AERODYNAMICS OF FLAPPING FLIGHT WITH APPLICATION TO … · a satisfactory explanation of the soaring flight of birds, and Walker (1927) gave a satisfactory quantitative discussion

[ 221 ]

AERODYNAMICS OF FLAPPING FLIGHT WITHAPPLICATION TO INSECTS

By M. F. M. OSBORNE

Naval Research Laboratory, Washington, 20 D.C.

(Received 6 November 1950)

(With Fifteen Text-figures)

INTRODUCTION

The problem of natural flight is one of long-standing interest, both intrinsically andin connexion with mechanical flight. Rayleigh (1883) was perhaps the first to givea satisfactory explanation of the soaring flight of birds, and Walker (1927) gave asatisfactory quantitative discussion of the flapping flight of birds in a particular case.Hoff (1919), using the data of Demoll (1918), attempted to bring the flight of insectsinto the domain of conventional aerodynamics by showing that lift coefficients ofreasonable value were required. He used only the linear velocity of flight to evaluatethese coefficients and ignored the motion of the wings. He also pointed out thesimilarity between the flow pattern around an insect and around a propeller oractuator disk, a similarity which must hold from momentum considerations, inde-pendently of the action of the wings (Bairstow, 1939). Demoll (1919) indicatedseveral errors and omissions in Hoff's work, so that in a number of cases the liftcoefficients required were inordinately high. Finally, a most thorough experimentalinvestigation was carried out by Magnan (1934), who was, however, unable to explainhis results theoretically.

The difficulty of high lift coefficients has not been removed by other investigationswhich have indicated lift coefficients larger than those normally expected in wind-tunnel measurements. It has, therefore, been believed that insects must utilize somespecial mechanism in flight not present in conventional aerodynamic phenomena.

In this paper the mechanism of insect flight will be examined in detail in order tofind some possible explanations for the high lift coefficients, as well as for othercharacteristic features of insect flight. General expressions will be obtained for theforce acting on a wing surface element moving in an arbitrary manner. This force willbe resolved into lift and drag components with respect to the relative wind, andintegrated over the surface of the wing and averaged over the period of a wing beatin order to determine the total average vertical and horizontal forces exerted by theinsect. The former is the required vertical force (weight of insect), the latter therequired thrust (drag of insect's body). Average values of the lift and drag coefficientsof the wings, CL and CD, can then be determined, since there are two known forcesgiven, and two unknown coefficients to be determined.

The scalar product of the vector force on the wing element into the vector velocityof the element with respect to the insect, integrated and averaged as before, gives themechanical power exerted by the insect in flight.

1 E B . 2 8 2 15

222 M. F . M . OSBORNE

The absolute magnitude of the total force on the wing element, integrated a m iaveraged, will give a lower bound to the force coefficient for the average total force,

The power, the lift, drag and total force coefficients computed for the twenty-fiveinsects for which data is available, will, when plotted against the other parametersof insect flight such as mass and ratio of flapping to linear velocity, reveal a numberof interesting and systematic characteristics of insect flight. The flight data, sum-marized in Table i (p. ), were taken from the work of Magnan, and were supple-mented by measurements on specimens of the U.S. National Museum.

For those who wish to draw conclusions from this work without verifying themathematical details, it may be said that the principal mathematical problem is tojustify replacing the instantaneous velocities by suitably chosen averages overposition and time. Once this process is admitted practically all of the qualitativeconclusions may be reached by drawing vector diagrams similar to those of Fig. 4,with these average velocities for any particular case. In fact, if the averaging preceptsare admitted, quantitative conclusions can be reached by drawing to scale, and withall small components such as v and w included, diagrams like Fig. 4.

DERIVATION OF THE FUNDAMENTAL FORMULAEIt is assumed that the lift force dFs on an element of wing surface c{r)dr is given by

dF.^pCÔÎlxWpfllxWyilxWl}, (1)and the drag force is given by

d F p = - ( | ) P C ^ ( r ) r f r | l x W | 2 { ( W - W . l l ) / | W - W . l l | } . (2)

The quantities which appear in these two formulae are defined as follows (see Fig. 1).Vectors are denoted by bold face, their scalar magnitude by two vertical lines.

1, m, n are mutually perpendicular unit vectors parallel to the axis of the wing,perpendicular to the axis of the wing in the instantaneous plane of beating, andperpendicular to this plane, respectively. The plane of beating is not the plane of thewing, but the plane in which its axis is moving, p is the air density. W is the relativewind or velocity of the surface element of the (right) wing with respect to the air, andisgivenby W = (nQxl r ) -U. (3)

Q, is the instantaneous angular velocity of the wing element, and U = (o, — (V+ v), —w)is the velocity of the air with respect to the insect's body. V is the velocity of flightand v, w are the induced velocities. c(r) is the chord, r and t are the independentvariables — distance along the wing and time. In the general case all of the quantitiesappearing in eqs. (1) and (2) (except p and c(r)) are functions of r and t. If the wingis assumed not to twist or bend, the unit vectors and Q will be independent of r.CD evidently represents the profile drag coefficient, since the relative wind includesthe induced velocities.

