinsects with higher Re, the LEV split can be observed even at lowA(see Lu et al.56), which can affect the variation in CL. Moreover, mostexperimental studies are conducted at higher Re due to limitationsin the force measurements at low Re. Looking at the large variationin Re across various past studies, an investigation of the combinedeffects ofA, Ro, and Re was thought necessary to reconcile thosestudies.

In our recent work,32 the individual as well as combinedeffects of these three important parameters were studied. First, theeffect of the span-based Reynolds number on CL was investigated.The increase in CL with Reb was very high in the lower range ofReb, as can be seen in Fig. 14(a). At higher Reb, CL changed byonly small amount. The increase in CL was due to the increasedLEV suction as a result of the higher vorticity magnitudes withincreasing Reb. However, at high Reb, the induced LEV burstingand splitting resulted in a loss of sectional suction, causing only asmall increase in overall suction. As expected, the variation in CLwas found to be also sensitive to Ro maintaining a similar trendwith Reb.

The individual effect of Ro was observed by investigating alarge range of Ro at fixed values of Reb. For the wing ofA = 2.91,extending Ro over a larger range showed a monotonic decrease inCL, as shown in Fig. 14(b). This decrease was due to the weakerLEV as a result of the reduced Coriolis acceleration17,18 at highRo. Here, A of the wing was maintained to be constant. How-ever, Bhat et al.32 found the LEV structures to vary withA, evenat a constant Ro. This is because, to maintain the same Ro, theoffset b0 had to be adjusted with a change in A [see Eq. (10)].Various wing root offsets caused the Coriolis accelerations to varyacross A even with a constant Ro. As Ro was insufficient in cor-relating the Coriolis acceleration with A, the scaling of Ro wasrevisited.

Since it was established that the wingspan was better cor-related with the LEV structure, the Navier-Stokes equations fora wing in a rotating frame of reference were rescaled using bas the reference length scale. In this case, the vector equationreduces to

DuDt

+(Ω/Ω2

)

(Rg/b)2 Ω × r +1

(Rg/b)2 Ω × (Ω × r) +2

(Rg/b)Ω × u

= −∇p +μ

ρUgb∇

2u. (11)

As expected, the viscous term scales with Reb = ρUgb/μ. Importantly,the Coriolis acceleration scaled with Rg/b, suggesting a span-baseddefinition for the Rossby number, i.e., Rob = Rg/b. Using this, it wasconfirmed that the LEV structure across variousA remains similarat a constant Rob.32

Using the modified scaling, the work of Harbig and co-workerswas extended to study wider ranges of bothA and Re. It can be seenin Fig. 14(c) that, at a constant Reb, CL increases by a small amountin the low range ofA and then suddenly decreases beyond a cer-tain value in a higher range. In other words, at any given Reb, CLis less sensitive to A only over a certain range. Interestingly, thisrange widens with an increase in Reb. If this range ofA is consid-ered to be optimum purely based on CL, the optimum range coversonly smaller A at small Reb and a wider range of A at higherReb. Coincidentally, these optimum ranges overlap the values ofAfound in nature at the respective Reb, which might be one of theimportant reasons behind the presence of a wide variety of A athigher Reb.

With an increase in A, with b as the reference, the meanwing chord decreases. This causes a reduction in the area avail-able for supporting the LEV. Beyond a certain critical value ofA,the LEV hits the trailing edge and interacts with opposite sign vor-ticity from the trailing edge (TEV). This interaction weakens theLEV and reduces the suction over the wing. Therefore, atA largerthan the critical value, CL suddenly drops. It should be noted that,at higher Reb, the LEV is narrower due to the increased span-wise vorticity transport. Therefore, the LEV remains less affected

FIG. 14. The coupled effects of Reb, Rob, andA on CL. Data extracted with permission from Bhat et al., “Uncoupling the effects of aspect ratio, Reynolds number andRossby number on a rotating insect-wing planform,” J. Fluid Mech. 859, 921–948 (2019). Copyright 2019 Cambridge University Press. The dashed lines indicate predictionsusing the polynomial fit to the data. The wing aspect ratio in (a) and (b) isA = 2.91. The Rossby number in (c) is Ro = 1.66.

