Download - QUICK ESTIMATES OF FLIGHT FITNESS IN HOVERING …Flight fitness in hovering animals 173 Table 1. Shape factor a for the second moment of the area of two whole functional wings (or

7. Exp. Biol. (1973). 59. 169-230 l 6 gWith 23 text-figures

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QUICK ESTIMATES OF FLIGHTFITNESS IN HOVERING ANIMALS, INCLUDING NOVEL

MECHANISMS FOR LIFT PRODUCTION

BY TORKEL WEIS-FOGH

Department of Zoology, Cambridge CBz ^EJ, England

(Received 11 January 1973)

INTRODUCTION

In a recent paper I have analysed the aerodynamics and energetics of hoveringhummingbirds and DrosophUa and have found that, in spite of non-steady periods, themain flight performance of these types is consistent with steady-state aerodynamics(Weis-Fogh, 1972). The same may or may not apply to other flapping animals whichpractise hovering or slow forward flight at similar Reynolds numbers (Re), ioa to io*.As discussed in that paper, there are of course non-steady flow situations at the startand stop of each half-stroke of the wings. Moreover, it does not follow that all hoveringanimals make use mainly of steady-state principles. It is therefore desirable to obtainas simple and as easily analytical expressions as possible which should make it feasibleto estimate the forces on the wings and the work and power produced. In this way onemay make use of the large number of observations on freely flying animals to be foundin the scattered literature. It may then be possible to identify the deviating groups andto approach the problems in a new way. This is the main purpose of the present studies,which both include new material and provide novel solutions.

Major emphasis must be placed on simplicity. This involves approximations sincethe true flight system is so complicated as to be unmanageable. However, when weconfine ourselves to free flight and make use of the most reliable flight data available,the task is neither as difficult nor the conclusions as unrealistic as one would expect,since it is possible to introduce simple corrections.

Although entitled ' Quick estimates', this does not mean that the approach is super-ficial, but rather that a procedure has been devised whereby the flight performance of agiven animal can be evaluated quantitatively as well as qualitatively on the basis of onlya few accurate observational data and a minimum of computation. In other words,given reliable information about bodily dimensions and wing-stroke parameters, themethod enables one quickly to arrive at a first-order approximation so as to assesswhether the animal makes use of well-established mechanisms or employs unusual ornovel principles.

The following procedure is, in essence, the strategy of the present investigation.From measurements of the size and shape of the wings, the geometry of the wingstroke, the frequency of the wing beat and the weight of the animal which is sustainedin hovering flight one can calculate, on the basis of steady-state aerodynamics, theminimum coefficient of lift which must be ascribed to the wings. From the same mea-surements one can also calculate the Reynolds number under which the wings operate,

170 TORKEL WEIS-FOGH

Vertical

wHorizontal

Chord, c(r)

Fig. 1. Simplified diagram of normal hovering flight. (A) The instantaneous forces.(B) The animal seen horizontally from the side, and (C) vertically from above.

and from this figure one can obtain (using published data on wings) the maximumcoefficient of lift which may be expected under these conditions. If the minimumcoefficient of lift for steady-state aerodynamics does not exceed the maximum coeffi-cient of lift obtainable at the Reynolds number in question one assumes that steady-state aerodynamics are adequate to explain hovering flight, and from then one can goon to calculate the power requirement and other parameters of the flight mechanism.If, on the other hand, the minimum coefficient of lift for steady-state aerodynamics isgreater than the maximum obtainable, it is clear that hovering flight cannot be explainedby steady-state aerodynamics and that a new approach must be made.

AERODYNAMIC RELATIONSHIPS

The main simplifications are that the animal is assumed to make use of steady-stateflow patterns only, that lift is produced at right angles to the direction of the relativewind, that the stroke plane is horizontal with no tilt, that the induced wind can bedisregarded because it is relatively small and, finally, that the wing movements aresinusoidal and of similar shape and the same duration for both the morphological

Flight fitness in hovering animalsdr

171

Fig. 2. Diagram of the wing parameters (A) and the movements during hovering (B).

upstroke and downstroke. As we shall see (p. 192), these assumptions lead to values forthe average lift coefficients CL needed to remain airborne which are slightly higherthan in the more complete treatment (Weis-Fogh, 1972). As to the specific aerody-namic power PJ, the estimates are too small by a factor of about 2 but, as we shall alsosee (p. 195), it is possible in a simple manner to make appropriate corrections. Theinstantaneous aerodynamic force on a wing element F(t, r) of chord length dr) and itsvertical (lift) and horizontal (drag) components are shown in Fig. 1 A. In Fig. 1 and inthe text, any quantity X which is a function of time t and distance r from the winghinge, or fulcrum, is written as X(t, r). Appendix 1 gives a list of symbols and constantsused.

The justification for the simplifications is that the majority of insects, birds and batshave been found to tilt the long axis of the body towards the vertical (Fig. 1B) whenhovering, so that the wings beat in an almost horizontal plane, usually symmetricallyabout the average positional angle y of 900 (Fig. 1C).

(a) Coefficient of lift

Fig. 2 A shows how the outline of a wing of total length R, and in particular thewing chord dj), varies with the distance r. In practice I have found that most wing


contours can be described quite faithfully by means of a simple mathematical functionlwhether the moving wing is morphologically two-winged as in Lepidoptera andHymenoptera or consists of only one part as in Diptera. It should be stressed here thatthere are hovering insects which make use of two pairs of wings which beat out ofphase, as is the case in Odonata and Neuroptera, and these insects are not included inthe present analysis.

The angular movement of the long axis of the wing in the horizontal stroke plane is

7(0 = y + W>sin(2nnt), (1)

where y is the average angle, <f> the stroke angle, n the wing-stroke frequency (numberof complete stroke cycles per unit time) and t is time. The angular velocity andacceleration are then

dyjdt = 77710 cos {zrnit), (2)

d*yldt2 = -2nW<f)sin(z7mt). (3)

The instantaneous force on a wing section at distance r from the fulcrum dependson the square of the instantaneous velocity of the section (v(t, r))2, on the force coeffi-cient, and on the area of the segment of width dr (see Fig. 2 B). In the case of theinstantaneous lift produced by the section, we then have

dL(t, r) = \px CL(t, r) x area x velocity2. (4)Thus

dUt, r) = \pxCL(t, r)xc{r)drxr\dyjdtf, (5)

where p is the mass density of air and GL(t, r) the instantaneous coefficient of lift atdistance r. This means that

dUt, T) = i/OT^V2 * CL(t, ')x c(r)ridr x cosi(2nnt). (6)

The four terms will be treated in turn. The first is a constant. The second (CL(t, r))will be taken as constant in this treatment for the following reasons. In large insectsand hummingbirds the wing is twisted linearly with respect to r so that succeedingsections have the same geometrical angle of attack (Jensen, 1956; Hertel, 1966). Insmall insects like DrosopMla the wing is rotated as a whole plate but the lift coefficienthardly varies at all with the angles of attack actually applied by that insect (Vogel,1967a). In both cases the angle of attack is set almost instantaneously at the beginningof each half-stroke and remains constant so that it is justifiable to consider the averagecoefficient of lift CL as a constant, as previously discussed (Weis-Fogh, 1972). Ofcourse this need not always be the case, but it is a reasonable approximation to beginwith. It also has the advantage that one can use steady-state lift/drag diagrams forfixed wings of known shape and aspect ratio later in the analysis.

The third term in equation (6) is the derivative with respect to r of the secondmoment S of the total wing area of all wings about the fulcra; it depends only onhow the area varies with the distance r from the base, i.e. it is a function of the contourof the wing. Since many biologists are unfamiliar with this expression it is treatedin Appendix 2. The general expression is

S = <rcR\ (7)

where a is a shape factor characteristic for the particular wing shape. It can be found bygraphical integration based on actual wings or from the mathematical function by

Flight fitness in hovering animals 173

Table 1. Shape factor a for the second moment of the area of two whole functional wings (ortwo pairs of wings) (eq. 7), shape factor rfor the third moment of two whole wings (eq. 50),together with the radius of gyration rg, the chord length ca at distance r^ the factor in eq.(11 b)for the calculation of Reynolds number for the wings shapes considered, and the factorin eq. (35)/or calculating A, the ratio stroke-arc I wing-chord.

(a)hape... Half ellipse

-

r, \Rc, 087cFactor in 8-7eq. (116)

Factor in °"575eq- (35)

Parabola '.

Tot

•R/VA0-6906-6

0693

(c)tectangle

R

1I

I-OOC

n-5

O-S77

wTriangle, base Triangle, apex

-

c

i

o-4ic3-3

O-995

^ ^

f

o-yicio-o

0-995

means of which the outline of the wing is described. The chord of the wing c is thendefined in each single case, as is explained in Appendix 2 and Table 1, where someshape factors are listed. It should be noted that the third term is a function only of rand not of t.

The fourth and final term in equation (6) is a function of time / and not of distance r,and we therefore have, for a complete quarter-stroke, that the average lift L is

L = X I*' c(r)rldrx [ ~ cosi(27mt)dt.J r-0 J t-0

(8)

A quarter-stroke is the time which it takes for the wing to move from an extremeposition and until it reaches the middle of its stroke path. We have already dealt withthe first integral, which depends only on the wing shape and is equal to S. The seconddefinite integral equals $. Since the movements and forces are assumed to be sym-metrical about the middle position (harmonic motion) the average lift over eachquarter-stroke must equal the weight G of the animal so that the average lift coeffi-cient is

(9)

(b) Reynolds number (Re)For a given aerofoil the coefficients of lift and drag depend not only on the angle of

attack but also on Reynolds number (Re). This is particularly so in tiny insects, whereit may be as low as 10 or even smaller (cf. Horridge, 1956). The task is to find a simplebut representative expression. The general expression is

fl V ' V '

where /* is the viscosity of the air and the ratio (i\p is the so-called kinematic viscosityv. In all systems of measurement, v has the dimension of (length)2/time. At 20 °C and

174 TORKEL W E I S - F O G H

normal atmospheric pressure v is o-i4cma/sec in the units used here. In ordinarffixed-wing airplanes the representative velocity is the airspeed and the length is somerepresentative wing chord. In our case we much make a specific choice. During hover-ing 80% of the lift is produced during the middle part of a half-stroke, where theaverage value of cos(27rcif) in equation (2) is 0-89. As to distance from the fulcrum r,we will choose the radius of gyration rg of the second moment of the area of the chosenwing shape (whether rg is determined graphically or analytically) and use the chordlength at that distance, cg. Some useful values of rg and cg are included in Table 1.We then have

=

Since rg and cg are functions of R and c, a practical expression for Reynolds number is

(Re) = factor x cRnfi, ( I r*)

where the factors relating to the wing shapes used in this study are also included inTable 1.

(c) Drag, aerodynamic torque and powerAs a corollary to equation (5), the instantaneous drag D(t, r) experienced by a

wing section isdD(t, r) = ip x CD(t, r) x c(r)r*dr x (dyjdtf. (12)

The bending moment Qa about the fulcrum, which this gives rise to, is the product ofthe drag and its distance r from the wing base, thus

dQaV, r) = lp* CD(t, r) x c(rydr x (dyldt)*. (13)

Since we have given reasons for using the value of the lift coefficient as averagedover both time and distance in this particular type of flight, we must do the same forthe conjugated drag coefficient so that the second term is a constant CD.

The third term is the derivative with respect to r of the third moment of the area ofall wings about the fulcra dT(r). The instantaneous aerodynamic bending momentof the wings which the thorax must act against is therefore

Qa{t) = \pCD x I]]**)** x {^j\ (14)The definite integral is the third moment T of the wing area about the fulcrum. It isnot normally encountered in engineering and it is therefore treated in Appendix 2,from which it is seen that T = TCR*, where r is the appropriate shape factor (cf.Table 1). The bending moment then varies with time as

Qa(t) = yC^cR^n^n^cos^znnt). (15)

The aerodynamic work done by the thorax on the air during a quarter-stroke is

which, when combined with equation (14), gives

f

Jt- in

cos3(znnt)dt. (17)1 - 0

PFlight fitness in hovering animals 175

ince the value of the definite integral is 1/(37771), the total work done by the wingsduring a complete wing-stroke cycle consisting of 4 quarter-strokes is

Wa = $pCDTc&n*n*p. (18)During steady-state hovering the average aerodynamic power which the thorax mustproduce in order for the animal to remain airborne is therefore

?a = nWa = $pCDTC&nW<p. (19)In addition to the quantities already discussed in equation (18) and (19), we must

find CD in order to apply the equations. In our case, and since we base the analysis onsteady-state flow, this is done by using experimental lift/drag diagrams of knownwings of suitable form and aspect ratio. Ideally one should use the actual wings of theinsect in question, but this is only possible in a few cases at present. The chosendiagram is stated in Table 4 for each species considered. The choice is based uponwing shape, wing profile, Reynolds number and, of course, availability. When CL

and (Re) are known, CD then follows, but apart from locust wings (Jensen, 1956), theother estimates suffer from the shortcoming that the lift/drag diagrams were obtainedin a linear air stream of uniform velocity. The rotational components, the variation inspeed from base to tip and the effects that this may have on details of the flow aretherefore disregarded. Future experiments may remedy this shortcoming but theeffect is not likely to be serious for a first approximation as attempted in this paper.However, there may be exceptions for some species as pointed out in the Discussion.

WING INERTIA AND DYNAMIC EFFICIENCY

If / is the moment of inertia of the mass of all wings about the fulcra, the accelerationof the wings gives rise to an inertial bending moment Qim According to equation (3)it is

SpyQJt) = l-± = -2/7rV0sin(27m*). (20)

During harmonic oscillations Q± will then vary linearly with the positional angle y andbe maximum at the extreme wing positions and zero at the middle of the stroke.Numerically the maximum value is

|£?i|max = 2lnW<l>. (21)According to equation (15) the aerodynamic moment is maximum when the wingspass their middle position, zero at the extreme positions, and

|£?«| max = kpCDTcRhT^(j>\ (22)In order to calculate the total work done by the thorax when both inertial and

aerodynamic bending moments are taken into account, we must integrate Q^+f, overa complete wing stroke. This was discussed in detail in a previous paper on the basisof some actual examples (Weis-Fogh, 1972). The present account is easier to followif the reader is familiar with the approach given in that paper, and it should be empha-sized that the following derivations are valid only if elastic torques inside the thorax orin the wings themselves are disregarded.