Average values for the induced velocities v, w, or velocity increments of the slip-stream, are given from momentum theory (Durand, 1935). They are independentof the mechanism of the wing action, and are obtained in first approximation by

Aerodynamics of flapping flight with application to insects 223

requiring that v and w at the insect have one-half the final value necessary to providethe necessary vertical force and horizontal thrust:

Lift (vertical force): L = Mg = nR2(V2 + w2)*210, (4)

Thrust: T=(\)CDbSbPV* = TrR?{V2 + w*?2v. (5)

R is the length of the wing, M is the mass of the insect, g is gravity, and Sb the bodycross-section area and CDb the body drag coefficient. Since, by division of the above,

Fig. 1. Quantities defining motion of insect in flight.

v\w = T/L, and for the insects considered the thrust T is small compared to thevertical force L. v is small compared to w, and will be neglected, since w turns out tobe small compared to the other velocities involved. The solution for w is given by

^ = P | [ - I + ( I + L2/n2 P

2R* F4)*]. (6)

Eqs. (1) and (2) express the assumption that the determining velocity (squared)for both lift and drag is the component of the relative wind perpendicular to the axisof the wing, which i s | l x W | = |W — W. 111. The subscripts s andp indicate perpendi-cular (senkrecht) and parallel to that component, W —W.U. The last factor in bracesgives the direction cosines of the forces. The drag is assumed to be perpendicularto the axis of the wing, hence the subtraction of the 1 component of W, W. 11 in the

15-2

224 M. F . M. OSBORNE

direction cosine. Forces along the length, or axis, of the wing tend to cancel for liftand thrust on the upper and lower half of a wing beat and overall between the rightand left wing, since they are in opposite directions for opposite wings. Also theydo not contribute to the power (eq. (8)) since they are perpendicular to the displace-ment of the wing elements, referred to the insect.

The assumption that the force is proportional to the square of the relative windcomponent perpendicular to the axis of the wing is the conventional one in helicopterand sideslip performance calculations, and applies primarily to steady flow. Steadyflow is far from realized in flapping flight, unless the variable or flapping componentsof velocity are small compared to the linear velocities, of the flapping velocity changesby a small fraction of itself over a displacement equal to the chord. The assumptionof the steady state force formula is made primarily so that the computed results forCL and CD can be compared with conventional aerodynamic data, as will becomeapparent from the subsequent discussion. However, there is indirect evidence givenbelow that this assumption is justified, at least in part, for the problem under con-sideration.

The total force on the surface element of the wing is

. (7)

The power expended in exerting the forces dF8 and dFp is

p, (8)

where PFs and PF are the power contributions due to CL and CD forces.The above formulae are general; the method of integrating and averaging them

would depend on the particular example to which they were applied. In case thenormal to the instantaneous plane of beating is the same for both wings, and remainsin the yz plane, as in Fig. 1, the unit vectors 1, m, n for the right wing are, in thex, y, z system, , . , .

J } l = (cos0 sim = ( - sin </>, cos <f> cos 6, - cos <f> sin 9), \ (9)

n = (o,sin0, cos0).

If 9, the inclination to the vertical of the normal to the instantaneous plane ofbeating (Fig. 1), is a constant, i.e. the wings beat in a plane, then Q. = d<j>/dt. Withthese two specializations, eqs. (1) and (2) give (bars indicate average)

Fs = (X., Ys, Zs) = vjjdFs (r, t) dr dtr,t

x (wsin<j>cosd +

r(d<f>/dt) sin 8 — w cos <j>,

V cos <f> + r(dcf>/dt) cos 8) (10)


p = (Xp, Yp,Zp) = vjjdFp(r,t)dr,j j p ( , ) , d tr,t

r,t

x (— r(d<f>/dt) sin <f> — Fsin <f> cos <f> cos 8 + w sin <j> sin 5,

+ r(d(f>/dt) cos <£ cos 0 + Fsin2 8 + Fcos2 <£ cos2 8 + w sin2 <f> sin 0 cos 0,

- r(d<f>/dt) cos <£ sin 8 + w cos2 &+w cos2 ̂ sin2 8 + Fsin2 <£ sin 8 cos 0). (i i)

The range of integration over r is the length of the wing; o to R, and in t over theperiod i/v of a wing beat, v being the frequency of flapping.

In eqs. (io) and ( n )

^m+^n^llxW^slW-W.ll^slW.mm + W.nnl2 *= r(d<f>/dt) + (Vcos8-wsm8)cos(/>, (12)

Wn = Fsin 8 + w cos 8,

and the denominators of the direction cosines in eqs. (1) and (2) have been cancelledagainst the velocity-squared factor. The factor 2 before c(r) takes account of left plusright wings.

The total average power* is from eq. (8)

P={wZs + VYS} + vJJdF^r, t). (n(d<f>/dt) x rl) dr dtr,t

T,t

x [r(d(f>/dt) + (Vcos8-wsind)coscf>]. (13)

Here the first term in the braces is the exact expression for PFs. The expressionfor power on the basis of the momentum theory (using eqs. (4) and (5)), is VT+zoL,which is the rate of change in kinetic energy in the slipstream on passing theinsect required in order to provide the net thrust T—{\)pCDbSbV

2 and lift L=Mg.Thus when CD=o ('ideal' fluid) the power determined from momentum theoryand the power determined by integrating force x velocity (i.e. first term in { } of(13)) for each element of the wing agree exactly, as they should.