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FIG. 15. The contours of CL, predicted using the polynomial model, are shown in the 3D space ofA, Reb, and Rob in (a) on theA − Reb planes at Rob = 0.6, 1, and 1.4and in (b) on theA − Rob planes at Reb = 500, 2000, and 3500.

by the TEV over a wider range of A, which results in a wideroptimal range.

Overall, this shows that CL has a coupled effect of Reb, Rob, andA. Based on our CFD data obtained from exploring a wide three-dimensional parameter space (of ∼100 points), a polynomial modelCL = f1(Reb, Rob,A)was derived using cross-validation and LASSOregularization. The predictions of CL using this model are plottedas dashed lines in Fig. 14. Using the same model, the values of CLwere predicted in the 3D parameter space ofA, Reb, and Rob. Thecontours of CL on various planes in this space can be seen in Fig. 15.Clearly, the maximum possible CL can be observed at a low Rob anda high Reb.

Finally, an attempt was made to reconcile the seemingly incon-sistent trends in CL withA from previous studies [see Fig. 11(a)]

by observing the relative variations in Reb and Rob. Using theirvalues of A, Reb, and Rob, the values of CL were predicted withthe polynomial model and compared with their data, as can beseen in Fig. 16. The model could predict the values of CL within10% accuracy, approximately, for most of the past results. Althoughthe predictions deviate from the data of Usherwood and Elling-ton22 and Carr et al.27 at a few points, overall, the model appearsto predict the values well. Some deviations in the predictions areobserved for the higher range of A in the cases of Luo andSun,23 Garmann and Visbal,25 and Jardin and Colonius.31 Thismight be owing to the fact that α used by Luo and Sun23 andGarmann and Visbal25 are 40 and 60, respectively, whereas thepresent model is based on the cases with α = 45. Moreover,the lift in the high-A cases of Jardin and Colonius31 had not

FIG. 16. The variations in CL reportedby previous researchers are comparedwith the predictions using the polyno-mial model, shown by open symbols anddashed lines of the corresponding colors.

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stabilized even after the rotation of 180, unlike that observed inother studies.

Broadly, it can be concluded that consideration of the com-bined effects of A, Reb, and Rob helps reconcile seemingly con-flicting past aspect-ratio studies. It is also clear from this studythat direct comparisons of the results from previous studies canbe made only when the values of their A, Reb, and Rob arematched.

It should be noted that the data of Shahzad et al. andPhillips et al. have not been included in the comparison due toambiguity in their CL values as mentioned earlier. Moreover, thepresent model does not predict the CL values well for Reb ina higher range (>105), such as those in the case of Han et al.(2015) and Kruyt et al.29 Further exploration of this range ofReb is necessary, first, to verify whether the span-based scalinggoverns the flow physics well even in this range and/or, second,to derive or modify the model to predict CL accurately in thisrange of Reb.

E. Future directionsIt should be noted that the present analysis is indeed lim-

ited to the hovering wings. In the case of the forward flight, theadvance ratio, i.e., the ratio of the forward speed to the mean wingtip velocity (J = V/U t), is considered to be an additional param-eter that affects the mean lift.6,61 Given a mean reference veloc-ity, an increase in J causes the mean lift coefficient to decrease.62

J and A have been found to have coupled effects on the meanlift coefficient and the LEV stability. Hence, the present modelwill have to be modified to include the effects of J. Moreover,the wing flexibility has been observed to enhance CL,63 whichshould be taken into consideration while predicting CL for flexiblewings.

The present model predicts broadly consistent CL in the inves-tigated ranges ofA, Reb, and Rob. However, at very high Reb, theLEV over high-A wings will tend to be more unstable, resultingin an early burst. Beyond a certain range, the effect of wingspanwill be insignificant and the flow will approach that over a two-dimensional wing, where the span-based scaling of LEV may notbe applicable. An extended range of Reb needs to be exploredin future to investigate the transition from the span-based scal-ing to the chord-based scaling. Moreover, at very low Reb, theL/D ratio drops rapidly and the viscous diffusion effects are verylarge, affecting the LEV structure.64 It might be interesting to seewhether the span-based scaling can be applied to the flow at suchlow Reb.