The aim is to deduce a simple analytical expression for the relative importance ofinertial forces as compared with aerodynamic forces in the absence of elastic defor-


I I

20 30 40Distance from fulcrum (mm)

50 60

Fig. 3. The distribution of the wing weight as a function of the distance from the fulcrumin a standard Sckutocerca gregaria. (A) One forewing (1) and (B) one hindwing (2). (C) is amodel of the mass distribution and applies to insects in general.

mations and storage of energy. This is most easily achieved if we use the ratio Nbetween the maximum inertial and aerodynamic moments in equations (21) and (22),according to which

It is now possible to find an expression for the dynamic efficiency 77 of the system,where T\ is the ratio between the total aerodynamic work and the combined inertialplus aerodynamic work. Again disregarding elastic deformations,

V = (24)

However, we must first find a value for / .

(a) Mass moment of inertia

In Table 4, / is the moment of inertia of all the moving wings about the fulcra, but itis simplest to consider a single fully unfolded wing. The mass of a narrow transversewing section at distance r is pc(r)d(r)dr, where p is the average mass density of thewing material, d(r) the average thickness, dr the width of the section and c(r) thechord.

I = p[~Rc{T)d{T)r*dT. (25)

In practice, I is usually found by cutting the wing into strips normal to the longaxis, estimating the mass of each section by weighing and finding the sum of the pro-ducts of mass times distance squared. In insects, however, I have found that an easierprocedure is often possible and sufficiently accurate for our purpose.


Table 2. Moment of inertia I of the mass of one fully unfolded fresh vringabout the main vmg hinge in a standard Schistocerca gregaria

(Ten old and 11 young adults of both sexes. The experimental results are comparedwith those calculated from equation (27). There is no significant difference.)

Length R (mm)Weight (mgf)

/ (gf cm sec1):ExperimentalCalculated

One forewing

57-1 ±°-i29-5 ±0-7

io~* (i-7i±o-o4)10-4 (1-63±005)

One hind wing

S3-i ±°-223-610-5

io~* (I-IO±O-O3)10-4 (1-1310-03)

Locust wings. The distribution of mass in fresh wings of Schistocerca gregaria is seenin Fig. 3 A and B. The insects were of different sizes but all results were recalculatedto a locust of average size, a standard Schistocerca, by multiplying lengths with theratio of the length indices, ///s, and weights by the ratio of the volume indices, v/v,,where /, and vs are the standard indices for length and volume and / and v those of theindividual insects used (Weis-Fogh, 1952). The standard deviations of the correctedfigures from 21 males and females of different adult ages are shown as vertical bars.When a straight line is drawn from the wing tip and adjusted to follow the curves asclosely as possible, it is seen that the mass tends to decrease linearly with distance rfrom the fulcrum both in the broad fan-shaped hindwings and in the stiff slenderforewings. The deviations occur mainly towards the base, where r is small and theeffect negligible. Consequently we can calculate / as being equal to that of a homogen-ous triangular plate of uniform thickness d and which is rotated about its base line c(Fig.3C).

In this case c(r) = c{R — r)/R, and a strip of width dr has the mass pdc(R — r)/R x dr.We then have

{y ^ (26)r - 0

Since the total mass of the plate is M = \pdcR,

1 = * M # » . (27)

This simple expression gives values for locust wings which are accurate within thelimits of the experimental estimates based upon weights and distances, as is seen inTable 2.

Other insects. It is often difficult to cut and weigh small insect wings, and for quickestimates it is obviously advantageous to use an expression like equation (27). Is itapplicable to other insects? In order to answer this I have calculated the moment ofinertia by means of equation (27) for the insects in which Sotavalta (1952, 1954) found/ by weighings. His results are of course not as accurate as those for the locust becausethe wings are small and only a few strips could be cut. Nevertheless, the conformitybetween the calculated and the empirical results is good, as is seen from Table 3, inparticular when the sources of error are taken into account. In fact, if accurate measure-ments are not readily available, the formula provides values which are entirely ade-quate for the present treatment. For flying insects considered as a group, the formula

ia EXB 59


Table 3. The moment of inertia of the wing mass about the wing hingein different insects

(The deviations of the values calculated from equation (27) and those foundby Sotavalta (1952, 1954) are given as a percentage.)

No. of Genera consideredInsect order examples (deviation in parentheses)

Odonata 3 Sympetrum ( + 22), Aetkna ( — 35)

Coleoptera 3 Mdolontha (o), AmpkimaUon (o), Cerambydd (+ 5)

Lepidoptera 8 Agrotii ( — 11), Ampkitrota ( — 5), Sideridu ( — 20),Amathet ( — 21), Poecilocampa ( — 8), Macroglossum( + 17), Pieris(-i9).

Hymenoptera 5 Apis (—17), Bombus ( —16), Vespa (—14),Opkion (-22)

Diptera 5 Tipula (-36), Trichocera (-15), Theobaldia (-28),Eristalis ( + 9), Calliphora ( + 7).

Averagedeviation

- i 6

+ 2

— II

- 1 9

is accurate and it must reflect some structural principle of general significance, to bediscussed in another context (Weis-Fogh, in preparation).

Effect of wing mutilation. In order to discuss some results reported in the literature(Danzer, 1956) we must deduce how / alters when the wings are mutilated. Weassume that / = ^MR2. Let the moment of inertia of the mutilated wing be Im. Itcan now be calculated by means of the theorem of parallel axes. In Fig. 4A let Mx

be the mass of the outer cut section whose centre of gravity is distance r from the ful-crum. The moment of inertia of the cut section about the fulcrum is now

Ix = I0+Mxr*, (28)

where /„ is the moment of inertia of the section about its own centre of gravity. It is70 = ^MjA:2, where x is the distance from the wing tip to the mutilation cut m. Letthe relative distance from the tip be £ so that £ = o at the tip and £ = 1 at the base.We then have that * = £R, Mx = ME? and r = R-^2£Ji) (see Fig. 4A). Thus

- (29)Ix =

For the mutilated wing7m = / - / x = £Mi?*(i-/(£)). (30)

The numerical value of the functions/(£) and (1 —/(£)) are plotted in Fig. 4B. Thesecurves are of general validity provided that the wing mass decreases linearly with r,i.e. provided that equation (27) is a sufficiently good approximation for the particularenquiry.

(b) Dynamic efficiency

As discussed earlier (Weis-Fogh, 1972), we shall not count work which is absorbedby the working muscles in the thorax by their braking action, the so-called negativemuscular work, but only the positive work imparted by the thorax to the moving wings.In order to do this our task is to integrate the function of the combined aerodynamicand inertdal bending moments Q^^ (7) with respect to y when the wings move fromone extreme to the other, i.e. over one complete half-stroke. This is sufficient since the

Flight fitness in hovering animals 179Fulcrum

0-8 -

a

I 0-6

B 0-4 -

0-2 -

N 1 1

-

-

r

.''' 1 1

1

i-AD

\

\

t

ii

1

1 1

/

(

\

\

i i

1 1 1 - - " T

*

/AD

-

-

1 1 T-——L

0-2 0-4 0-6 0-8

Relative distance from tip,£

10

- 0-8

- 0-6

- 0-4

- 0-2

10

Fig. 4. The effect on the moment of inertia / „ of mutilating the wing of an insect by means of atranverse cut (m), as explained in the text and in equation (30).

two half-strokes are similar with respect to work. More precisely, we want to find theratio between the following two integrals, eventually expressed in terms of the ratio N:

QM<ir. (31. 32)

An exact solution is lengthy and tedious to use. The procedure adopted here isneither exact nor elegant but it leads to a simple approximation which is accurate towithin a few per cent, which is more than adequate for our purpose.

Since we are dealing with ratios rather than absolute figures, let the angular move-ment represent 1 angular unit so that the wing moves from y = o to y = 1 during theupstroke in Fig. 5. Let the maximum aerodynamic moment in equation (22) be repre-sented by +1 unit. The maximum inertial moment in equation (21) is then +N unitsat the beginning of the upstroke and — N units at the end. While Qi varies linearly,Qa varies according to the thick curve ABC. The combined curve Qfa+f) is repre-sented by the thin curve from + Af to E'. It is the area between the latter and the baseline AE' which represents the total positive work during one half-stroke.

If the aerodynamic torque also varied linearly from minimum to maximum and frommaximum to minimum, the work during the first quarter-stroke would correspond tothe area of triangle ABD and be \ work units; during the second quarter it would be

(epresented by DBC and also amount to £. In this case the combined work J ^1

i8o TORKEL WEIS-FOGH

+3

+N

+2

I-.-

- 2 -

- 3

A D

\ \\ \\ \\ \X \

\ \\ \

\ \\ \

\ \\ \

N. \

\ \

-N

7 = 0-5 J W = lPositional angle y (relativo units)

Fig. S- The work diagram of a hovering animal and the method of integrating the total workduring one half-stroke, as explained in the text.

would be represented by the area of the quadrangle ANBE, which is

-NJ4{N+i)) work units.

However, we have to add the two dotted areas above the lines NB and BE' respec-tively. From equation (17) we already know the exact value of the aerodynamic workduring one quarter-stroke. In our case the area is \ work units so that it is ^ unitlarger than the triangular area during the first quarter. During the second quarter thearea to be added is approximately equal to the densely dotted area EBE", leaving outthe small horizontally hatched area EE'E'. The dotted area is equal to the ^ unit timesthe ratio DE/DC, which is i/(iV+1). We then have for one half-stroke that

N 124(iV+i) '

For large values of N the error is insignificant and for very small values it is alsounimportant, amounting to 0-2 % for N = o, i.e. when the inertia forces are zero. Theerror may rise to a few per cent at intermediate values; this does not matter here.

Since Wa amounts to $ work units per half-stroke, the dynamic efficiency is

V =Wa

(34)


Fig. 6. The relationship between the dynamic efficiency rj (in the absence of elastic forces)and the ratio N between the maximum inertial and aerodynamic bending moments inequation (33).

N is easily found from equation (23) and the relationship between 77 and N isplotted in Fig. 6, from which the dynamic efficiency can be read. When the inertialforces are vanishingly small and N = o, the efficiency is of course i-oo. When the twobending moments are equally large it is 0-85, but for N = 5, which is quite common,V is 0-37.

EXPERIMENTAL METHODS AND MATERIAL

Most of the flight data derive from critical examination of the literature, but it wasessential to obtain new information concerning both large and very small insects.

(a) Syrphinae hovering in nature

The true hover-flies belong to the subfamily Syrphinae within the large family ofSyrphidae. So far, the best studied genera have been Volucella and Eristalis, whichrepresent two other subfamilies; they hover in a way similar to that of wasps and bees,i.e. in accordance with our main assumptions (see the definition of 'normal hovering'in the next section). However, the Syrphinae fly forwards, sidewards and hover withthe body axis horizontal and with the wings oscillating in a non-horizontal plane andthrough a surprisingly small stroke angle. Obviously, they are exceptions to the generalrule; the same applies to the Odonata (damsel flies and dragonflies). While I have beenunable to study hovering dragonflies in nature, I have recently succeeded in recordingthe wing-stroke frequency of Syrphinae of the genera Syrpkus, Sphaerophoria andPlatychirus together with the body weights and wing parameters of the same individuals.The animals hovered in front of flowers or between vegetation where there was nowind on calm summer days at Tibirke in Denmark. The flight tone and therefore thefrequency was picked up by means of a Briiel & Kjaer 25 mm Condensor Microphone


Type 4145, with battery-driven Pre-amplifier Type 2619 and Power-supply Typl2804. The signals were recorded on a Nagra IV-S battery-driven tape-recordersupplied with a special noise-reduction system which ensured flat response characteris-tics between 100 and 970 Hz. The steep cut-off at either end corresponded to a reduc-tion of 20 dB at 30 and 2000 Hz respectively. This was essential in order to preventinterference from other sound sources. The individuals were narcotized instantan-eously while hovering by means of a strong jet from a Sparklets CO2-spray filled withchloroform and caught in a small net. The weights of the body and the wings weredetermined by means of an electronic microbalance immediately on return to thelaboratory. The insects were identified according to Coe (1953).

(b) Free hovering in the laboratory

Large dung beetles. The flight of medium-sized and of small lamellicorn beetles likeMelolontha vulgaris (cockchafer), Rhizotrogus (now AmphimaUon) and Cetoma aurata(rose chafer) have been studied both in nature and in the laboratory (Sotavalta, 1947,1952), but since the papers by Osborne (1951) and Bennett (1966, 1970), the aero-dynamic mechanisms involved have been in dispute. In particular, Magnan's (1934)data for the large stag beetle Lucanus cervus have been taken as evidence for substantialcontributions by non-steady flow. Thus Osborne (1951) found an average lift coeffi-cient CL of more than four. The beetle weighed 2-6 gf and had a' wing load' of 3*2 kgf/ma. I could not obtain this species but Professor J. W. S. Pringle, Oxford, broughtsome large elephant-dung beetles of the genus Heliocopris back from Kenya. He kindlymeasured the frequency acoustically during near-hovering, free flight and also gave meweights (8-13 gf) and wing dimensions; the wing load was 3-4 kgf/ma and thereforecomparable to that of the stag beetle. As in other lamellicorn beetles, the stroke angle <j>is about 1800.

Large hawk moths. Dr P. J. Wilkin, Brunei University, has recently shown me(personal communication) recordings of wind-tunnel experiments with the Floridatobacco hornworm moth, Manduca sexta (Johannson), which was flying suspendedalmost horizontally from an aerodynamic balance. His results indicate that, undercertain conditions, these insects can produce lift coefficients considerably in excess ofsteady-state values, and he seems to have confirmed this in model experiments. Thequestion is how and to what extent freely flying sphingid moths make use of unusualaerodynamics - for instance, when feeding normally like hummingbirds in front of aflower. Caterpillars of Manduca sexta (Sphingidae, subfamily Acherontinae; earliername Protoparce sexta Johannson) were grown in the laboratory on a semi-syntheticdiet (Hoffman, Lawson & Yamamoto, 1966). Adults were isolated and hovered spon-taneously in front of an artificial flower but only shortly after ' dusk' and in subduedlight. The flower was scented with the synthetic attractant isoamyl salicylate and thefeeding solution contained 10% sucrose in water. The temperature was 26 °C and therelative humidity 90%. The flight chamber measured 71 x 71 x 62 cm3, and when themoth was hovering in the correct place near the centre of the almost dark cage twocrossed light-beams of low intensity were interrupted by the insect. This blanked twoC.D.S. phototransistors which triggered a time-delay circuit which then started asound-insulated high-speed 16 mm motion camera (Hitachi HIMAC Type 16 HM,rotating-prism type, with speed control unit HM 502 and time switch to prevent


iverride). It also triggered two 1000 W quartz-iodine lamps, and the timing was suchthat the lamps did not light up until the film had reached a sufficient speed, i.e. about3000 frames/sec. This elaborate arrangement was essential because Manduca sexta is anocturnal insect which often makes a dive when struck by intense light. However,since this reaction does not take effect for a few wing beats, normal hovering as well asaerobatics and fast forward flight could be studied, at least in brief sequences.