This is indirect evidence that the assumed formula for Fs (eq. (1)) gives a correctresult when integrated in a case of periodic motion. The agreement between theformulae for the power computed by the two different methods corresponds, in thecase of a simple lifting airfoil of finite span, to the agreement between the workdone against induced drag, and the increase in kinetic energy of the trailing vortexsheet.

• By this is meant only the mechanical power (force x distance/time) expended by the insect.There is, of course, a much larger power loss in heating and chemical changes within the insect, asis true for any thermodynamic engine. No account is taken of this here.


In practice errors would be introduced by the fact that v and w are not constantin space and time, but the assumption of their constancy is a standard one in aero-dynamic practice, and the error committed is not large.

The integration of eq. (7) can usually be carried out exactly, and by replacing thetotal force by (T2+L2)i one can determine (Ci + Cf,)*. The result so obtained isa lower limit to the value of (Ci

L+C%)i, since the integration effectively adds all theforce elements in the same direction. One thus obtains

+ Wl)drdt. (14)r,t

The explicit integration of eqs. (10), (11), (13) and (14) is tedious and the resultssomewhat lengthy. The steps will not be given here, but only an outline of thenecessary additional assumptions and approximations. Eq. (14) can be integratedexactly.

The beating cycle is divided into two parts, the down beat, subscript 1, and theup beat, subscript 2. This was done since fairly constant conditions, particularly asto velocities, may obtain on a down or up beat alone, but not for the entire beatingcycle. Hence averaging methods and approximations may be used for a single downor up stroke not valid for the entire cycle. Also, the explanation of some of thephenomena of flapping flight require differentiation between the up and down beat.Over the up or down beat 0, CL, CD are assumed to be constant with respect to rand t, and to have the values

(15)

If P is the period of a wing beat, the subscripts 1 and 2 refer to the time intervalso < t <Ti\oix (down beat) and TT/O^<t<P (up beat), respectively.

<f> is assumed to vary sinusoidally over the up or down beat, where

</> = Asma)t, a>1 = w(i— rf), a>2 = w( i+? / ) , 7H = 2TTV(I — i)2).

A is the amplitude of motion in the plane of beating, and r/ = (a — i)/(a+1), where ais the ratio of duration of down beat to up beat, tj and 80 are assumed small, but £,, t,are not so restricted. The double sign for C& takes account of the possibility thatleading and trailing edge of the wing, or pressure and suction side, may be inter-changed on the up and down beat. This does not affect the sign of CDi.

The principal difficulty at this point is to obtain a satisfactory approximateexpression for the radical (W^ + Wffi; once this is obtained the integration is straight-forward. No single approximation could be found which was satisfactory for allvalues of the ratio of the flapping to linear velocity (rd<f>/dt)IV and the time. Thisratio varies from zero to much greater than unity over the length of the wing andperiod of a wing beat, and becomes both positive and negative.

One method is to replace the variable terms in eqs. (10), (11) and (13) by averages.Two preliminary remarks are first necessary:


fp(1) If we have a complicated expression to integrate, f(t) dt, and we can guess

Join some way an average value / , then Pf is a good guess for value of the integral.If circumstances are such that we can guess average values over two parts of therange of integration o to a, and a to P, then the average value for the entire interval

TC

J

In what follows, P corresponds to the period of a wing beat and the two intervalsto the period of the down and up beat, v\o>x and TT/O)2.

(2) Suppose c(r) is a simple function, o at the end of an interval o to R, and positivein between, like a semi-elipse or a rectangle. We wish to integrate an expression ofthe form ,R

7= (a0c{r) + a1rc{r) + a2r*c(r) + a3r*c{r)+...)dr. (16)Jo

The a's are assumed such that the expression converges. Now if 5 is the areaR

c(r) dr, I can be writteno _ _

I=a0S + a1fS+a2r*S + aar3S+.... (17)

Now for a simple geometrical figure as above f, r2*, r3*, etc., are all not so very differentand one could approximate the desired integral by substituting a single averagevalue for r for all those appearing, that is

IâoS+a^ S+a2r*S+a37*iS+.... (18)

The best choice would be the one for which the corresponding a was the largest.In the above case presumably a2 was the dominant coefficient.

To return to the integration of eqs. (10), (11) and (13). These are simply integratedby replacing the various variable terms by shrewdly chosen averages, the averagesbeing chosen over two stretches as above, the down and up beat. The choice foraverage values is made in the following way. (W^+ Wffi was expanded under thetwo separate assumptions that the ratio (rd</>/dt)/V was (1) small, and (2) large, andthe resulting expressions could then be integrated. In this way expressions corre-sponding to (16) above were obtained in which the coefficients a0, alt a2, a3>... wereconstant or easily integrated trigonometrical functions of time. Average values werefixed for r(r2* in (1 o) and (11), r3* in (13)) which fitted both approximations moderatelywell. Also averages for cos <f>, sin </>, cos cot were chosen which fitted both approxi-mations moderately well. (Both expansions (1) and (2) above are still fairly goodwhen (rd<f>/dt)/V~i.) These selected averages were then substituted directly intothe original eqs. (10), (11) and (13). As will be seen by comparing the equationsbelow with eqs. (10), (11) and (13), this process simply amounts to replacing all theterms in eqs. (10), (11) and (13) by shrewdly chosen constant values and integrating intwo steps. When this is done the results are as given below. S is the total area of bothwings. Let the average velocity components in parentheses in eqs. (10) be indicated by