Furthermore, the present model is developed based on theconstant angle of attack (α = 45). Some previous studies23,25

have incorporated different α, for which the present model maynot predict the forces accurately. Moreover, Kruyt et al.29 haveshown that, at high Re, large-A wings benefit from lower anglesof attack to achieve high L/D ratios. Hence, the present modelmight be improved by combining the results, for example, of Saneand Dickinson65 to involve the effect of α on the aerodynamicforces.

Finally, the understanding of the physics behind the LEVdevelopment, its growth, and burst is important in understandingits effects on the aerodynamic forces. The analysis of various terms

in the vorticity transport equation applied to the LEV for vari-ous A, Reb, and Rob might help linearize the functions f(A),g(Reb), and h(Rob) (see Appendix C), which in turn will result infewer polynomial terms of the model. Hence, the present empir-ical polynomial model might be simplified using this detailedanalysis.

IV. CONCLUDING REMARKSThis review presents a comprehensive overview of previ-

ous studies on insect-wing aspect ratios. Measurements of realinsect wings show that there is a wide variety of aspect ratiosfound in nature. In terms of flow structures, studies present-ing the effects of A over rotating or flapping wings are broadlyconsistent. The leading-edge vortex (LEV) over a wing, which isresponsible for the high lift, is observed to split at higher A.Moreover, the increase in the LEV cross-sectional size along thespan causes the LEV to reach the trailing edge without reattachingand then weaken through the influence of the opposite signvorticity from the trailing edge for A > 3. Thus, some studiespredict A = 3 to be optimal based on the mean lift coefficientof the wing (CL). On the other hand, other studies present dif-ferent conclusions. The inconsistencies in these trends have beenreviewed by further studies, which indeed reveal that they aredue to the coupled effects of the simultaneous but different vari-ations in Reynolds and Rossby numbers between sets of experi-ments. Taking this further, recent work shows that the flow struc-ture over an insect wing better correlates with modified Reynoldsand Rossby numbers based on wingspan. Moreover, this scalingalso helps better reconcile apparent differences in the CL-A trendsrecorded in a variety of previous studies. This review highlightsthat the direct comparisons of the previous results showing Aeffects on the wing aerodynamics are only possible if the valuesof the aspect ratio, Reynolds number, and Rossby number arematched.

ACKNOWLEDGMENTSThe authors acknowledge a generous computing time alloca-

tion from the National Computational Infrastructure (NCI) andPawsey Supercomputer Centre (NCMAS Merit Grant Nos. n67 andd71) and support through the Australian Research Council (GrantNos. DP170100275 and DP190103388).

APPENDIX A: DATA FROM PREVIOUS STUDIESThe data for the geometrical and kinematic parameters of wing

models in various aspect-ratio studies have been shown in Table I.The symbol ∗ denotes the values calculated based on the availabledata.

APPENDIX B: INSECT WING GEOMETRIESAND KINEMATICS

The data for the geometrical and kinematic parameters of vari-ous insects mentioned in previous studies have been summarized inTable II.

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TAB

LEI.

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0.57

72.

00.

150

0.71

1∗1.

422∗

0.61

8∗45

578

035

75∗

714

9∗0.

600∗

1.57

Extr

acte

dfr

omtim

etra

ces

(201

5)(r

otat

ing)

0.57

73.

00.

100

0.66

6∗1.

997∗

0.60

5∗45

578

034

99∗

1049

6∗0.

580∗

1.58

Extr

acte

dfr

omtim

etra

ces

0.57

74.

00.

075

0.64

3∗2.

574∗

0.59

9∗45

578

034

59∗

1383

8∗0.

550∗

1.54

Extr

acte

dfr

omtim

etra

ces

Phys. Fluids 31, 121301 (2019); doi: 10.1063/1.5129191 31, 121301-14

Published under license by AIP Publishing

Page 15

Physics of Fluids REVIEW scitation.org/journal/phf

TAB

LEI.