Small chalcid wasp. The flight of very small insects is virtually unknown (Horridge,1956). Pupae of the parasitic chalcid wasp Encarsiaformosa Gahan (family Eulophidae,subfamily Aphelinidae) were obtained from a commercial dealer; this species is usedfor biological control of the greenhouse white-fly, Trialeurodes vaporariorum West-wood (see Speyer, 1927; Burnett, 1949; Parr, 1968). The tiny adults (wing length0-05-0-07 cm) both fly actively and make strong jumps. The procedure therefore wasto take a high-speed film (as above) whenever the wasps became particularly active,as could be brought about by gently tapping the flight chamber. The plastic chamberwas 5-4 cm high, 3-5 cm wide and 1 -6 cm deep, so that the insects flying near the centrewere within the range of focal depth (2 mm) and aerodynamically almost free of walleffects. The insects are diurnal and are attracted to light. A 50 W fibre-optical verticallight beam was constantly on and tended to concentrate them in the vertical centreaxis of the chamber, while three 150 W quartz-iodine lamps were triggered simultan-eously with the camera (6-8000 frames/sec); they did not appear to interfere withflight. The lens was a Carl Zeiss 63 mm Luminar,/1:4'5, used at low focal opening,and the magnification on the film was one to four times. The temperature was 26-27 °Cand the air almost saturated with water vapour. It was usually easy to distinguishbetween jumps, flight initiated by a jump, and free 'unaided' flight. Only the latterscenes are used here.

RESULTS

As in the previous study (Weis-Fogh, 1972), I shall use the technical force-length-time system, i.e. gram force-centimetre-second or gf-cm-s system, but the relevantSI units and conversion factors are given in Appendix 1.

(a) Normal hovering

All birds and bats, and many insects, make use of fast forward flight for transportover long distances; locusts and other migrating species are good examples. This typeof flight is now reasonably well understood in birds and insects on the basis of ordinaryaerofoil action (Weis-Fogh & Jensen, 1956; Jensen, 1956; Weis-Fogh, 1956; Penny-cuick, 1968 a, 1969). However, many small birds and bats and most insects also practiseslow forward flight and hovering on the spot when they forage, approach the nest,perform sexual displays or explore the surroundings. The explanation is not only thatthis, the most strenuous form of flight, is less demanding in small animals than in largeanimals (because of the surface/weight ratio), but presumably also that the conquest bymeans of active flight of the small niches of the aerial biosphere has opened up anendless variety of ecologically different niches to which the response has been intensespeciation, best exemplified by the 750000 known species of winged insects. Hoveringflight is therefore of general significance, particularly in insects. The main problemsare whether it is based on a common mechanism or not, and what the main principle is.


(b)

(c)

Fig. 7. Typical body and wing postures in some hovering animals, as drawn from flashphotographs of (a) the hummingbird ArchUochuz colubru (Greenewalt, i960), (6) the sphingidmoth Deilephila elpenor (Nachtigall, 1969), (c) the sphingid moth Manduca sexta during normalhovering (present study), (d) a bumble-bee Bombus sp. before landing on a flower (Schmidt,i960), (e) Manduca sexta during a quick manoeuvre (present study), and (J) the cockchaferMelolontha vulgaru during vertical take-off (Lane, 1955).


We have already seen that the hovering of hummingbirds and of Drosophilainvolves the same type of wing movements and can be understood on the basis ofsteady-state aerodynamics, although non-steady periods and effects undoubtedlyexist and may play a significant role under some conditions (Weis-Fogh, 1972). Bothtypes of animal perform what I shall call normal hovering. It can be defined as (a)active flight on the spot in still air by means of wings which are moved (b) through alarge stroke angle and (c) approximately in a horizontal plane, while (d) the long axis ofthe body is strongly inclined relative to the horizontal, sometimes almost to a verticalposition. Fig. 7 shows some examples of normal hovering. It has been drawn fromflash photographs of freely flying animals. The first question is how common normalhovering actually is.

All the animals listed in Tables 4 and 5 are known to hover, and if the name appearswithout an asterisk the species uses normal hovering. The examples have been greatlyextended by direct observations of free flight in nature which I have made during thepast few years. In the case of nocturnal species a high-pressure mercury-vapour lampwas used (Philips HP 125 W, 4000-7200 A).

Normal hovering was observed in many species of Isoptera in East Africa (swarmingtermites), Plecoptera (stone-flies), Thysanoptera (thrips), Hemiptera (leaf-hoppers,aphids, backswimmers, may bugs), and Neuroptera (lacewings). The Coleoptera orbeetles deserve more comment. Although the elytra (forewings) are usually spread outand flap to some small extent, they contribute next to nothing during hovering flight(Nachtigall, 1964) and the hind wings can be considered as the sole movers, beating ina horizontal plane (cf. cockchafer in Fig. 7/) and sweeping through a large angle ofabout 1800 on each side. However, because the body is broad, the two wings do notcome into]close contact. Normal hovering is perhaps more easily studied in Coleopterathan in any other group except the large crane-flies (Tipulidae), and most species orgroups make use of it, the tiger beetles (Cicindelidae) being possible exceptions.

Among Lepidoptera the moths generally appear to move their wings in accordancewith the common pattern. This is indicated by my direct observations and also appearsfrom flash photographs taken in nature (Fig. 76, Deilephila elpenor) and it was provedby the slow-motion film9 of the sphingid moth Manduca sexta hovering in the labora-tory (Figs. 8, 9). In both cases the moth was photographed vertically from above.Fig. 8 shows one complete wing stroke during normal hovering on the spot. It oftenhappens that the wings of one side are leading slightly, relative to those of the other,and in this case the left wings were leading. The morphological underside of the wingsi9 black. The sequence started when the left wings began the morphological down-stroke (Fig. 8 a). The film speed was 3000 frames/sec, increasing to 3900 at the end ofthe stroke (Fig. 8p). Only every 10th frame was traced. From the apparent length of thebody axis it is seen that the body is inclined about 45° relative to the horizontal. Thisscene, and films taken horizontally from the side, show that the wings beat almosthorizontally like those of hovering hummingbirds, that they are twisted in a similarway and that upstroke and downstroke are of equal duration (cf. Weis-Fogh, 1972),the tilt /? of the stroke plane being about io°. It is apparent from the films that thefore- and hindwings move together and that a cleft between them does not normallyopen, although this happens during certain brisk manoevres. Furthermore, duringnormal hovering the acceleration of the wings does not lead to any significant elastic

i86 TORKEL WEIS-FOGH

V .

(0 (0

V(m) («) (») (p)

0 5 cm

Fig. 8. Normal hovering of the sphingid moth Mcmduca sexta as traced from every ioth frame ina slow-motion film taken vertically from above at 3000 (start) to 3900 (end of stroke) frames/sec.The undersides of the wings are drawn in black.

bending of the long axis of the wing (Figs. 8a-p; jc) but, again, substantial bendingcan occur during quick escape reactions (Fig. je). We are not concerned with thelatter situations here; the main point is that the elastic storage of energy is confinedto thoracic structures during normal hovering as well as during fast forward flight,as is apparent from other films.

Fig. 9 is an example of part of a complicated manoeuvre where the moth started fromnormal hovering in front of the flower, but after the light and camera had been on for ashort time it began to move backwards with vertical body axis (Fig. ga; 3000 frames/sec) and during the stroke the axis tilted backwards and the speed was also increasedin the backwards direction. The morphological underside (black) of the wings istherefore seen during the main part of the stroke, whereas the upper side wouldpredominate during fast forward flight, underlining the supreme manoeuvrability ofthe insect as well as the general principles of flight. On occasions like this the manoeuvreeven resulted in a backwards somersault.

The flight of true butterflies (Papilionidae) is different from that of sphingid moths,partly because the wing surfaces often come into direct contact with each other at thetop of the upstroke (a ' clap') and partly because it looks as if the angular movementsdeviate appreciably from the sinusoidal case. They have therefore been marked by adouble asterisk (••) and need to be studied separately in another context.

The Mecoptera (scorpion flies), Trichoptera (caddis flies) and the large group ofHymenoptera (sawflies, swarming ants, wasps and bees) all appear to use normalhovering, at least as the main principle. It is well known that the honey-bee (Apis

MFlight fitness in hovering animals T87

00 CO (?) (A)

(0

V("0 ' ' (») ' l («) ' v

(P)I M M I0 5cm

Fig. 9. Same as Fig. 8 but during the beginning of a backwards somersault,a» explained in the text.

mellifica) can change the inclination of the stroke plane and that this is almost hori-zontal when the insect is hovering in front of a flower (Neuhaus & Wohlgemuth, i960;Nachtigall, 1969). It is more difficult to observe this in the bumble-bee (Bombus)because of the transparent and relatively small wings. However, my own observationsand Fig. y(d) indicate that they also use normal hovering. This was easy to observewith certainty in the large carpenter bees of the genus Xylocopa, which have deeplystained wings (X. violacea in Italy, 1972; and an unidentified species in Kenya, 1972).It may have been thought that the very small chalcid wasp Encarsia formosa wouldrepresent an exception but this is not the case, as is apparent from Fig. 14 and thedescription later. The reason for the triple asterisk (***) against its name in Tables 4and 5 is that in every single wing stroke the two pairs of wings come close together in adistinct dorsal clap before the morphological downstroke starts. The significance of thiswill be analysed in the Discussion (p. 214), but the same phenomenon appears to bepresent in Drosophila virilis according to Vogel (1965, fig. Ill , 1), who did not com-ment upon it however. It has therefore been similarly marked.

The large and diverse group of truly two-winged insects Diptera presents specialproblems. Whereas crane-flies, mosquitoes and other Nematocera as well as manylarge Brachycera and Cyclorrhapha undoubtedly use normal hovering in most cases(this applies for instance to Eristalis, Volucella and probably Calliphora), the dorsalclap seen in Encarsia and Drosophila probably occurs in many other small flies andinsects from different orders. In addition, the true hover-flies, the Syrphinae, performsteady hovering flight according to principles which are unknown so far and which

i88 TORKEL WEIS-FOGH

(b)

(c)

00Fig. 10. Examples of wing shape and applied contour functions in (a) the hummingbirdAmaziUafimbriatafltwiatUU, (b) the fruit-fly Drosophtlafimebris, (c) the hover-fly Syrphus ribem,(d) the hornet wasp Vespa crabro, («) the sphingid moth Manduca sexta, and (/) the cabbage-white butterfly Pieris brasticae. The hatched areas are discussed in the text.

also seem to apply to large dragonflies, Odonata, as already pointed out. Their flightwill therefore be treated separately.

Finally, some small bats make regular use of hovering, and although the main move-ments are consistent with the normal pattern the wings are somewhat flexed during themorphological upstroke (Eisentraut, 1936; Norberg, 1970), as is also the case in smallpasserine birds (Zimmer, 1943; Brown, 1951; Greenewalt, i960). A single asterisk (*)is therefore put against Plecotus auritus in Tables 4 and 5.

The term normal hovering is then well justified, the most important known excep-tions being Papilionidea, Syrphinae and Odonata.


(b) Wing shape and contour function

During hovering it is the outer two-thirds of the wings which matter. At a firstglance it may seem difficult to find a simple geometrical function which describes thewing contour in sufficient detail for a realistic estimate of the second and third momentof the area, S and T. However, it is not difficult in practice, and in most cases the wing(s)can be described by means of a semi-ellipse where the wing length R corresponds tohalf the major axis and the minor axis is adjusted to give the best fit for the outer partof the wing; this is the value for the adopted wing chord c in Table 4. The procedureis illustrated in Fig. 10 for both two-winged and four-winged animals. In some casesthe major axis is tilted by a few degrees relative to the line from the fulcrum to thetip in order to give the best fit, as is seen in the case of Syrpkus (Fig. 10c). This doesnot introduce any significant error. It may be thought that the hatched areas betweenthe wing contour and the ellipse justify a substantial correction, but this is not thecase. If, for example, we consider a wing in which the inner part is constricted, leavinga large hatched triangular area, as in Drosophila and Syrphus, with a base equal to\c and a height of \R, S will be overestimated by 3% and T by only 1-2%. The liftcoefficient and the aerodynamic power will then be underestimated by similarly smallamounts and this is far too little to worry about in the present approach. The poorestfit is undoubtedly that of Pieris in Fig. io(/) because the outer areas count most.However, the deviation is not large enough to be of real concern since much largererrors are introduced because our basic assumptions do not seem to apply to this group.In quantitative terms, the fit for the hummingbird AmaziUa and for Manduca isalmost perfect, taking into account that the contour does not remain really constantin any animal during actual flight.

The special problem of Encarsiaformosa will be discussed in connexion with Fig. 13.

(c) Flight data: Table 4

Only the most reliable data (or free flight have been accepted and only if they belongto one set for each specimen included in the list. The data for Drosophila virilis are anexception since they derive mainly from Vogel's wind-tunnel studies, as discussedelsewhere (Weis-Fogh, 1972). As to Encarsiaformosa, the weight G is the average fromsome ten individuals and the first example refers to a particular flight of a particularlylarge specimen, whereas the second entry represents average values. Apart from bodyand wing dimensions, the most critical parameter is the wing-stroke frequency nbecause it varies greatly with size and it enters into the equations in the second or thirdpower. However, in the selected examples the values can be considered to be accurate.The same does not apply to the stroke angle <fi, but it cannot normally amount to morethan 1800 = n = 3-14 rad, and <j> is usually about 1200 = 2-09 rad in both large andsmall hovering animals.