Wgx = a>s'm<f> cos 6 + Vsin (f> sin 6, W^ = (rd<j>/dt) sin 6 — co cos <j>,

W8Z = Vcos<f> + (rd<f>ldt)cos6,


and similarly for Wpx, Wpv, Wpz'm. eq. (u ) . r, r2, r3 were determined by numericintegration over the wing form of each insect. Eq. (10) gives

= CLA

Eq. (n ) gives

Wmi = (

The quantities appearing in these two equations are defined as follows:7mx

= (r2* Acojz*) + (V cos 81 — w sin 8X) JVnj = V sin 81+w cos 8X,

^2/2*) + (V cos da — w sin 0S

+ w cos 02,

Jz^sind^wJÂ),>*) sin 82-wJ0(A),

(21)

+ VJl(A) cos2 8X + w(i6Jl(A)/7T2) sin 8X cos 8X,

WPVi = ( - r2* ̂ w2/2*) / 0 ( ^ ) cos 6>2 + Fsin2 (92

+ VJl{A)cos2d2 + K)(i6/f(^4)/7r2)sin(92 cos (92,

WpH = ( - r2* ̂ cVa*) sin 0! + a; cos2 0X

+ a ; / 2 ^ ) sin2 dx + V(i6Jl(A)/n*) sin 0X cos 0x,

Wpzi = ( + 7 2 i ^co2/2i) sin 02 + w cos2 02

+ wJl(A) sin2 02 + F( 16Jl(A)/n2) sin 02 cos 02. ,I are Bessel's function of the first kind obtained from the expansion

of cos </>, sin (f>, where <£=A sin cot.

(20)


If the expression in braces in eqs. (19) and (20) is abbreviated by A, B, C and Done imposes the condition that the sum of the horizontal forces Yp + Ys provide therequired thrust T=(\)pCDbSbV

2 and that the total vertical forces Zs + Zp providesthe lift L = Mg, then r = ^ + ^ C , L^C^ + C^. (22)

Then by Cramer's rule,

(23)

This gives, using the substitution indicated for A, B, C, D, the expression for theaverage lift and drag coefficients CL and CD in terms of the observed flight parameterfor any given insect.

In a similar manner one finds that the total power is

TL

AB

CD

TL

1/1/1/

ABAB

CD

CD

1

*

•J

P=wZs+VYs

* )*

^nf + (Fcos 6X - to sin 6J o-8J0(A)],

Wmi =

( 2 4 )

^n)* + (Vcos61-w sin 9±) o-8f0(A),

z{%n? + {Vcosdz-w sin 6,) o-8f0(A),Wni = V sin 0X + a; cos dly

Wn2 = V sin 6Z + w cos 62.

Notice that these are not the same averages as for eqs. (10) and (u) . Cubic termsare more important in the power, square terms in the forces. The averages for r andthe trigonometrical functions are chosen accordingly.

CD arid Ys, Zs in (24) are computed from the eqs. (23) and (19).Eq. (14) for the minimum total force coefficient can be integrated exactly. The

result is

2( Fcos d1 - w sin ̂ ) f Aco^/n) (fo(A) + (f )f2(A))

+ (Fcos 62 - w sin d2

Wlx = (Fsin 61 + w cos ^) 2 ,

2f\{A)),(25)

Note that this is again a different set of values for Wm and Wn, which here could beintegrated exactly.


A second method of integrating (10), (11) and (14) replaced (W^+ Wffi bysum of the absolute values of Wm and Wn, times a correction factor k which wasa very slowly varying function of the ratio Wm\Wn, taken as constant. Thus

From that step on the resulting integrations could be carried out almost exactly.The resulting formulae are exceedingly long, since a different set of formulae isrequired for each of the different sign possibilities for Wm, Wn and CL on the up anddown beat.

This second method has the advantage of showing analytically and directly theeffect on the vertical force, thrust and power, of varying the flight parametersCL> CD> & an<l w o n t n e UP a nd down beat. The effect of independent variationsappear as linear terms in g, £, 77 and 88, while the interactions appear as cross-productsof £,£,77 and 80.

Check computations by the two methods gave results in fair agreement. All ofthe numerical results given in this paper were obtained by the first method.

DETERMINATION OF FORCE COEFFICIENTSApplications of eq. (23) to determine CL and CD for the twenty-five insects given inTable 1 were made under two assumptions: £ = £ = 0, or equal force coefficients onthe up and down beat, and £ = £ = 1, or zero force coefficients on the up beat. It wasfelt that these represented the two limiting cases between which the actual practicemust lie. The results are shown in Figs. 2 and 3, which give the average force coeffi-cients for the entire wing beat or the value on the down beat, respectively. Thelength of the horizontal line represents the range of solution as the drag coefficientof the body varies between zero and unity, the points being plotted for zero-bodydrag coefficient.

It will be observed that for the case of zero force coefficients on the up beat theratio of lift to drag does not exceed 3 or 4 to 1, which is quite acceptable aerodynami-cally. Hence large lift coefficients are always associated with large drag coefficients.All coefficients are positive, the largest lift and drag coefficients being 5-4 and 2*6,respectively. In the case of equal-force coefficients on the up and down beat, thereare two instances of negative lift and drag coefficients, and six with zero or negativedrag coefficient. The largest lift and drag coefficients are 11 and 20, respectively.