(Con

tinue

d.)

Lite

ratu

reW

ing

shap

eEX

P/C

FDr 2(S)A

b 0R g

/bR g

/cR g

/(b

+b 0

)α

Ret

=U

tc/ν

Re=

Ugc

/νRe

b=

Ugb

/νC

L tC

LRe

mar

ks

Kru

ytRe

ctan

gle

EXP

0.57

72.

00.

317

0.86

6∗1.

732∗

0.65

8∗45

1100

072

36∗

1447

3∗0.

605∗

1.40

Extr

acte

dfo

rα=

45

etal

.(r

otat

ing)

0.57

73.

00.

211

0.76

7∗2.

302∗

0.63

4∗45

1100

069

71∗

2091

2∗0.

536∗

1.34

Extr

acte

dfo

rα=

45

(201

5)

0.57

74.

00.

158

0.71

9∗2.

875∗

0.62

1∗45

1100

068

26∗

2730

6∗0.

538∗

1.40

Extr

acte

dfo

rα=

45

0.57

75.

00.

127

0.69

0∗3.

450∗

0.61

2∗45

1100

067

36∗

3368

1∗0.

556∗

1.48

Extr

acte

dfo

rα=

45

0.57

76.

50.

097

0.66

4∗4.

313∗

0.60

5∗45

1100

066

51∗

4323

0∗0.

548∗

1.50

Extr

acte

dfo

rα=

45

0.57

78.

00.

079

0.64

7∗5.

177∗

0.60

0∗45

1100

065

96∗

5276

9∗0.

539∗

1.50

Extr

acte

dfo

rα=

45

0.57

710

.00.

063

0.63

3∗6.

330∗

0.59

5∗45

1100

065

48∗

6548

2∗0.

518∗

1.46

Extr

acte

dfo

rα=

45

0.58

41.

50

0.58

4∗0.

876∗

0.58

4∗45

400

233∗

350∗

0.86

01.

53D

irec

tdat

apoi

nts

Shah

zad

Beta

CFD

0.58

43.

00

0.58

4∗1.

734∗

0.58

4∗45

400

233∗

693∗

0.80

01.

76D

irec

tdat

apoi

nts

etal

.fu

nctio

n(r

otat

ing)

0.58

44.

50

0.58

4∗2.

632∗

0.58

4∗45

400

233∗

105

3∗0.

780

1.85

Dir

ectd

atap

oint

s(2

016)

0.58

46.

00

0.58

4∗3.

514∗

0.58

4∗45

400

233∗

140

6∗0.

750

1.86

Dir

ectd

atap

oint

s0.

584

7.5

00.

584∗

4.37

8∗0.

584∗

4540

023

3∗1

751∗

0.71

01.

82D

irec

tdat

apoi

nts

0.57

71.

00

0.57

7∗0.

577∗

0.57

7∗45

577

577∗

0.48

4∗1.

45D

irec

tdat

apoi

nts

Jard

in0.

577

2.0

00.

577∗

1.15

5∗0.

577∗

4557

71

154∗

0.55

7∗1.

67D

irec

tdat

apoi

nts

and

CFD

0.57

73.

00

0.57

7∗1.

732∗

0.57

7∗45

577

173

1∗0.

579∗

1.74

Dir

ectd

atap

oint

s

Col

oniu

sRe

ctan

gle

(rot

atin

g)0.

577

4.0

00.

577∗

2.30

9∗0.

577∗

4557

72

308∗

0.57

7∗1.

73D

irec

tdat

apoi

nts

(201

8)0.

577

5.0

00.

577∗

2.88

7∗0.

577∗

4557

72

885∗

0.56

0∗1.

68D

irec

tdat

apoi

nts

0.57

76.

00

0.57

7∗3.

464∗

0.57

7∗45

577

346

2∗0.

537∗

1.61

Dir

ectd

atap

oint

s0.

577

7.0

00.

577∗

4.04

1∗0.

577∗

4557

74

039∗

0.52

8∗1.

58D

irec

tdat

apoi

nts

0.57

31.

80

0.57

31.