The ratio A between the stroke arc and the wing chord deserves special attentionbecause, when an ordinary wing starts to move, it takes 1-2 chord lengths for circula-tion F to build up, i.e. for full lift to be produced. Consequently, I have calculated thestroke-arc/wing-chord ratio A at the distance from the wing base corresponding to theradius of gyration rg for the second moment S of the wing area about the fulcrum

Tab

le 4

. Fla

ght

data

use

d in

this

dy

&

Th

e m

ost r

elia

ble

valu

es m

easu

red

du

rin

g free h

over

ing

flig

ht, s

low

forw

ard

flig

ht o

r, in

a f

ew c

ases

, in

win

d-tu

nnel

exp

erim

ents

. Th

e re

fere

nces

are

giv

en in

par

enth

esea

. In

thia

and t

he f

ollo

win

g ta

ble

the

num

bera

mean: (

I) S

otav

alta

, 19

52, (

2) S

otav

alta

, 19

54, (

3) V

ogel

, 19

66, (

4) W

eis-

Fog

h, 1

956,

(5)

Wei

s-F

ogh,

19

61, (6

) Wei

s-F

ogh,

19

72, (

7) H

ocki

ng,

1953

, (8)

Neu

haua

& W

ohlg

emut

h, 1

960,

(9)

Nor

berg

, 19

70, (

10)

Wei

s-F

ogb,

unp

ubli

shed

, (1

1) P

nn

gle

, per

sonal

wm

mu

nic

atio

n,

(12)

Sot

aval

ta,

1947

, (13

) G

reen

ewal

t, 1

960,

(14) Gre

enew

alt,

196

2, (1

5) p

rese

nt s

tudy

, (16

) La

si

d, W

eath

ers &

Bem

tein

, 19

67, (

17) P

ran

dtl

& T

eitj

em,

1934

, (18

) Jm

sm, 1

956,

(19

) N

acht

igal

l, 1

967,

(20)

Vog

el, 1

9676

, (21

) H

orri

dge,

195

6, (2

2) H

erte

l, 19

66, (

23)

La

aie

d,

1963

, (24)

Cha

dwic

k &

Gil

mou

r, I

W, (

25)

Sot

aval

ta &

Lau

laja

inen

, 19

61,

(26)

Th

orn

& S

war

t, 1

940.

Th

e et

roke

-arc

/win

g-ch

ord

rati

o h

= ~

Jc

,

is in

clud

ed as

a ba

ais

for

disc

uasi

on o

f no

n-st

eady

aer

odyn

amic

s. O

ne

aste

rin

k (+

) mea

m that

the

spec

ies

hove

r in

a w

ay s

omew

hat d

iffe

rent

fro

m th

e 'n

orm

al'.

Tw

o a

ster

&

(a*)

in

dica

te b

utt

efie

a an

d th

ree

aate

riek

s (*

*+) against E

nuns

ia fo

nnos

a an

d D

rosu

phila

oiriliz mean t

hat

the

uppe

r wing s

urfa

cea

may

app

roac

h each o

ther

clo

sely

at

the

top

of t

he

etro

ke, h

ere

call

ed 'c

lapp

ing'

, al

thou

gh t

he wings

beat

hor

izon

tall

y an

d a

re

mov

ed b

asic

ally

aa

a&&

ned

&th

e th

eory

.

Bat

s: P

lcco

tus

m't

us*

Birds: h

umm

ingb

irds

, T

mh

ili

Arc

- co

lubT

ir A

magiL

ia ji

mb

riat

a j-7

1Niacilir

Pal

agm

a gi

gas

Col

eopt

era:

bee

tlea

M

elol

onth

0 *m

is

Am

phim

ollo

n ro

lzci

tiolt

He

ha

pn

i sp

. H

. sp.

Cct

oltio

mu

ata

Cer

amby

cid species

Lep

idop

tera

: bu

tter

tlie

a &

mot

ha

pieriz ,+

*+

P. b

rars

icac

**

sphi

nx l

igur

tn'

Man

duca

sex

to

M. s

wta

M

ocro

glou

ra s

tclla

tono

n A

mat

ha

bico

lola

go

Wei

ght,

G

0

9.0

3'7

5-1

20'0

0.60

0.

29

7'8

I 2.

8 0.

46

0.14

0.04

0.

14

1-6

2'

12

1'1

2

0.28

0.

09

Win

g-

Win

g st

roke

S

trok

e L

iftl

dras

diagram,

ahap

e fr

eque

ncy,

an

gle,

Sm

k"a"

refe

renc

es,

Win

g-ch

ord

aspe

ct

rati

o

Mo

men

t of

Ref

eren

ces

on

stro

he

(9)

Tab

le 4

(co

nt.)

Hym

enop

tera

: was

ps, b

een

vespa

crab

ro

v. w

lgm

ir

Bom

bur

tme

rtri

r B

. lap

idm

iur

Api

r m

efiji

ca

Ew

sia

fonn

osa

(Cha

lcid

was

p)+

++

E

. fm

sa

(Cha

lcid

was

p)+

++

Wei

ght,

G

(gf)

0.60

009

0.88

0.

48

0'1

0

25 x

10-

25 x

10-

0.02

8 0'0

10

0.00

1 0.

15

0.05

6 0

'00

2

Win

g-

Win

g et

roke

Stroke

ahap

e frequency

angl

e,

Str

oh

arc

(c

f. n

Win

g-ch

ord

Tab

le I

) (sec-')

(rad

) A

Moment o

f inertia,

I (g

f an 8'

) x

10

-I

6 I 2.7

15.8

13

.1

0.85

-

-

6.6

0.06

3

X 1

0' 2.7

07

8

I x

IO-'

Tab

le 5

. Mai

n re

sults

: co

e~

ts

of Iif

i and

dra

g, s

peci

fic a

erod

ynam

ic p

ower

and

dyn

mtr

ic e

$%am

q

Th

e v

alue

s of CL an n

ot c

orre

cted

be

ca

w th

in w

ould

mak

e li

ttle

dif

fere

nce.

Aa

to t

he

epec

if~

c aero

dyna

mic

pow

er c

, both t

he uncorrected (equation

19) an

d th

e co

rrec

ted values are g

iven

, an

d th

e va

luer

, for N

and

th

e dy

nam

ic e

ffic

ienc

y q

are

both

bas

ed o

n th

e corrected

figur

es.)

Bat

s:

Ph

hu

mrt

hu

B

irds

: hum

min

gbir

ds, T

di

AT

- w

krb+

ir A

nrao

iliof

incb

ricr

tapu

oiot

ilu

Pata

gona

gigas

Col

eopt

era:

bee

tla

MeI

dont

ha v

dgo+

ir A

mph

imal

la s

olrt

lria

lis

H-

sp.

H.

ap.

Cei

onio

ma

to

Cer

tunb

ycid

spe

cies

Lep

idop

tem

: bu

tter

fliee

, moths

Finir

nap?

P. brassicae++

sphi

nx li

gurtri

Mcm

duca

sexta

M. rexto

Mac

mgl

ourc

mrt

ella

tmro

n A

nrat

iru bicolorago

Hym

enop

tera

: w

asps

, be

ee

Vupa c

rabr

o v. Ou

cpm

ir B

ornb

ur im

&

B. l

npid

miu

r A

pir

mrU

if;ca

E

neor

rin fonrtara (

Cha

lcid

was

p)+

++

E

. fancua (

Cha

lcid

was

p)+

++

Spe

cifi

c ae

rody

nam

ic p

ower

P:

(per

gf

Wy

wei

ght)

(E

q. 1

9)

23)

34)

L

Dyn

amic

Sim

ple

the

ory

(7.0

)

11.7

10

.9

10.5

11'3

9'

4 18

.5

20.2

15

'7

8.2

-

(5.9

) 7'

3 6.

2 5'

3 6.

9 -

9.2

7'2

I 6.

4 r 1

.5

9"' (9.7

) (1

2-41

3'7

15.3

23

'5

9'3

9'9

corr

ecte

d

-

-

3.1 -

4'3

3.6 -

-

-

3'4 -

(0.4

) 3 '

4

3.6

8.0

5'8 -

6-2

5'9

3'1

5'0

6.

3 -

-

4.6

4.1 1'7

5-6

2.8

Me

tah

lic

ra

te d

urin

g actu

al f

ligh

t (cal h

-l) per

gf body

wei

ght


. equation 7). We must of course use the chord length cQ at the same distance so that

A = <t>rg\cg = factor x 4>Rjc. (35)

The factor in equation (35) is given in Table 1 for the different wing shapes. Except insome butterflies, A is usually larger than 2-5 and varies between 3 and 5 in most of thegroups, which indicates that ordinary aerodynamics applies. However, in butterflies(Papilionidea) and in true hover-flies (Syrphijiae, Table 9) A is often as small as 1-4.Under these conditions one must expect non-stationary flow to be of major importance.It should be noted that the small chalcid wasp has a relatively high value of about 4, butspecial mechanisms are definitely involved in this case. The reference numbers in thelegend of Table 4 applies to Table 5 also.

(d) Gross results: Table 5

The weight of the animals ranges from 20 gf to 25 /Jgf and the wing length from13 to 0-06 cm, the corresponding Reynolds numbers (Re) being 15000 and 15 respec-tively. It is possible to obtain lift coefficients approaching i*o during normal steady-state flow down to (Re) between 10 and 100, and the square-law relationship betweenforce and velocity holds down to 10 (Thorn & Swart, 1940). It is therefore meaningfulto apply the analysis to the entire size spectrum in the tables. However, in the range of(Re) between 10 and 100 the drag tends to be larger than the lift, for (Re) = 10 by afactor of about 3, and normal flight would be both difficult and expensive in energy.Above 100-200 the lift/drag ratio has improved sufficiently for normal principlesto be operative.

The major conclusion from Table 5 is that most hovering animals move their wingsin accordance with steady-state principles and at Reynolds numbers well above thecritical range, and so that the average lift coefficient CL is relatively small, seldomexceeding i-o. This applies to all the lamellicorn beetles hitherto considered excep-tions (Osborne, 1951; Bennett, 1966, 1970) and to most wasps and bees. Apart fromEncarsia the highest value among Hymenoptera is for Bombus terrestris, which manyauthors in the past have denied the 'right' to fly. Yet, the average lift coefficient of aparticularly heavy specimen was only 1-2. It is also seen that the hovering sphingids(Sphinx and Manduca) fall within what can be achieved without involving new prin-ciples. This does not exclude that during take-off or escape reactions non-steadysituations may arise and be important, as will be discussed later, but it is hardlywithin the groups mentioned so far that we are likely to find the really interestingexceptions.

There are, however, some important deviations like Pieris napi (and probablyP. brassicae) and, of course, the chalcid wasp Encarsia formosa. In both cases CL is muchtoo large and in both types the known flight differs from that of normal hovering.For these reasons, the results are shown in parentheses because the simple analysis ishardly justified. As we shall see, the same applies to Syrphinae and Odonata. Theresults for the bat Plecotus auritus are also in parentheses but they are presumably notfar from the true values in spite of the fact that the wings are somewhat flexed duringthe upstroke.

13 EXB 59


Table 6. Corrected values for specific aerodynamic power P* as compared with the resultof simple calculations and with a more complete treatment {Weis-Fogh, 1972), togetherwith ratio w\vg.

P* (cal h"1 per gf weight)

Amazilia fimbriataManduca sextaVetpa crabroTkeobalduj ortftulotcAides aegyptiDrosophila viriUs

ratio0-22o-is0-15O-IIO-I2O-2O

Lift/dragratio6:16:16:17:27 H7:4

Simpletheory

10-96-89 3

15-323-SJ4-5

Correctedvalues

241 2

182 1

2819

1972values

2 2—

—

—

—

24

(e) Corrected results: a simple approach

In the simplified treatment there are two main variables which have been omitted,the tilt /? of the stroke plane relative to the horizontal and the induced wind w. As weshall now see, the tilt is of relatively small significance whereas the induced wind is amajor factor when estimating the aerodynamic bending moment Qa and the power Pa.

During true hovering the direction of the induced wind w is vertical, its magnitudedoes not change significantly with time and, at the level of the beating wings, it equals

w* = GI(2npR2). (36)

So far, we have assumed in equation (5) that the true relative wind V(t, r) whichmeets a wing segment is equal and opposite to the instantaneous flapping velocityv(t, r) but, obviously, it is the vector sum of w and — v(t, r). In order to arrive at asimple method for estimating the accuracy of the results and to introduce correctionswhen needed, we must know the ratio between w and v(t, r). In the following we shalluse the radius of gyration rg of the second moment S of the wing area as a represen-tative distance, and therefore the flapping speed at that distance vg(t). We then have

Moreover, we shall only consider the most common case of a semi-elliptical wing andonly during the middle half of the stroke when 80 % of the lift is being produced.Under these conditions, equation (37) is reduced to

w\vg = 2-52 x io» x J(G)l(</mR*). (38)

Some numerical values are listed in Table 6. It is seen that a ratio of 0-2 is commonbut in some small insects it may approach 0*1. In similarly built animals the weight Gis proportional to the cube of a representative length /, i.e. G oc P, and since (f> tends tobe constant and independent of size, the ratio in equation (38) will be size-independentprovided that n oc l~i. It will increase in proportion to /i if noc /-1. As will be shown inanother paper, most insects vary the frequency with size as n oc l~\ which tends togive a small w/vg ratio for small insects. In Drosophila, on the other hand, frequencyand size vary as n oc l~i. The ratio is therefore similar in hummingbirds andDrosophila.

In Table 6 the lift/drag ratio is also given, and the correction procedure is best

Flight fitness in hovering animals

Vertical

Horizontal

Drag D, simple

Fig. I I . The effect of the induced wind to and of the tilting angle /? on the force system actingon the wing of a Drosophila during the middle part of the stroke and at the distance from thefulcrum corresponding to the radius of gyration r, of the second moment S of the wing area, ascompared with the simple system in Fig. i . The L/£> ratio ia 1-75, tt>/r, is 0-20 and ft is 200.

illustrated with reference to Fig. 11 for Drosopkila virilis, where wjva = 0-2, L/D =7:4 and /? = 200. In the simple case L and D are vertical and horizontal respectively,the resulting wind force being a vector from the origin to point A. However, — va isactually tilted by 20° and the vertical w must be added to give the resulting relativewind velocity Yg. The true resulting wind force F then has a vertical component whichis 7 % larger than L and a drag component in the direction of the wing movement,i.e. in the stroke plane, which is 30 % larger than D, in this case. At the same timeVg is 5 % smaller than — vg, and the velocity squared is therefore 10 % smaller than theassumed value. The combined result is that CL has been estimated almost correctlywhereas the aerodynamic power Pa has been underestimated by 30%. The exactcorrections vary with /?, L\D ratio, and wjvg ratio, but for the material as a whole thereis no justification for correcting the CL values, which appear to be valid within ±15%.However, the power is always systematically underestimated, and some representativevalues are compared in Table 6. In this and in the previous table, we use the specificaerodynamic power, or power per unit body weight P j . It is seen that the correction issubstantial and that the corrected figures compare reasonably well with the resultsobtained in a more comprehensive treatment of AmaziUa fimbriata and Drosophilavirilis (Weis-Fogh, 1972). It should be noted that the corrections are largest when theLjD ratios are high.