Under both assumptions, therefore, somewhat larger force coefficients are derivedthan are met with in conventional aerodynamics, but the assumption of equal-forcecoefficients on up and down beat leads to the additional anomalies of zero or negativedrag and lift coefficients. It is, therefore, concluded that the assumption of zeroforce coefficients on the up beat must represent a much closer approximation to thetruth than equal force coefficients on the up and down beat. In other words, theinsect derives the great majority of the useful flying force on the down beat (Guidi,1938; Hoist & Kuckemann, 1942). This conclusion is reinforced by comparingpoints for the same insect on Figs. 2 and 3. The required lift coefficient for forcesacting on the down beat alone is in most cases only slightly greater instead of twice

Tab

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sa a

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(-14, -3)(-20, -5)

I-2 -1

Co,'

Fig. 2. Average lift a. drag coefficient for equal-force coefficients on up and down beat. f=£ = o.Points with arrows fell outside the range of co-ordinates of the figure. The co-ordinates are inparentheses.

5 -

4 -

3 -

Fig. 3. Average lift v. drag coefficient for down beat for zero-force coefficients on up beat, f = £ = i.


great, as might be expected, than when force acts on the wing during the entirebeating cycle. In a few cases, it is less. Except for the ' anomalous cases', the dragcoefficient is always a great deal less for the down beat alone than for the entire cycle.

There is a simple explanation for the conclusion that the great majority of theuseful flying force must come on the down beat. One draws the components of theaverage (with respect to r and t) relative wind in the yz plane (Fig. 4). It is specifiedthat the mean flapping velocity VF = (rd<f>/dt)2i = f2i Aw/2i is the same order ofmagnitude as the velocity of flight V, say, within a factor of 3 larger or smaller. Thisrange covers most of the insects considered. Fig. 4 illustrates a number of suchdiagrams showing various choices (1/3, 1/1, 3/1) for the ratio of flapping velocityto the velocity U, VF/U, and the inclination 8 of the plane of beating. Fig. 4 isessentially a geometrical interpretation of eqs. (19) and (20). In this way one candetermine to within a quadrant, referred to axes parallel and perpendicular to therelative wind, the direction in which the resultant force Fs + Fp on the wing duringthe up or down beat must lie. Now the average resultant force for both up and downbeat must provide the necessary vertical force Mg and thrust. This is a force which ispredominantly up, with a small forward component. If it turns out, as is usually thecase, that no matter how 8, the inclination of the plane of beating is chosen, thatfor the up beat any force within the 900 uncertainty has small or negative componentsin the direction in which the average resultant must lie, then any force acting duringthe up beat is undesirable. The insect will therefore tilt his wings to achieve as nearlyzero force coefficients on the up beat as possible.

If the flapping velocity is larger than the linear velocity and 8 is ~3O° or less(Fig. \e), there is some advantage for the insect to reverse the direction of thelift (CL force) on the up beat, by interchanging either the leading and trailing edgeon the up and down beat or the pressure and suction side. In the calculations thisassumption was made in those cases in Fig. 2 for which 8 was taken to be 30°—all except the six on the CD = o axis. There is some evidence that this 'reversedflapping' actually occurs, judging from the drawings and photographs in the literature.For 'reversed flapping' calculations the negative in the double sign in eqs. (15) and(19) is chosen.

When reversed flapping takes place there is also some advantage to the insect tomove his wings in a tilted figure 8, with the steep side on the down beat (80 > o). Inthis way the average resultant forces on the up and down beat more nearly approachparallelism, and the resultant on the up beat is more nearly in the desired direction.Observation is in agreement with this (Magnan, 1934; Schroder, 1928).

Fig. 4 also serves to show that the force perpendicular to the relative wind or CL

force is the one which most commonly provides the useful or flight-enabling force.Observations indicate (Guidi, 1939; Kukentahl & Krumbach, 1927-34, pp. 573-5;Magnan, 1935, pp. 61, 97,104,112, ii4;Marey, 1890) that 8 is usually 900 or slightlyless when the ratio (rd<f>/dt)2i/V=f2iA(D/2iV=VF/V is small and can decrease asthis ratio increases. Drawing the relative wind diagrams, as in Fig. 4, will show thatthe CL force is more nearly in the direction in which the average resultant must lie;the CD force is at best usually at right angles to the necessary resultant. For an

234 M. F. M. OSBORNE

\ cos(5V

Fs,

uFir.

(c)

(d)

Fig. 4. Graphical determination of forces on up and down beat in YZ plane and average resultantforce. Vrt or Vpt — root-mean-square flapping velocity. F,, Fv — ioxces perpendicular andparallel to relative wind. F,, Ft = average resultant force on down and up beat. F = averageof Fi and Ft. U=average velocity of air with respect to insect. The magnitude of the inducedvelocity u> is exaggerated in the vector diagram to the right of the insect. It is always very smallcompared to at least one of the velocities Vr or U, and is so drawn in (a) to (e).

£ = £ = 0. The necessary thrust is not provided unless Fl^Fl.

08 = 30°. Reversed napping.

(«)(b)

(«)(«0(«)

VT

VF

V,

— U, t

— U, t

~ [7/3SU/3

0 = 9 o o .

> = 9 o ° .