031∗

0.57

3∗45

167∗

300

0.49

2∗1.

50D

irec

tdat

apoi

nts

Bhat

0.57

32.

90

0.57

31.

667∗

0.57

3∗45

103∗

300

0.49

2∗1.

50D

irec

tdat

apoi

nts

etal

.Fr

uitfl

yC

FD0.

573

4.0

00.

573

2.29

1∗0.

573∗

4575∗

300

0.46

9∗1.

43D

irec

tdat

apoi

nts

(201

9)(r

otat

ing)

0.57

35.

10

0.57

32.

921∗

0.57

3∗45

59∗

300

0.44

3∗1.

35D

irec

tdat

apoi

nts

0.57

36.

20

0.57

33.

545∗

0.57

3∗45

48∗

300

0.42

0∗1.

28D

irec

tdat

apoi

nts

0.57

37.

30

0.57

34.

169∗

0.57

3∗45

41∗

300

0.40

3∗1.

23D

irec

tdat

apoi

nts

0.57

71.

50.

667

1.20

2∗1.

803∗

0.72

1∗45

1400

1010∗

1514∗

0.52

01.

00∗

LEV

lift

Phill

ips

EXP

0.57

73.

00.

333

0.88

2∗2.

646∗

0.66

1∗45

1400

926∗

2778∗

0.58

01.

33∗

LEV

lift

etal

.Re

ctan

gle

(flap

ping

)0.

577

4.5

0.22

20.

778∗

3.50

0∗0.

636∗

4514

0089

1∗40

09∗

0.60

01.

48∗

LEV

lift

(201

5)0.

577

6.0

0.16

70.

726∗

4.35

9∗0.

623∗

4514

0087

2∗52

31∗

0.65

01.

68∗

LEV

lift

0.57

77.

50.

133

0.69

6∗5.

220∗

0.61

4∗45

1400

860∗

6448∗

0.63

01.

67∗

LEV

lift

Phys. Fluids 31, 121301 (2019); doi: 10.1063/1.5129191 31, 121301-15

Published under license by AIP Publishing

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TABLE II. Insect data obtained from the work of Weis-Fogh,4 Ellington,35 Ennos,66 and Zanker.46

Literature Animal n (Hz) 2ϕA (rad) b (mm) A Re r2

Plecotus auritus 13 2.09 120 2.09 14 000 0.500Archilochus colubris 52 2.09 45 3.00 6 400 0.500Amaxilia fimbriata fluviatilis 35 2.09 59 2.95 7 500 0.500Patagonia gigas 15 2.09 130 3.02 15 000 0.500Melolontha vulgaris 62 3.14 28 2.80 4 700 0.500Amphimallon solstitiale 78 3.14 20 2.86 3 000 0.500Heliocopris sp. 38 3.14 70 2.92 17 000 0.500Heliocopris sp. 41 3.14 77 2.85 23 000 0.500Cetonia aurata 103 3.14 22 3.14 4 300 0.500Cerambycidae species 80 3.14 15.5 3.23 1 600 0.500Sphinx lingustri 30 2.09 50 2.17 6 300 0.500Manduca sexta 27 2.09 54 2.16 6 700 0.500

Weis-Fogh (1973) Manduca sexta 29 2.09 50 2.17 6 100 0.500Macroglossum stellatarum 73 2.09 21 2.10 2 800 0.500Amathes bicolorago 50 2.09 16 1.45 1 600 0.500Vespa crabro 104 2.09 24.3 2.64 4 200 0.500Vespula vulgaris 143 2.09 13.2 2.75 1 600 0.500Bombus terrestris 156 2.60 17.3 2.37 4 500 0.500Bombus lapidarius 143 2.60 16.6 2.37 3 700 0.500Apis mellifica 240 2.09 10 2.33 1 900 0.500Encarsia formosa 400 2.36 0.62 2.70 15 0.577Tipula sp. 53 2.09 17.3 3.76 770 0.500Theobaldia annulata 262 2.09 6.3 3.94 480 0.500Aedes aegypti 600 1.83 2.5 3.57 170 0.500Eristalis tenax 182 2.09 12.7 2.65 2 000 0.500Calliphora erythrocephala 159 2.09 9.7 2.62 1 000 0.500