It is helpful to consider the more general diagram in Fig. 12, which applies to anLjD ratio of 6:1 and a wjvg ratio of 0-2 for semi-elliptical wings. It is seen that theoptimum tilt of the stroke plane with respect to the horizontal is about + 200 and that

13-3

[96 TORKEL WEIS-FOGH

Vertical

-15°

Fig. 1 a. General diagram for correcting the average lift coefficient CL and the aerodynamicpower Pa when ft varies in animals with semi-elliptical wing contours, an LID ratio of 6:1, anda w/v, ratio of 0-3. The true drag in the stroke plane is seen to be about twice as large as thedrag D predicted from the simple theory whereas the lift L is affected to only a small extent.

this does not alter the calculated CL significantly. However, the true drag in the strokeplane is about double that estimated from the simple theory. Consequently, in thefollowing we shall use the corrected values for P%, N and t] in Table 5. They hardlydiffer by more than 20% from the true values. It should be noted that quite largevariations in /3 have relatively little effect and we are therefore justified in disregardingthis parameter in a treatment whose aim is a first approximation based on simpleexpressions.

(/) Specific aerodynamic power PJ

Of the 30 examples from 28 species in Table 5 the available information was incom-plete for the bat Plecotus auritus and the moth Amathes bicolorago, which also had anunusually high lift coefficient (i-6). For the two Pieris butterflies and the chalcid waspEncarsia formosa we have reasons to believe that the basis assumptions are invalid,so that one cannot include them in the general survey but must discuss themseparately.

The most surprising result is that the power requirement of the 23 remaining species(24 examples) is very similar and does not seem to vary systematically with size, the


Table 7. Evidence for an elastic system in flying insects

The total mechanical power ouput in the absence of elastic forces P*(B+Ois compared with the known metabolic rate (cal h"1 per gf body weight.)

Amaztiia fimbriata(hummingbird)

Vespa crabro (hornet wasp)Apis melUfica (honey-bee)Aides aegypti (yellow-fever

mosquito)Eristalis tenax (drone-fly)Drosopkila virilis (fruit-fly)

Dynamicefficiency

(V)

0-51

0 3 10-300-70

o-34°'95

P*lttH>

43

58604 0

562 0

Metabolicrate

2 0 0

1 0 0

3 0 0

ICO

us1 2 0

Minimummechanicalefficiencyof muscle

0-22

0-58O-200-40

O-49O I 7

Physiologicalfeasibility

Possible

ImpossiblePossibleImpossible

ImpossiblePossible

average being 21 cal h"1 gf-1. The upper limit is 40 for the heavy lamellicorn beetleHeliocopris and the lower limit is 11 for the large sphingid moth Manduca. This mustbe accepted as an empirical fact and both the absolute magnitude and the range areconsistent with earlier results (Osborne, 1951; Weis-Fogh & Jensen, 1956; Weis-Fogh,1964). It does not immediately follow from equation (19), but this is because one mustalso consider the effect of Reynolds number (Re) on the drag coefficient CD, as will bediscussed elsewhere (Weis-Fogh, in preparation). It is also consistent with the factthat the specific chemical power consumption, the metabolic rate, is of the samemagnitude in small and large flying insects (Weis-Fogh, 1964), and probably in smallbirds and bats as well. This obviously requires more discussion.

If we accept that the small insect Encarsia formosa could produce the necessary liftcoefficients shown in parentheses and that the drag coefficients are of equal magnitude,its aerodynamic power requirement would be 10-12 cal h - 1 gf-1 (uncorrected) andtherefore be of the same magnitude as in other hovering animals. This came as asurprise.

(g) Dynamic efficiency and elastic forcesTable 5-clearly shows that the ratio N = \Qi\m*xl\Qa\ max is high in almost all

insects analysed, with the exception of butterflies and Drosopfala. This means that thedynamic efficiency t] is low and that most species must spend 2-3 times more mechani-cal energy than needed for flight alone, provided that there is no elastic system presentin which the kinetic energy of the wings can be stored and later released. In fact,Table 5 provides strong evidence that an elastic system must be present in insects. Ifwe consider the metabolic rates of the six species for which the data are sufficientlygood, Table 7 shows that, as to energetics, flight would not be feasible in Vespa crabro,Aides aegypti and Eristalis tenax unless they could make use of elastic bending momentsin the thorax or wings similar to those in the locust Schistocerca gregaria, the mothSphinx Ugustri and the dragonfly Aeshna grandis (Weis-Fogh, 1961, 1972). As to thehoney-bee, Apis meUifica has the highest continuous metabolic rate measured in anyinsect and we do not known how efficient its wing muscles are; it is a borderline case,while in Drosophila the inertial forces are too small to be significant. The availableevidence amounts to a circumstantial proof that as a group, flying insects possess anddepend upon elastic forces in order to store and release the kinetic energy of the oscil-

TORKEL WEIS-FOGH

B

Fig-* 3 • (A) The morphology of the chalcid wasp Encarsiaformosa Gahan, used here for the studyof hovering flight in very small insects, (a) Tegula, (a)-(6) length of submarginal vein (d), (c)the marginal vein, and («) the tip of the anal vein. On the left-hand side the triangular contourfunction used in the first estimate of Ci is drawn in broken lines while the rectangular area wasused for the second estimate in Table 5 and later. (B) The related but larger species Coccophagustpectabilii Compere, re-drawn after Compere (1931).

lating wings. Under conditions of normal continuous flight the elastic system residesin the thorax and not in the wings because the wings are only bent elastically and to asignificant extent during exceptional circumstances. This does not apply to Encarsiaformosa.

(h) Hovering flight of a very small insect

The Problem. The calculated average lift coefficient of about 3 for Encarsiaformosain Table 5 is much too high to be compatible with steady-state aerodynamics. It maybe argued that the flight data in Table 4 are incorrect or misleading. To some extentthis may be true because the morphological upstroke is of shorter duration than pre-dicted from an harmonic movement of the wings due to the 'clap' period towardsthe end of each upstroke (cf. Figs. 14, 15), but the effect could not reduce the liftcoefficient to less than about 2. We are therefore faced with the problem that these tinyinsects do in fact hover and produce an average lift coefficient of 2-3 at a Reynoldsnumber of between 10 and 20. It should be emphasized that we have not taken intoaccount the brim of hairs and bristles seen in Fig. 13 but only the size and shape of themembranous parts of the wings. At present, we can hardly do better since the hairsare invisible on the available films, and it is doubtful to what extent they play a directrole in the process of lift production, as will be discussed when a new mechanism isproposed in the Discussion (p. 217).

The Insect. As to morphology, Fig. 13 shows that Encarsiaformosa Gahan resemblesan ordinary four-winged wasp. The most outstanding deviations are shared with othervery small insects among Hymenoptera and Coleoptera, in particular the brim of long


marginal hairs, the 'stalked' wings, and the lack of any significant wing veins apartfrom a strong reinforcement of the leading edge (costal veins). In addition, the winghinge is so constructed that the 'stalked' elastic wing bases can be swung above thedorsum and attain a positional angle of more than 1800 (cf. Fig. 15). The effect is thatthe two pairs of wings can be ' clapped' closely together above the dorsum in spite ofthe relatively large distance between the wing fulcra of opposite sides. I have oftenseen this dorsally ' clapped' posture of the wings in dead or preserved specimens, butit is always present towards the end of the upstroke during active flight, whether theinsect moves horizontally forwards or backwards in the air, hovers or is engaged inrefined manoeuvres.

Some details deserve attention since they may be of functional significance. Fig. 13was constructed from a set of photomicrographs of specimens embedded in glyceroljelly after mild treatment in lactic acid. Staining with methylene blue revealed thatresilin hardly plays any significant role in the wing system so that the main elasticitymust reside in the flight muscles and in the hard cuticle of thorax and wings, as inlarger Hymenoptera. The inner part of the forewing, from the axillary sclerites andthe tegula (a) to point (b) where the marginal vein (c) starts, is not ordinary wing mem-brane but a rather solid plate of hardened cuticle. I have not been able directly tocompare Compere's (1931) description of the forewing in the genus Coccophagus,whose members are somewhat larger (Fig. 13 B), because in Encarsia there is no costalcell and the submarginal vein (d) is the strongly reinforced leading edge itself. It ishollow and carries a row of what appear to be campaniform sensilla at its posteriormargin, but the most important characteristics are (1) that it can be twisted about itslong axis as a thick-walled tube (pronation and supination) and (2) that it is connectedwith the posterior part and the short anal vein (e) by means of sclerotized elastic cuticle.The wing base from (a) to (b) is therefore a functional unit which is twisted in basicallythe same way as the whole wing is in large insects, but the twisting does not necessarilyresult in a uniform twist of the main wing surface. On the contrary, the marginal vein(c) is not confluent with the submarginal vein (d) and there is a transverse, elasticbending zone from (b) to the tip of the anal vein (e). This means that the outer, mainpart of the wing surface may tend to be turned as a whole plate (pronated or supinated)when the basal submarginal 'stalk' is twisted.

Both the upper and the lower wing surfaces of both pairs of wings are covered withsmall bristles, about 20 /jm long. They may not have any aerodynamic importance butthey would prevent the wings from adhering to other surfaces, including the opposingwing surfaces, during the ' clap' period. Finally, the distance between the wing basesof the stalked hindwings is smaller than that of the forewings. This and the fact thatthe forewings are moved both up and down and forwards and backwards relative tothe body axis require a sliding coupling between the two wing pairs. The couplingconsists of the rolled-in hind margin of the forewings (called the retinaculum by Com-pere, 1931, broken line) and the two upwardly directed hooklets at the tip of the mar-ginal vein of the hindwing. In preserved specimens the hindwings are usually unhookedand held at a different positional angle from that of the forewings, indicating that thehindwings do not merely follow the forewings passively although they move togetherduring flight.

On the left-hand wings in Fig. 13 A the triangle drawn in broken lines corresponds to

2 0 0

Stroke 2 ends

TORKEL WEIS-FOGH

Stroke 3 starts

16

15 16

14

Stroke 4 starts

17

Horizontal 1J2mm

Fig. 14. Tracings from a slow-motion film of a hovering Encarsiaformosa which flies slowly back-wards and upwards, as described in the text The film speed, 7150 frames/sec, was notsufficiently high to permit unambiguous tracings of the wing contours during the fastest partof the wing stroke, as indicated by the broken lines. The frequency n was 403 see"1.

the contour function used in the first example for Encarsia in Tables 4 and 5. Itshould take care of the possibility that the marginal hairs may be considered as part ofthe lift-producing surface. More realistically, the hatched rectangle represents the bestsimple fit for the case that only the membranous parts matter significantly. It should benoted that only the outer two-thirds of the rectangle is used in the calculations becausethe wings are 'stalked'. In fact, this refinement makes little difference as to CL but isimportant for the further treatment of lift production.

The films. About 50 slow-motion films were obtained, which assisted in building upa picture of free flight, but only ten gave quantitative information about true hovering.

IFlight fitness in hovering animals 201

t was a surprise to find that the movements of Encarsia wings basically resemble thoseof other insects, small or large. During fast forward flight the body angle is small andthe wings beat obliquely up and down. During hovering on the spot, or when flyingslowly backwards, Fig. 14 shows that the body is almost vertical and the wings beatalmost horizontally. A detailed description is necessary but it must be emphasizedthat the film speed (7150 frames/see) is insufficient for accurate tracings of theoutlines of the wings when they move fastest or are ' shadowed' by the body.

In the film no. 47 B a freely flying wasp moved slowly into the field of view from theleft to the right and so that it also climbed at an angle of about 450 (Fig. 14). Over fourcomplete strokes the frequency was 403 sec"1 and the translational speed of the bodywas 9 cm sec"1 in the backwards-upwards direction, or sufficiently small to be ignoredin relation to the speed of the wings themselves relative to the air, which exceeds150 cm sec"1 at the radius of gyration during the downstroke and is significantlyhigher during the upstroke. However, the fact that the insect climbed not only showedthat it could produce lift in excess of what is needed for hovering but it also made itpossible to estimate during which part of the stroke lift is produced, by means ofmeasuring the vertical oscillations of the body. The tracings in Fig. 14 are fromwingstroke no. 3 and relate to the movements of the wings relative to the body. Thevertical and horizontal axes are indicated; and frames are numbered so that no. o isthe beginning of the morphological downstroke. The animal is seen at a somewhatoblique angle with its long axis slightly tilted by about 200 towards the reader and sothat the left-hand side of the head is above the plane of the paper. The right pair ofwings (on the far side of the insect) then appears at or slightly above the level of thehead, and the left pair beneath. Sometimes it was not possible to distinguish fore-and hind-wings, but whenever the interpretation is in doubt, broken lines are used tooutline the wings. The anterior or costal edge of the forewings is drawn in heavy line.