' = 45° ., 0 = 9OC

, <?i = 45


Ixception to this statement, see d, Fig. 4, down beat. Here both CL and CD forcesare nearly equal and equally useful. Thus the insect, in common with all successfulflying machines, primarily depends on the force at right angles to the relative windfor flight. Magnan (1934) sketches one exceptional insect, the Syrphus fly, for whichthe CD force is the useful one. Use of the CD force is one way to obtain very largeforce coefficients at the price of considerable power. Other examples of the use of theCD force will undoubtedly be found when more experimental data is obtained. Theywill occur when 6 is near 900 when hovering, or greater than 900 in forward flight. In

o1

10

10

0-1

I I I

I I

I

-

-

-

A

-

o• B

D

I I I I I

•i-M6

a

I I I I

•>*iit *

I

1

©

1

A

i i i i I i i0-1 10 10

Fig. 5. Minimum total force coefficient on down beat coefficient v. ratio ofroot-mean-square flapping velocity to velocity of flight. 8 = 300, | = { = 1.

any case the significant force, perpendicular or parallel to the relative wind, canalways be determined for any given mode of flight by drawing the relative winddiagram as in Fig. 4.

There remains to explain the occasional large lift and drag coefficients which appearin Figs. 2 and 3. To this end reference is made to Figs. 5 and 6 in which the minimumtotal force coefficient, determined from eq. (25), is plotted against the ratio of theroot-mean-square (with respect to r and t) flapping velocity to the velocity of flight.This ratio measures the importance of the acceleration or inertia forces; when smallthe inertia forces are negligible. It will be observed that there is a distinct correlationbetween this ratio and the total force coefficient; when this ratio is small the forcecoefficient does not exceed those values normally expected in aerodynamic practice.


Thus the abnormally large lift coefficients derived by previous investigators must 1attributed to acceleration effects. Both theoretically and practically there is no upperlimit to the force coefficient referred to the velocity squared if acceleration forcesare permitted. Thus for an impulsive start from rest, as in the case of an oar which isjerked, the terminal velocity may be very small and the force large. Hence the force'coefficient referred to the square of the average velocity will be very large. Thisconclusion holds for both potential and non-potential flow, though in the formercase the motion must not be perfectly cyclic.

10 p=

_ E

10 -

0-1

I I I I I

-

A

: ^

i i i

t•«

©

i i i i i

*

A

t4 v

H

i i

i

• *

o

i

Yi

I i I I i i I

10 10

Fig. 6. Minimum total force coefficient on down beat v. ratio of root-mean-squareflapping velocity to velocity of flight. 0 = 60°, £ = £ = 1.

Part of the high lift coefficient may be due to a slotted wing or flap effect. For thoseinsects with two pairs of wings which overlap, the anterior always overlay theposterior, which is the correct arrangement for a slotted wing. In some cases theposterior is hinged by hooks to the anterior, in which case the posterior may actas a flap.

In Figs. 5 and 6 the vertical lines run up to the values of (C\+ C2D)*, as determined

from the separately derived values of CL and CD given in Fig. 3. Computations of(Cx + C!))*^. were also made for the case of equal force coefficients on the up anddown beat (£ = £ = o) and for assumed 8 = 900 and o° (not given here), and all led toessentially the conclusions above. In the case of the remaining figures in this paper,


ey all refer to zero-force coefficients on the up beat (£ = £= 1). Calculations werealso made for £ = £ = o but are not given, since they indicated in some cases too smallor negative values for the power, corresponding to the zero or negative values of CD.

Since the observations did not provide values of 6 they had to be assumed, thoughin some cases they could be inferred from the drawings of Magnan. This assumed

10

_ E

10 -

-

-

-

1

- 1

- X/ \/ \

-

- ' ©

a

•

-0

1 1

1

11

-X-

V?1 /1 XYAJ

0 ;

©

00DB

a

1 1 1

v»1t1

•*X

1

YA

|

a©

8D

a

1 1 1

V

1

^++yn

©

c0

a

••

1 1

0° 30° 60° 90°

Fig. 7. Minimum total force coefficient on down beat as a function of the assumedinclination 6 of the plane of beating, f = £ = i.

value influenced the value of the double amplitude zA of the wing motion in theplane of beating, since the double amplitude projected on the vertical (xz) plane, 2</swas the observed quantity given by Magnan. Hence the smaller the assumed d, thelarger the derived 2A, which would vary between its projected value and 1800 as 6varied from 900 to o°. The effect of the assumed 6 on the derived minimum total

TEB.28,2 16


Iforce coefficient is shown in Fig. 7, where it can be observed by tracing out a curve™for a single insect that the force coefficient can at most increase by a factor of 4 or 5to 1 as 0 varies from o to 90°. The variation is much less for those insects witha projected double amplitude close to 1800 and for those insects for which no pro-jected amplitude was given, so that a constant amplitude in the plane of beating hadto be assumed. The range of solution for o < CDb < 1 is not given in Fig. 7; the rangewas negligible. Assumed values of 8 and A in Table 1 are in parentheses.