Amaxilia fimbriata fluviatilis 58.5 3.90 6 100 0.487Ficedula hypoleuca 100 2.21 11 000 0.503Manduca sexta 51.8 2.74 5 400 0.499

Ellington (1984) Apis mellifica 9.8 3.37 1 600 0.544Tipula obsoleta 13.7 5.23 630 0.603Eristalis tenax 11.4 3.58 1 600 0.534Aeshna juncea 47.4 5.82 1 900 0.567

Tipula paludosa 59 1.95 17 5.24 5 975 0.598Bibio marci 99 2.43 11.2 3.34 4 304 0.552Conops strigata 144 2.51 7.7 3.95 2 053 0.593

Ennos (1989) Eristalis tenax 183 1.50 11.5 3.65 2 399 0.534Calliphora vicina 117 2.62 9.2 2.93 3 416 0.540Simulium 183 2.60 3.26 2.59 445 0.519Drosophila melanogaster 254 2.37 2.02 3.14 142 0.545

Zanker (1990) Drosophila melanogaster 218 1.22 2.47 2.91 101 0.573

APPENDIX C: POLYNOMIAL MODEL FOR CLPREDICTIONS

Previous studies have shown that the flow structure and themean lift coefficient of a wing changes with A, Re, and Ro. Ourprevious study32 showed that the effects of A on the flow struc-ture can be decoupled using the span-based Reynolds and Rossby

numbers, i.e., Reb and Rob, respectively. In that study, predictionsof CL were obtained over a three-dimensional parameter space ofranges 1.8 ≤A ≤ 7.3, 75 ≤ Reb ≤ 4000, and 0.5 ≤ Rob ≤ 3, coveringthose for most insects and wing models in the literature. These datawere used to fit a polynomial function P such that

CL(pred) = P[ f (A), g(Reb), h(Rob)],

Phys. Fluids 31, 121301 (2019); doi: 10.1063/1.5129191 31, 121301-16

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Page 17

Physics of Fluids REVIEW scitation.org/journal/phf

where

f(A) =A, g(Reb) = log(Reb), and h(Rob) = log(Rob).(C1)

Here, f(A), g(Reb), and h(Rob) are the functions of A, Reb, andRob, respectively, used to linearize the variables with respect to CL.In order to construct the model P, first, a polynomial model ofthe 4th order is built with all the possible terms in f(A), g(Reb),and h(Rob) as variables. Furthermore, the LASSO (least absoluteshrinkage and selection operator) method is used for the variableselection and regularization. This method helps fitting the datawith only selected variables, rather than using all of them. As aresult, the polynomial function P reduced to a linear combinationof 15 variables with their corresponding coefficients, as shown inTable III.

TABLE III. The polynomial model P was derived as a linear combination of thefollowing terms.

Term Coefficient

f(A) −0.020 131g(Reb) 0.179 068h(Rob) −0.015 914

g(Reb)h(Rob) −0.063 995f(A)g(Reb)h(Rob) −0.008 485

f(A)[h(Rob)]2 0.021 444

[ f(A)]4 −0.000 008[ f(A)]3g(Reb) −0.000 086[ f(A)]3h(Rob) 0.001 084[ f(A)]2[h(Rob)]

2 0.003 366f(A)[g(Reb)]

3 0.000 063[g(Reb)]

4 −0.000 066[g(Reb)]

2[h(Rob)]

2 −0.003 224g(Reb)[h(Rob)]

3 0.032 481[h(Rob)]

4 −0.056 562

FIG. 17. The comparison of the actual CL values and the predicted values usingthe polynomial model.

Approximately 100 datapoints were used to fit the model. Thecomparison of the predictions and the actual values can be seen inFig. 17. The rms error of this model was 0.604, and the R2 value ofthe model was found to be 96.9%. Using a lower-order polynomialresulted in higher rms errors and lower R2 values. The predictionsusing the polynomial model followed nearly the same trends, whichwere expected with the given range of variables, as can be seen inFig. 14.

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