The toing stroke. Although forewing and hindwing on each side can separate andsometimes do so, basically they move as a single unit, the pair of hooklets ensuringa sliding anchorage to the retinaculum of the forewing. The stroke in Fig. 14 iscovered by 18 frames and starts in no. o with the wings held closely together as twoplane plates above the dorsum and with the leading edge at about the same level as theforehead. During the next o* 1 msec the two pairs of wings are ' flung open' like a book,the hindwings representing the back of the book (nos. 1 and 2). The two pairs thenseparate and move almost horizontally during the morphological downstroke as if theywere ordinary aerofoils working at moderate angles of attack (nos. 3-5), until they passthe midpoint where y = 900. The angle of attack is then increased and the wings con-tinue to move for another 300 but are also swung upwards until the end of the down-stroke is reached somewhere between nos. 7 and 8. The morphological upstrokestarts partly with a rapid reversal of angular movement which actually leads to anelastic bending of the wings at the transverse bending zone (nos. 9 and 10) andpartly with a quick ' flip' gf the main wing surface whereby it is rotated as a wholethrough a large angle so that the co9tal edge again leads (nos. 7-9). It resembles theflipping of a pancake in the air, hence the term. The rapid morphological upstrokethen follows with a tilt y? of the stroke-plane angle of about 300 and elastic straighteningof the wings; it ends in a' clap' when the wing tips are well below the level of the head(nos. 13-14). The 'clap' lasts about one quarter to one fifth of the entire cycle and


Table 8. Vertical movements (incmx io~8) of a freely hovering, slowly climbing Encarsiaformosa, as analysed during three successive complete wing strokes (strokes 1-3), eachrepresented by 17-18 frames (nos. 1-18) and filmed at 7150 framesI'sec

Stroke 1 Stroke a Stroke 3Downs troke

FramesClimb

'Upstroke'FramesClimb

'Clap'FramesClimb

Total climb (cm x io-*)

0 - 73

8-133

14-170

6

0 - 77

8-126

13-163

16

o-73

8-133

14-170

6

cannot produce any useful aerodynamic forces, because the two pairs are held closelytogether, their flat surfaces are vertical, and they are moved vertically upwards untilthe tips reach the level of the head again (no. 17).

Detailed analysis of the movements of the body relative to a co-ordinate systemfixed in space showed that in three successive wing strokes in this sequence (strokes1-3), the vertical movements followed the same pattern, as listed in Table 8. Thehorizontal movements of the body are of less significance since the speed in this direc-tion is small, the wind resistance is therefore small, and the kinetic energy would tendto blur any variations. The division between ' upstroke' and ' downstroke' cannot bemade very accurately, but comparison with Fig. 14 should clarify doubts as to theinterpretation. It is seen that both the horizontal downstroke and the tilted upstrokeproduce lift in excess of the body weight in all three strokes, and in approximatelyequal amounts. Also, the 'clap' period does not appear to contribute to the verticalforce; the apparent exception in stroke 2 is probably caused by the' up shoot' followingthe exceptionally strong lift during the preceding two phases. In any case the resolutionof the photographs is too small to permit any further conclusions about the ' clap'period. However, another important observation is that a substantial amount of climband therefore of lift is produced between frames o and 3, i.e. well before the wingsreach maximum angular velocity during the ' downstroke' and immediately followingthe 'flinging open' phase. Similarly, it looks as if hit builds up immediately after the'flip', which initiates the 'upstroke', but the rapid angular movements and the elasticdeformations of the wings during this phase prevent further analysis at present.

The wing-stroke cycle and the angular movements of the long axis of the forewingare summarized in Fig. 15, which derives from another specimen in film 47A. Theinsect was photographed from in front and almost along the long axis of the body, thewing-stroke frequency being 374 sec"1. During the period of' clap' the positional angley is larger than 180°, as already explained, and the 'downstroke' follows a course notfar from that of an harmonic movement at the overall frequency of oscillation. On theother hand the 'upstroke' is considerably faster since the half-cycle includes both theupwards movement and the ' clap' period.

Conclusion. Both these and other films were used to estimate the rate of rotation ofthe wing plane during the periods of 'flinging open' and of 'flip', but these figures arg

Time: 00

Flight fitness in hovering animals20 40

203

60 (ms)

T3

2

00aa•aao

\3

- 10

12 16 0 4Frame number

12 16 0

Fig. 15. The angular movements of the long axis of the wing relative to the body in Encartiaformota with an indication of the clap, the fling and the flip phases. The frequency was lowerthan in Fig. 14, 374 sec"1, and the film speed 7300 frames/sec.

more relevant when we test the new proposed mechanism for lift production in theDiscussion to which these observations lead (p. 216). What should be kept in mind atthis stage is that the wings are moved essentially as if ordinary aerodynamic principlesapply; the useful force is produced almost at right angles to the movement, in contrastto a principle based upon pushing or rowing, which have often been invoked to explainthe flight of very small insects. Our problem can then be re-defined as follows: how isan apparent conventional aerodynamic cross-force, i.e. aerodynamic lift, producedby means of flapping wings under conditions which prevent the building up ofsufficient circulation F in the usual way?

(1) Hovering flight of syrphid flies and dragonflies

The flight. The true hover-flies belong to the subfamily Syrphinae. Apart from thefrequency quoted by Magnan (1934) for Scaeva pyrastri (190 sec"1) and by Rohden-dorf (1958/59, 131-170 sec"1) for various Syrphus species, next to nothing is knownabout their flight. They are of course easily observed; but, in contrast to the distincthum produced by flying species of the subfamily Eristalinae (Eristalis, HelophUus,Myiatropa, etc.) and Volucellinae {Vohicella), whose wing-stroke frequencies haveoften been recorded, the flight sounds emitted by the Syrphinae are very weak. It isalso characteristic that they retain an almost horizontal body axis whether they hoveror dash off in fast forward flight when disturbed, as mentioned earlier (p. 181). Bothin Syrphinae and in Odonata this is probably an adaptation to an adult life entirely

TORKEL WEIS-FOGH

Time: 10 ms between dotsFig. 16. Sound tracings from some of the freely hovering syrphid flies recorded in nature andused in Table 9. (a)-(c) Syrphus balteatus hovering in still air at wing-stroke frequencies 177,123 and 100 sec"1 respectively. (<f) Syrphus ribetti at 208 sec"1, (e) Sphaerophoria scripta at309 gee"1 and (/) at 315 sec"1, (g) Syrphus corollae at 163 sec"1, (h) EristaUs arbustorum at217 sec"1, and (t) EristaUs tenax at 162 sec"1. Distance between dots on the time calibrationaxis represents 10 msec.

dependent upon flight for feeding and sexual display. In both groups no time is wastedby adjusting the body angle or the visual field when instant manoeuvres are called for.I have seen a male Sphaerophoria scripta hover motionless in front of a female with thebody axis tilted head downwards. During typical hovering in still air the stroke angle,<f>, is surprisingly small, estimated to be about 6o° and not exceeding 75° in any case,but no accurate measurements exist. This means that the stroke-arc/wing-chord ratio,A, in equation (35) is only about 1*4 in Syrphinae. The wings appear to beat up anddown with a tilt of about 45 ° to the horizontal and the tips definitely do not follow theusual horizontal figure-of-eight; but, again, future slow-motion films or stroboscopicobservations of freely flying insects are needed to elucidate the details. In contrast, myunaided observations on foraging EristaUs, Helophilus and VohtceUa indicate thatrepresentatives of these subfamilies make use of normal hovering. In fact, Eristaliscan be quite difficult to distinguish by sight or sound from a foraging bee.

Wing-sound tracings. The sounds produced by hovering Syrphinae are barelyaudible to the human ear and they are complex, as is seen from Fig. 16. The upperthree recordings (a-c) are from the expert hoverer Syrphus balteatus, which can vary itsfrequency appreciably. It is 177 see"1 in (a), 123 sec"1 in (b) and only 100 sec-1 in (c).The distance between dots on the time-calibration strip represents 10 msec. At all

Tab

le g

. The

smal

lest

pos

sible

lift

coe#

amt

€& in

how

ring

Syr

pWps

RF

(Syrphidae)

Th

e ei

ght species o

f we

hove

r-fl

ies,

Syrp

hin

ae, use a

str

oke

angl

e of

75'

or

l-. T

he

two

spec

ia, o

f dr

one

free

, Eri

scal

inac

, use a

larg

er a

ngle

of

about

120"

. Th

e d

urn

in p

are

nth

m relate

to s

trok

e an

gles

not

norm

ally

seen

duri

ng free

hove

ring

in n

atu

re. E

ven

under

th

e mast

favo

urab

le c

ondi

tion

s, s

tead

y-s

tate

aer

o-

dy&

cs

ie i

nsuf

Eci

-mt to

erp

l&

the

hove

ring

of

Syr

phw

and

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frequencies the pattern is very complex, and this applies to all the recordings includingthose of Syrphus ribesii (d), S. coroUae (g) and Sphaerophoria scripta (e, / ) . The latterclearly demonstrates that the true wing-stroke frequency of 315 sec"1 in (/) would beestimated to be twice as high if a purely auditory method was used, the so-called tenorerror discussed by Sotavalta (1947).

No attempt is made in this study to interpret the sound tracings. They are merelyused to determine the frequency of the beating wings during true hovering of eachindividual included in Table 9, but they may offer important clues about the flightmechanism at a later stage.

The aerodynamic problem. If an animal were restricted to the use of conventionalaerofoil action for hovering, the best possible solution would be (a) to make equal useof upstroke and downstroke and (b) to beat the wings almost horizontally, as duringnormal hovering. Any substantial deviation would require higher lift coefficients.If therefore we calculate the average CL for the syrphid flies on this assumption, knownto be untrue for the Syrphinae, we would arrive at the minimum figures seen in Table 9.The contour function of the wings was a semi-ellipse, and c was estimated from the bestfit for each individual insect. The figures in parentheses are the results when the strokeangles <p differ appreciably from the true values. It is seen that in all Syrphinae of thethree morphological types CL should be as large as 2-3, i.e. incompatible with steady-state aerodynamics by a factor of at least 2 or 3. In fact the real discrepancy is largerpartly because A is so small (1-4) and partly because the wings do not beat in a hori-zontal plane. Clearly, the non-steady flow patterns present at each end of the stroke inany hovering animal (Weis-Fogh, 1972) must be utilized and must have become domi-nant in Syrphinae, rather than the steady-state phases which play a major role in mostother groups.

This extreme situation is probably correlated with the relatively low figures for thewing loading of Syrphinae, 0-7 kgf m~2, which contrasts with those for Eristalinae andVolucellinae, where the load figure is 1-2 kgf m~2 (Magnan, 1934; Rohdendorf,1958/9). According to Table 9 the larger and relatively more heavy Eristalinae seem torely mainly on conventional principles. However, within flying insects, and in parti-cular within Diptera, there must clearly be a whole spectrum ranging from the ordinaryto the exceptional mode of lift production.

As to Odonata, calculations similar to those in Table 9 for Aeshna juncea (based onSotavalta, 1947) and for Aeshna grandis (Weis-Fogh, 1967) show that hoveringdragonflies must rely on non-steady aerodynamics because in both species each of thefour wings should operate with a minimum CL of 2-3. Again, the wing-load figure forOdonata is low, o-i-o-6 kgf m~2 (Mullenhoff, 1885).

In hovering Syrphinae and Odonata there is no 'flinging open' phase, but at eachend of the wing-stroke there must be a rapid rotation of the wing surface about itslong axis, or a phase corresponding in principle to the 'flip' of the main part of thewing surface in the chalcid wasp.

DISCUSSION

This investigation represents a first-order approximation so that refinements andsupposedly second-order effects have been avoided deliberately in order to make thesurvey manageable in terms of labour and also to discover new principles and solutions


Imich may have been overlooked in previous studies. In fact, the conclusions arrived athere and in the previous sections should make it clear that we are now beginning tounderstand the major mechanisms in hovering flight in terms not only of conventionalaerodynamics and mechanics but also of new and maybe hitherto unknown principles.In any given animal species the net result may depend on a complicated combination ofmechanisms, as is so often the case in biological systems where countless generationshave been selected for fitness in the course of evolution. Since late Devonian, about350 million years ago, nature has explored aerial locomotion in all its aspects and overa great range of size and form. Of the 1 million known animal species, living and fossil,750000 are winged insects.

(a) Fast forward flight

During recent years there has been rapid progress in our understanding of thegliding and soaring flight of birds (Pennycuick, 1968 b, 1972 a; Tucker & Parrot, 1970),bats (Pennycuick, 1971), of the extinct pterosaurs (Bramwell, 1971) and of some insects(Jensen, 1956; Nachtigall, 1967; R. A. Norberg, 1972). Similarly, powered flight hasbeen studied in detail in birds (Pennycuick, 1968a, 1969; Tucker, 1968; Bilo, 1971,1972; Nachtigall & Kempf, 1971), in bats (Norberg, 1970; Thomas & Suthers, 1972)and insects (Weis-Fogh & Jensen, 1956; Jensen, 1956; Weis-Fogh, 1956; Nachtigall,1966; Wood, 1970, 1972). The list is far from complete, but three main conclusions areeither borne out or inherent in most results and arguments presented so far: (1) thatthe basic mechanism is similar to that first proposed by Lilienthal (1889) for large birdsand later found in locusts (Jensen, 1956) as discussed elsewhere (Weis-Fogh, 1961),(2) that, as to quantities, fast forward flight in animals is based mainly upon steady-state aerodynamics (Brown, 1948, 1951, 1953; Weis-Fogh & Jensen, 1956; Jensen,1956; Pennycuick, 1968a, b), and (3) that many refinements are present within theframework stated under (1) and (2) which can be considered to be consequences ofrefined kinematics seen in birds, bats and insects (cf. Nachtigall, 1966; Nachtigall &Kempf, 1971; Ruppel, 1971; Bilo, 1972). In a recent book Pennycuick (1972ft) hasprovided a synthesis of several aspects of vertebrate flight. Although some of his pointsare arguable, the general conclusions and line of thoughts provide an admirably clearpicture of the present state.

In contrast to this there have been claims in the literature that some insects with highwing loadings, in particular lamellicorn beetles, must make use of non-steady princi-ples to a major extent (Osborne, 1951 ;Bennett, 1966,1970). We have seen that, althoughno new principles have actually been proposed but must be related to a high rate ofwing twist in some way (Bennett, 1970), the relatively heavy beetles and bees need notnecessarily use non-steady aerodynamics, not even when they hover. One of the resultsof the present study is that exceptions to this rule (i.e. steady-state aerodynamics) are tobe found among the insects (and birds?) with a low rather than a high wing loading, asis clearly demonstrated in Table 10.

In spite of considerable progress in recent years it still remains true that the flyingdesert locust Schistocerca gregaria is the best understood and analysed example of fastforward flight, and we should therefore briefly consider Martin Jensen's results from1956 on the basis of our analysis of the wing movements seen in Fig. 17 (not previouslypublished). It represents traces of the sequence of 18 equally spaced flash photographsj 1 o~6 sec) from stroboscopic film I. The wing-stroke frequency n was 17-5 sec"1 and all


Table 10. The load figure, i.e. {total weight) I (sustaining wing area),in flying animals

Based upon MUllenhoff (1885), Magnan (1934), Sotavalta (1947), Greenewalt (1962) andpresent study. (•) indicates that most of the species make use of normal hovering, (••) thatunusual principles are likely to be important (in the case of DrotophUa to an unknown extent.)