The actual values of 8 as used for the computations of CL, CD (Figs. 2 and 3) andthe remaining figures involving the power were chosen as indicated in Table 1.In general, for (r(d<j>/dt))2i/V> 1, 8 was assumed to be 300, whereas if this ratio weremuch less than i, 8 = 6o° was assumed. The values of 8 chosen by this rule werethe least values compatible with the available drawings of Magnan and others. Theconsequence of such a choice is that A is a little too large, and the derived values ofCL, CD and GDICL and the power a little too small. This is considered to be 'advan-tageous' to the insect in that the necessary power is a minimum. The range of solutionas 8 is varied can be estimated from Fig. 6.

From another standpoint also the derived average values of C'L, CD and {G\+ Cf,)*are minimum values. Since the actual instantaneous values of these quantities mustvary with r and t, at some time the instantaneous values must exceed their averages.

According to Table 1, col. 8, it will be observed that the down beat is slower thanthe up beat, or OJ1<O>2 (see also Marey, 1895, p. 228; Guidi, 1938, p. 1108). This isan ' advantageous' mode of flight to the insect, in that it tends to reduce the powerfor fixed frequency of flapping and fixed requirements of vertical force and thrust.Since the down beat must provide most of the force, and since the force goes up as thevelocity squared of the wing while the power goes up as the velocity cubed, it requiresless power for the same average force when a smaller .force is exerted over a longertime (w1 < w2) than a larger force over a shorter time (a^ > a>2).

GEOMETRICAL SIMILARITY AND POWERThe degree of geometrical similarity of the insects considered can be examined inFigs. 8 and 9. Here the wing length and area are plotted against the mass. The dottedline in each figure indicates the expected slope for geometrical similarity, i.e. thequantities plotted should be proportional to M*, M* respectively. Geometricalsimilarity is realized with considerable dispersion except possibly for the wing area,which seems to increase slightly faster than M}. Fig. 9 also shows that the Lepidopteraand Neuroptera (squares and circles), which tended to have smaller force coefficients,have relatively larger wing areas for their masses.

In contrast to the large individual departures from geometrical similarity for thewing area, the power as a function of the mass shows surprisingly small dispersion(Fig. 10). The power is very nearly proportional to the mass (dotted line indicatesproportionality). This proportionality is reasonable from the standpoint of meta-bolism, and is also in keeping with the observation that the wing area increasesslightly faster than Mi. From aerodynamic considerations it can be shown that

001

Fig. 8. Wing length v. mass of insect. The dotted line indicates the slope for geometrical similarity,RozMt.

10 -

001

Fig. 9. Wing area v. mass of insect. The dotted line indicates the slope for geometrical similarity,SccMi

16-2

i?•ii.

la-

10*

105

1 1

11

Mi

l

-

-

-

-

1 1 1 1 1 1

1 i

/ ,001

o

1 1

1/

1

l a/

1 1 M i

0-1

<;

/

I

T° ̂

•

i i i i i

/

/

11110

X

/

I I I I I 11110

Fig. io. Power v. mass, £ = £= i. The dotted line indicates the slope for simple proportionality.

I I I I I I I I I I I I I I I I I I

10

Fig. I I . Power per gram as a function of the total force coefficient. £ = £ = I.

-2S

10

0-1

001

-

:

r DO

—

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-

-

i I I I

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a

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XX

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&H

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7

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0-1 10 10

Fig. 12. Mass v. ratio of root-mean-square flapping velocity to velocity of flight.

10

Fig. 13. Total force coefficient on down beat v. mass. $ = £= 1.


other things being equal, either the wing area must increase faster than is require^by dimensional similarity, the power increase faster than the mass, or both, as seemsindicated in the present case. The uncertainty of the data renders this conclusion asyet only tentative.

Fig. 14. Ratio of power expended against drag to power expended against lift forces v. ratioof root-mean-square flapping velocity to velocity of flight. £ = f = 1.

Fig. 11 gives the specific power, or power per gram, as a function of the total forcecoefficient. They increase together, again with surprisingly small dispersion. Fig. 11indicates about io~2 horse-power per lb.—better than a man, scarcely in a class withaircraft engines!

There is a very slight negative correlation of mass with ratio of flapping velocityto velocity of flight (Fig. 12). There is also a slight negative correlation between total


force coefficient and mass (Fig. 13), which does not support Rashevsky's conclusionthat CL should increase with the mass (Rashevsky, 1944). In general, both lift anddrag coefficients for insects are large, larger than those for birds, in the few caseswhere they have been evaluated (Demoll, 1919; Walker, 1925).

In Fig. 14 is plotted the ratio of the power expended against Fp (forces in CD,second term of eq. (13)) to power expended against Fs (forces in CL, first term ofeq. (13)) as a function of the ratio of the root-mean-square wing-flapping velocity

DlPTERA

Tabinus bovinus

Sarcophaga carnaria L.

Musca domestics

Volucella pellucens Meig.

NEUROPTERA

Brachytron pratense Mull.

Calopteryx splendens Harr.

Pyrosoma minium Harr.

Panorpa comunis L.