Range (kgf/m«)Bats

Large Up to 8Medium 2-3Small* 1-2

BirdsSome ducks, Otis, GrusHens and fowls

Vultures and eaglesPigeons and crowsSmall passerinesHummingbirds*Swifts, swallows, bee-eaters*

InsectsColeoptera, Lamellicomia*Hymenoptera, Vespoidea, Apoidea*Diptera, large Brachycera and Cyclorrhapha*Lepidoptera, Sphingidae*Lepidoptera, Noctuidea*Coleoptera, many groups*Diptera, Syrphinae**Odonata**Drotoph&a virilis**Chalcid wasp, Encarsia formosa**Lepidoptera, Rhopalocera"

11-178-14

S-io3-52-S2 - 3

1-3-2-5

1 •2-4-10-8-4-s0-5-2-00-4-1-20-3-0-6o-i-o-60-3-1-1

o-i-o-6o-35O-I2

O-O4-O-2

other flight parameters also appeared to be normal for steady horizontal flight at3' jm sec"1. The average vertical lifting force was 2-23 gf as measured on the aero-dynamic balance, or 97% of the body weight, while the detailed analysis based uponsteady-state aerodynamics resulted in 2-17 gf. This discrepancy (3%) is well withinthe accuracy of measurements and the error arising horn, predictable non-steady effects(Jensen, 1956). The same applies to the measured and calculated average horizontalthrust which is much smaller (o-iogf and 0-14 gf respectively). We can thereforeexpress confidence in the flight situation itself as well as in the method of approach.

The numbering of the frames starts with the left forewing at its top position (no. 1)when the hind wing has already started its downstroke. The forewing is marked bymeans of two very fine white hairs placed normal to the long axis of the wing, one nearthe tip and the other further towards the fulcrum, where as distinct rear flap is presentin the form of the stiff vannal area of the forewings (corresponding to the flexiblefan-shaped vannal membrane of the hind wing). The hair was cut at the natural lineof bending to permit free operation of the flap. Apart from the downward angularmovements of the whole wing the downstroke is characterized by a nose-down twist ofthe wing, a pronation. This deformation is the result of active movements at the wingbase and results in an almost linear twist 8 with distance from the fulcrum. Further-more, when the forewing approaches the horizontal position and the angular velocityhas reached its maximum and begins to decline again, the vannal flap is automaticallytilted downwards, altering the wing profile and increasing the lift coefficient (Jensen


B

Fig. 17. Exact tracings from our original him used Dy Martin Jensen (.1950) tor his analysis offlight no. I in a female Sckistocerca gregaria. The numbering starts when the forewing begins thedownstroke and is being pronated, i.e. when the hind wing has already advanced considerably.(A) Downstroke and pronation, (B) upstroke and supination. The upper wing surfaces are dottedand the vannal flap of the forewing is indicated by transverse hatching. The indicators on the wingsare minute hairs with white tips and are clearly visible in the films. A typical example of fastforward flight (330 cm s~l) in a large insect (weight 2-30 gf). Note that the elastic bending of theforewing is only slight and can be ignored in the aerodynamic analysis whereas the vannal areaof the hindwing is moulded by wind and mass forces.

1956). The upstroke starts with a rapid reversal of the twist, a supination (between nos.12 and 13), a flattening out of the wing (no. 13), and shortly after an upwards tilt of theflap (nos. 14-16). This gives rise to the so-called Z-profile which offers little lift anddrag (Jensen, 1956). These deformations are always present whenever the locust fliesactively, but may of course become exaggerated in still air and under other adverseconditions, as seen in Fig. 18. In the latter case the large stroke angle and the airresistance which meets the wing of a struggling tethered animal can result in visibleelastic bending, but this is not the case during normal forward flight (cf. Fig. 17(14)and Fig. 18/).

In principle, the hind wings are moved in a similar way although the flexible vannustends to obscure the basic similarity. In an extremely detailed study of the movementsof the forewings of the migratory locust (Locusta migratoria) by means of stereo-photo-graphs taken from above, Zarnack (1972) has recently claimed that Jensen's kinematicanalysis is invalid. What he means is presumably that there were second-order effectsnot taken into account, such as various elastic bendings, and this is undoubtedly so.However, Zarnack failed to demonstrate the much more important flap and the Z-profile, both of considerable aerodynamic importance, and my own conclusion is thathis method cannot be as accurate as he himself claims since innumerable direct obser-vations on three species of locusts (Sckistocerca gregaria, Locusta migratoria and Notna-dacris septemfasciata), stroboscopic films from the side (Fig. 17) and flash photographs

14 EXB 59

2 1 0 TORKEL WEIS-FOGH

Downstroke

Upstroke

Fig. 18. Extreme deformations of the forewing in Schittocerca gregaria when started in still airand flapping at high stroke angle and frequency. Traced from flash photographs. The trans-verse lines are white markings of cellulose paint. Elastic deformation is now appreciable at thebeginning of the upstroke (cf. Fig. ye).

of ScJristocerca gregaria (Fig. 18) never fail to show these features. There are next tono differences in morphology and flight between the three species.

The net result of the movements relative to the air is illustrated in Fig. 19 (repro-duced from Jensen, 1956, by combining his figs. I l l , 6 and III, 8). The tracings referto the tip and to the middle section of a forewing as seen on the unfolded ellipticalcylinder, whose axis is the horizontally moving wing base indicated by the broken line.It is seen that the geometrical angle of attack of each section is small and carefullyadjusted to the path. It should be noted that Jensen's calculations are based on the truethree-dimensional movements and that the mutual interference between the wings wastaken into account in estimating the aerodynamic angles of attack. The diagram ispresumably valid in principle for most types of fast forward flight. It is also reminis-cent of a similar diagram in Fig. 20 for the two-winged fly Phormia regina flyingforwards at 2*8 m sec"1 and analysed by Nachtigall (1966), the main differences being(1) a much higher rate of wing twist ddjdt of about 6000 rad sec"1 as against maximally500 rad sec"1 in Schistocerca, (2) high geometrical angles of attack towards theextremes of each half-stroke where the high twist rates also occur, and (3) a larger ratioof flapping speed to forward speed. These points are all relevant in the discussion ofhovering flight but at this point it is obvious that the flight of these two very differentinsects is based essentially on the same type of movement, and that one can envisagea continuous evolution from the one to the other. This also means that the sameindividual can utilize basically identical mechanisms for fast and slow forward flightas well as for true hovering.

Flight fitness in hovering animals 2 1 1

18

13

18

Fig. 19. Developed picture (unfolded elliptical cylinder) of the movements of the tip (A) and ofthe middle part (B) of the forewing relative to the air in a flying Schutocerca gregaria (repro-duced from Jensen, 1956, flight no. I).

(b) 'Delayed' elasticity

The high rate of wing twist may mean that an elastic deformation is propagated at aspeed which is relatively small compared with that of the rate of deformation from wingbase to tip. In the absence of external forces, the velocity of propagation u of a torsionaldeformation is

«2 = Glp, (39)

where p is the density of the material and is about 1 g cm"3, while G is the shearmodulus. The relationship between G and the Young's modulus E is

G = £/[2(i +/»)], (40)

where /i is the Poisson ratio and can be taken to be close to 0-4 in a material like insectcuticle. According to Jensen & Weis Fogh (1962) E for the solid cuticle of locustforewings is 870 kgf mm"2 so that G is 310 kgf mm"1 (about 3100 N mm"1), The maxi-mum speed of propagation in a wing is then u = 5-1 x io3 cm sec"1. The actualdeformation will reach the wing section later than indicated by this figure if the move-

14-3


Fig. 20. A picture similar to that in Fig. 19 but from a forward-flying large Dipteran flyPhorrma regata (redrawn from Nachtigall, 1966, Fig. 36). The upper part of the leading edge isindicated by a triangle.

ment is damped by air or other loads and if the elastic modulus is smaller, as it must bein the soft and flexible parts of the wing membrane.

How does this influence flight? In the case of a locust with wing length 5 cm andwing-stroke frequency 20 sec"1, each stroke lasts 50 x io"3 sec and the propagationtime is only 2% of the period. The effect will be insignificant. In the fly Phormiaregina the duration of one stroke is 8 x io"3 sec, of which the twist phases at top andbottom last about 0-5 x io"3 sec each. With a wing length of 1 cm the propagationtime is at least o-2 x io"3 sec. The same applies to syrphid flies, and it means that if thewing twisting itself is instrumental in creating the aerodynamic flow upon which liftdepends in these insects, i.e. if the flip is of importance, the action must be propa-gated from base to tip with a delay which is appreciable or even of the same magnitudeas the duration of the pronation or supination. This may have an important and bene-ficial consequence for hovering flight in Syrphinae and other insects, as will bediscussed later.

In the case of the small Chalcid wasp Encarsia formosa, the marginal veins are0-035 cm long and the total wing length is 0-062 cm. The minimum propagation timesare then 0-7 x io~5 and 1*2 x io"6 sec respectively. During the 'flinging open' phase,the two wings open by at least 6° in io"5 sec and probably by io° on the average (seelater). We therefore cannot neglect the propagation time of the torsional movementwhen dealing with the rapid changes at either end of the wing stroke, but it is unlikelyto be of importance during the middle part.

It should be noted that in geometrically similar animals the relative importance ofthe propagation time is independent of size {t oc /°) when n oc /-1, as is usually but not

Clap

Flight fitness in hovering animals

Fling End of fling

2 I 3

B <J'

'—-.AT&*

r=0 E

Fig. 31. The fling mechanism in the chalcid wasp Encartia formosa. (A) The start, middle andend of the fling in diagrammatic form, to be compared with Figs. 13-15. (B) The 'flingingopen' seen in the direction parallel to the long axes of the wings, during the clap (1), in the pro-cess (2) and towards the end with the resulting circulations (3) and shed vortices (4). (C) Thebeginning of the horizontal, morphological downstroke after the separation of the two pairsof wings and the shedding of tip vortices. (D) The geometry of the opening wings during thefling, as referred to in the text. (E) The propagation of the cleft between the two wing pairscaused by ' delayed' elasticity. The air movements are indicated by thin lines.


always the case. The significance of this factor is therefore potentially the same i |small and large flapping animals.

(c) A novel mechanism for lift production in Encarsia: the fling

The idea. The low Reynolds number makes high lift coefficients impossible and yetCL was found to be about 3. It should also be recalled that lift in excess of the bodyweight is produced early after the 'flinging open' phase seen in Figs. 14 and 15, here-after called the fling, namely between frames 3 and 4 or considerably earlier than theattainment of corresponding maximum angular velocity by the horizontally beatingwings. This is not consistent with ordinary aerofoil theory and practice. Could it bethat the fling is a mechanism for creating circulation round the wings before the rightand left pair separate and start to move in a horizontal plane during the morphologicaldownstroke in Fig 21C? At first sight this seems improbable because what is needed isthe setting up of a circulation in the sense indicated in Fig. 21 A, and the twisting move-ments (pronation) are of the opposite sense. We are therefore not dealing with anordinary Magnus effect. However, the possibility exists that a transient circulation ofthe desired sense could be induced. Once established it would immediately producelift normal to the direction of the horizontal translational velocity vt and in an upwardsdirection. It could then be reinforced to some extent during the stroke, but there wouldalways be a tendency for it to approach the low value corresponding to the smallsteady-state lift coefficients at the relevant Reynolds number. The mechanism thendepends on two opposed wing surfaces placed in air of zero velocity and vorticity.They suddenly split apart along the leading edges by an angular movement around therear edges which can be represented as a 'hinge' (Fig. 21 A) so that air must flow intothe space created between the wings from in front. The opposite flow is prevented bythe adjoining rear edges forming the 'hinge'. Immediately after this fling, which cor-responds to the pronation in normal flight, the wings on either side separate, each carry-ing a bound vortex with it. The question is, will this movement result in a circulationof the necessary strength T and the correct sense? It is a problem of flow relative to asolid body which suddenly starts to move (impulsive start) in a viscous incompres-sible fluid; the resulting motion of the fluid must then be irrotational and withoutcirculation. The point is that the body then breaks into two, each of which carries itsown circulation of opposite sign but of equal strength.

Feasibility. If the flow pattern is consistent with the formation of two bound vorticesof opposite sign, circulation can be created without the formation of any startingvortices and therefore without interference or delay due to them, i.e. without anyWagner effect. However, apart from the flow pattern, a number of other conditionsmust be fulfilled. During the flinging phase where air must flow into the vacatedspace from above, i.e. in the right direction, Reynolds number must be sufficientlyhigh so that the energy and vorticity are not degraded into heat almost immediately.Also, the boundary layer must be thin relative to the space available. During an im-pulsive start the thickness S of the laminar boundary in zero at the start and increaseswith time approximately as

$ » V(«), (41)where v is the kinematic viscosity (Prandtl, 1952).

Examination of the films show that the fling lasts about 3 x io~* sec as a maximum..


This corresponds to a thickness of 65 fim. Although the hairs (20 jim long) may in-crease the effective thickness, the boundary layer should not constitute a seriousobstacle.

As to Reynolds number (Re) we must analyse the system in more detail during thefling itself. For reasons given later (p. 217) we disregard inflow from the tips and con-sider only the air streaming in from the area delineated by the two leading edges andthe arcs at either end, the leading frame (two-dimensional flow in Fig. 21D). The rateat which the volume of instreaming air changes with time when passing the leadingframe is dV/dt and the velocity at the leading frame is uB and at distance r, from theedge it is «,. We have

dVldt co) rco , ."< = — = 2 0 ; ^ = Yd- (42>

Since we may consider ddjdt = was constant and 6 = tot, we have

ur = r\tt. (43)The velocity of the entering air is then zero at the trailing edge, where r = o and, moreimportant, inversely proportional to time t. A representative distance r for a triangularsection would be r = 2^/3, so that

(Re) = & = »£ ( 4 4 )

The major part of the fling occurs in 1-4 x io~* sec and the chord is c = 2-2 x io~a cm.After io~B sec, 7 x io~5 sec and 1-4 x io~* sec (Re) is then 75, 10 and 5 respectively. Inother words, at the start of the fling the velocity and (Re) are high, energy can be pum-ped into the flow system at a high rate by the action of the wings, and the flow is notdominated by viscous forces. It should therefore be possible to calculate the circulationaccording to the principles for vortex-free flow.