—

•\X

o©

o9

Orthetrum caerulescens Fabr. O

Aeschna mixtra Latr. •

COLEOPTERA

Melontha vulgaris Fabr. X

Cetonia aurata - f

Lucanus corvus • *

Telepborus fuscus \

HYMENOPTERA

Xyhcope Wo/acea A

Bombus terrestr/s Fabr. A

Vespa germanica £>

Vespa crabra L. A

Apis mellifica L V

Amonophila sabulosa V. Del. ̂ f

LEPIDOPTERA

Papi/Jo podalirius •

Vanessa atafanta L. I

Pieris brassica L B

Macroghssa stellatorem L. 3

P/usia gamma L. H

Fig. 15. Identifications of points for all figures.

to the velocity of flight. Fig. 14 indicates that the fraction of the power expendedagainst Fp increases with the importance of the inertia forces. This shows the in-creasing price in 'wasted power' paid for the use of inertia forces in flight, since CD

forces are usually in an undesirable direction, as Fig. 4 showed.In Fig. 14 the dispersion is also surprisingly small, in view of the large individual

departures from geometrical similarity. The small dispersion of quantities involvingthe power, as opposed to quantities involving the geometrical dimensions, indicatesthat as power plants they are essentially similar, despite their great diversity in form.

The range of solution for quantities involving the power (Figs. 10, 11 and 14)as the drag coefficients of the body CDb varies between o and 1 is shown by thelength of the vertical lines, the points being plotted for CDb = o. The dependence ofthe power on CDb is considerable, a consequence of the dependence of CD on CDb.

There are four pairs of similar star-type points on the figures, connected by dottedlines. These are the Coleoptera. The upper point of each pair refers to calculations


ignoring the elytra as wings, the lower includes them as a part of thewings. This was done since the contribution of the elytra to flight was not known,but it was felt that these two cases were the limiting ones.

Fig. 15 identifies the points in all the figures, and Table i summarizes the pertinentflight data.

SUMMARY

1. General formulae are derived giving the lift, thrust and power when the wingmotion is specified. The formulae are applied to twenty-five insects for which quan-titative data are available. Average values for lift and drag coefficients, CL and CD,are derived by equating the weight to the vertical force and the thrust to the horizontaldrag of the body.

2. The large drag and lift coefficients obtained for insect flight are attributed toacceleration effects. There is a distinct correlation between (C£,+ Cf>)* and the ratioof the flapping velocity of the wings to the linear velocity of flight. When this ratioand therefore the accelerations are small, the force coefficients do not exceed thoseto be expected for flat plates. Owing to the nature of the assumptions and approxi-mations made, the values derived for CD, CL and CDjCL are minimum values.

3. Other characteristics of insect flight are discussed. In general, insects fly insuch a way as to minimize the mechanical power required. In most, but not all cases,the useful force is the one perpendicular rather than parallel to the relative wind.The wing tips should move in a figure 8, the down beat should be slower than the upbeat, and the majority of the necessary force must be supplied on the down beat.

4. Figures are given using the data from the twenty-five insects considered, showingaverage relations between power, specific power, mass, acceleration forces, forcecoefficients and geometrical dimensions. The power per gram, the 'wasted power',and the force coefficients all increase as the importance of the acceleration forcesincreases.

5. When plotted as functions of mass, quantities involving the power show muchless dispersion than quantities involving the geometrical dimensions. This is takento mean that despite the diversity of insect form, as 'power plants', they are allessentially similar.

6. A table of the observed or adopted flight parameters (frequency of beating, mass,wing area, velocity of flight, amplitude and orientation of wing motion) is appended.

REFERENCESBAIRSTOW, L. (1939). Applied Aerodynamics, p. 621. London: Longmans, Green and Co.DEMOLL, R. (1918). Der Flug der Insekten und der Vogel. Jena: Gustav Fischer.DEMOLL, R. (1919). Der Flug der Insekten. Naturwissenschaften, 8, 480.DURAND, W. (1934). Aerodynamic Theory, 4, 318. Berlin: Julius Springer.GUIDI, G. (1938). A study of the wing beats of pigeons in flight. J. R. Aero. Soc. 42, 1104.GUIDI, G. (1939). The wing beats of pigeons. J. R. Aero. Soc. 43, 457.HOFF, W. (1919). Der Flug der Insekten. Naturwissenschaften, 7, 159.HOLST, E. VON & KUCKEMANN, D. (1942). Biological and aerodynamical problems of animal flight.

J. R. Aero. Soc. 46, 39.KUKENTHAL, W. & KRUMBACH, T. (1927-34). Handbuch der Zoologie, VII, II. Halfte. Stresemann,

E., 'Aves', pp. 573—75. Leipzig: Walther de Gruyter.

Aerodynamics of flapping flight with application to insects 245IIAGNAN, A. (1934). La Locomotion chez les Animaux. I. Le Kol des Insectes. Paris: Hermann et

Cie.MAREY, E. J. (1890). Le Vol des Oiseaux. Paris: Libraire de l'Academie de Medicine.MAREY, E. J. (1895). Movement, p. 228. New York: D. Appleton and Co.RASHEVSKY, N. (1944). Flight of birds and insects in relation to their form. Bull. Math. Biophys.

6, 42.RAYLEIGH, LORD (J. W. STRUTT) (1883). The soaring of birds. Nature, Lond., 27, 534.SCHRODER, C. (1928). Handbuch der Entomologie, 1. Jena: G. Fischer.WALKER, G. T. (1925). The flapping flight of birds. J. R. Aero. Soc. 29, 590.WALKER, G. T. (1927). The flapping flight of birds. II. J. R. Aero. Soc. 31, 337.

Documents

AERODYNAMICS OF FLAPPING FLIGHT WITH APPLICATION TO … · a satisfactory explanation of the soaring flight of birds, and Walker (1927) gave a satisfactory quantitative discussion