Circulation. My first and somewhat naive approach was to estimate the strength Fof the circulation by calculating the amount of kinetic energy Ek which the fling couldachieve by accelerating the air from zero velocity until the widening space between thewings was filled and the wings separated. This could be taken as a minimum figurewhich was then related to the kinetic energy Ee of the air in a bound vortex of strengthF. Since the lift L per unit length of the wing span is related to the translationalflapping velocity vt as

L = pvtT, (45)vt could be estimated for the given wing length R, or more accurately for 2/3.R,choosing the flapping velocity at the middle of the rectangular area in Fig. 13. Theresults were sufliciently encouraging in so far as the calculated velocity vt needed tolift the insect was 260 cm sec"1 while the observed maximum value for the wings isabout 180 cm s"1. This means that even according to a crude estimate F is of the rightorder of magnitude and could more than double the lift coefficient; but it was clearlynot sufficient, as could be expected from such a crude approach. However, the analysisalso showed that the kinetic energy Ek of the inflowing air

Ek oc tojt. (46)This means that the time sequence is important. If the fling is faster, at the start thanindicated by the average values used, the performance would be improved. This istrue also in a more sophisiticated approach.


At this stage I consulted Professor Sir James Lighthill, who most kindly discussedthese ideas with me and wrote (10 December 1972):

I have made calculation of the circulation at the end of the fling phase on the followingbasis: [cf. Fig. 21B]. The angular velocity (radians per second) with which the two wingsap and aq are opening about the point a is taken as w. Thus I first calculate the circulationaround the contours shown [B3], neglecting shedding of vorticity during the fling. This is aclassical calculation for vortex-free ('irrotational') flow. The answer is exact: for the geo-metry shown the circulation is 0-69 we* round both the pencilled contours.

If for the moment this value be acepted, then we may consider what happens when a smallgap appears between the two wings. This should not make any significant difference to theflow because that flow has zero velocity at the point a on both sides. The circulation roundthe pencilled contours [thin lines in Fig. 21B] should be unchanged, therefore; and shouldremain unchanged as they move apart by the usual Helmholtz-Kelvin arguments.

I have considered the effect of any vorticity shed at the points p and q during the 'fling'phase. At first sight the shedding of vorticity in the sense indicated [see Fig. 21, B 4] is goingto diminish the circulation around the contours shown in the previous figure [B3]. Since thecalculation has to be made while the wings are still connected at a, however, those circulationsare not around truly closed circuits and so we cannot say for certain whether they are dimin-ished by the exact amount of the circulation around the shed vorticity. A calculation needs,in fact to be made; and this (which I have done) shows, interestingly enough, that »/thevorticity shed has passed a considerable distance away from the edge by the end of the'fling' phase then the circulations around the contours shown in the first figure [B3] arediminished by only a moderate fraction of the circulation around the shed vorticity...This leads me to believe that considerable circulations around the two wings would remainat the end of the 'fling'.

Apart from shed vorticity (to which we shall return later, p. 217) and viscous effectswe then have, for the Encarsia wing immediately at the end of the fling period, whenseparation of the wings occurs, for each wing approximately

(47)

During the first 2̂ 4 x io~* sec the angle d is 1300, so that 1200 in Fig. 21 (B 3) and in thecalculations is a conservative estimate. The average angular velocity To is 8 x io3 radsec"1 during this period and (11 x io3 rad sec"1 during the first 1-4 x io~* sec). For achord c = 2-2 x io~2 cm this gives T = 2-67 cm2 sec"1.

According to equation (45) and the values for R and vt used there, the flappingspeed vt at which the total lift equals the body weight is then 93 cm sec"1. This speedis achieved at about frame 3 in Figs. 14 and 15, i.e. at the time when the films show thatthe body begins to gain height (cf. Table 8). There is therefore hardly any doubt thatthis novel method of creating circulation is sufficient and essential for making a smallinsect like Encarsia formosa airborne.

Details of Professor Lighthill's analysis and calculations, which include some con-siderations of three dimensional aspects of the fluid flows, will be published elsewhere(Lighthill, 1973).

Some possible refinements. According to the relationship in equation (46), the principleof which should remain true, the initial period of the fling is crucial as far as energytransfer is concerned and also with respect to Reynolds number (equation 44). Wehave already seen that the minimum time needed for a torsion of the anterior sub-marginal and marginal veins (cf. Fig. 13) to travel from the wing base to the distal end


0-7 x io~5 sec; and if the wing surface is made of a similar material it will take 1-2 xio~° sec to reach the wing tip. During the initial io~* sec the average value of 0) is11 x io3 rad sec"1 so that io~* sec corresponds to an angular opening of 6°. During theinitial and energetically most important phase there is therefore no doubt that thecleft between the wings must open gradually from base to tip, as illustrated in Fig. 21E.The implication is that there is no chance of shedding tip vortices until the cleft hasreached the tip.

In this context it may well be that the marginal hairs have a specific function - thatof preventing or delaying tip-vortex formation. It is instructive to note the differencebetween the hair brim of Encarsia and that of the closely related but larger Coccophagus(Fig. 13 B). Also, Horridge (1956) has pointed out that the smaller the insect the largerand more prominent the hair brim until the wing of the smallest known winged insects(R = 0-02-0-04 cm) consists of a rod with a brim of hairs. We do not know how suchwings are moved or whether these insects can fly as freely as Encarsia (R = 0-06 cm),but one should remember that if circulation can be created by means different fromthose of ordinary aerofoils, the actual shape of the wing is of little importance. What isneeded is an elongated solid body which carries a bound vortex and which is movedrelative to the stationary air.

The hind wings and the prominent hairs at their rear edge in Encarsia may improvethe mechanism because it should be possible to fling open the forewings not as oneopens a book, but by pulling the two parallel wing surfaces apart while the hind wingsseal the rear entrance. This could be the case during the initial phase and should alsomake it possible for the animal to exert more control than is otherwise possible. Inaddition, the long hairs on the hindwings could counteract the unavoidable inflow ofair at the proximal wing margin.

Other flying animals. It was noted that DrosophUa virilis appears to have a clap and afling period and for that reason it is marked by a triple asterisk (***) in Tables 4 and 5.Although it is just possible to analyse and understand the flight of Drosophila on thebasis of steady-state principles (Weis-Fogh, 1972), the usual mechanism does notleave any safety margin, and it appears highly likely that this and other small insectsmake use of the fling principle, thereby gaining extra lift and manoeuvrability. Thecost involved is unknown at present.

As to butterflies (Lepidoptera, Rhopalocera) they take off with the wings often heldin the clap position at the start, as already demonstrated in Vanessa io by Magnan(1934, p. XIV). The fact that the lift coefficient of Pieris napi in Table 5 is much toohigh (2-2) for wings of very low aspect ratio strongly indicates that this group ofinsects regularly utilize the fling method.

As to birds, it is easy to hear one or two distinct claps when a pigeon suddenlytakes off from a perch when disturbed. Dr R. H. J. Brown has kindly shown me un-published slow-motion films of flying pigeons from which it appears that at very lowair speeds the wings touch each other at the top, when the sound is produced. Theyare then flung open in a way similar to the fling in Encarsia. It would be interesting tostudy this phenomenon in large insects also, and in other birds where it might assistflight particularly during take-off in emergencies. Could it explain some of Dr Wilkin'sresults referred to on page 182 above?

Dimensional considerations. The lift of a wing is L oc G oc P oc vtn, where / is a


representative length and vt oc nl<j>. In the case of a fling mechanism T oc «/2, wherlw a n . We then have L oc P oc n8/4^, or for the specific lift

L* = (L/G) oc /° oc tfyl. (47)

In a series of geometrically similar animals, and if <j> is constant, the usefulness of thefling mechanism is therefore independent of size (L* oc /°) when the frequencyvaries as n oc /~i. This also happens to be the relationship needed for remaining air-borne by conventional means, as is seen from equation (9). The fling method is there-fore of equal potential use in animals of different sizes. Most animals have to increase nfaster than this when they become smaller because the lift coefficient and the lift/dragratio decrease at low Reynolds number, as has been discussed above (p. 195), andthe most common relationship for a given series of animals is n oc l~x. In this caseL* oc t~x and small animals will benefit more from the fling than large ones. It is'clearly desirable to confront these deductions with the various modes of flight seen innature, but they do indicate that the fling mechanism as a regular component offlapping flight should be more favouable in small than in large animals and most bene-ficial when the wing loading is relatively small. However, in emergencies and othersituations of particularly strain the fling mechanism can be used throughout the sizerange.

(d) The flip mechanism in hover flies

Obviously the fling mechanism whereby two sets of wings start by forming onestructure which is then suddenly deformed before the sets rapidly move apart cannotexplain the flight of the true hover-flies, the Syrphinae. However, there are some simi-larities, and although the following account is much more speculative than the pre-ceding one it is concluded that we are confronted with a special mechanism which Ibelieve to be novel and which I shall call the flip mechanism.

Consider an ordinary insect wing which is stretched out from the body. During aperiod of twist at the extreme positions, be it pronation or supination, the leadingedge is suddenly swung actively, nose-down during pronation and nose-up duringsupination. In many insects the posterior part of the wing is often flexible and does notfollow the rapid movement of the front part at all closely. In Encarsia each effectivewing surface on each side consists of a large strong forewing and a smaller hind wing.It appears from some of the films as if the hind wing functionally corresponds to theflexible part in a two-winged insect, and I have therefore adopted the descriptive term' flip' to describe the sudden twist both in Encarsia and in Syrphinae.

The wings of Syrphinae (Fig. 22 A, B) are extremely interesting in this context anddiffer from other Dipteran wings in several respects. (1) The wing itself has a con-centration of strong veins going from base to tip in the anterior part while the posteriorpart is poorly supplied; this is the' aderarme' wing of Rohdendorf (1958/9). (2) Thereis a special reinforcement known only in the family Syrphidae, the so-called venaspuria (v.s.). It tends to reinforce the anterior almost triangular part and to providelinear reinforcement parallel to which the wing must bend during pronation and supi-nation, separating the anterior, and the posterior parts funtionally. (3) I have observedthat the cuticle of the posterior part is extremely thin, soft and pliable in the livinginsect, particularly at its free border. (4) One could add a functional peculiarity,namely that I have never caught a flying hover-fly which had damaged wings. This


Head( = 0 r=io- /=2xlO-4s

Fig. 22. The flight of hover-flies, Syrphinae. (A) The small fly Platychirui pdtatut Mg., showingthe large alula (a/.). (B) The wing of Syrpkus balUatus with a large pterostigma (pt.) and the'false' vein, vena spuria (v.s.). (C) The proposed ./&£ mechatwm during pronation preceding thedownstroke, as resolved in time so that the effect of 'delayed' elasticity becomes apparent;viewed obliquely from above.

contrasts sharply with the common observation that Hymenoptera, Lepidoptera andmany other hovering insects are often seen to hover with tattered or worn wings. (5)Finally, the small rear flap at the wing root, the alula (al. in Fig. 22 A), is unusuallylarge and can be bent upwards or downwards at right angles to the remaining wingsurface, presumably by means of a pleuro-axillary muscle. It is probably used forcontrol and manoeuvres, particularly in gusty wind, but since it cannot be of import-ance for lift generation we shall not analyse it further.

As a first approximation the wing can be considered as a plate with a stiff anteriorpart of roughly triangular shape which can be twisted actively from the base andanother posterior part which is pliable and is indicated by transverse hatching inFig. 22 C. The wing length R in a typical hover-fly is 1 cm, and the propagation of atoreional movement from base to tip therefore lasts at least 2 x icr4 sec. In fact, it islikely to be longer because the wing has an unusually big pterostigmatic area (Fig. 22 B,pt.) recently shown to be more dense and heavy than the remaining part of the wing, atleast in Odonata (R. A. Norberg, 1972). At the typical frequency of 150 sec"1, the


wing-stroke period is 6-7 x io"3 sec of which 7% is used for twisting at either enl(Nachtigall, 1966, in Phormia), or 5 x io"4 sec for pronation and supination respec-tively. There must therefore be a significant delay between the onset of twisting at baseand tip. Also, the soft posterior membrane has a much lower elastic modulus so thatthe deformation cannot reach this region until after the active twist is completed, i.e.6-20 x io~* sec after the onset. The air at the hind margin is then at rest during thetwisting phases, the margin representing a stagnation line.

On this basis I have attempted to illustrate the probable deformations in Fig. 22 Cat the onset of pronation (1) when t = o, (2) when the wave has reached the middlepart of the wing at t = io-*sec, and (3) when it arrives at the tip at about t =2 x io~* sec. I have also indicated the likely air movements. At the start the air is at rest.In (2) the middle part is suddenly bent in a fashion reminiscent of the fling in Encarsiabut a circulation in the form of a bound vortex cannot be established unless an oppo-site posterior vortex of opposite sense is created. It will be noted that at this stage, thetip is not moved so that any shedding of vorticity at the tip will be insignificant. Thepropagation of the twist therefore introduces economy. In (3) the entire wing hasundergone twist and has acquired what corresponds to a bound vortex although someuseful circulation must now be lost due to the unavoidable tip vortex. When the wingthen begins to swing down, leaving the system of posterior vortices behind as if theywere ordinary starting vortices, it has already acquired circulation of a strength un-related to the flapping speed and may produce lift at much lower speeds than envis-aged in a steady-state system. It should be mentioned that the vertical induced wind istoo small to remove the trailing vortex significantly during the time occupied by thetwisting movements.

In this mechanism the soft posterior membrane is essential for the initial distribu-tion of the two vortices relative to the wing, the anterior one bound to the wing andthe posterior one free. If the posterior part were stiff and the wing a simple torsionplate, the rear part would swing up as the leading part swings down, the system wouldbe symmetrical about the neutral axis and no useful circulation could build up beforethe translation starts.' Once the real wing is in movement the soft rear flap will alsoincrease the downwash and prevent separation or stall.

Without quantitative proof I propose that this is the mechanism which enablesSyrphinae and Odonata to obtain the high lift coefficients of 3 or more during hoveringflight. The principle could be used by any animal with similar wing characteristics. Itdiffers in principle from the fling mechanism in the way that the posterior vortex can-not be used and tends to counteract the useful bound vortex until the two have becomeseparated in space, and the induced wind is too small (about 50 cm sec"1) to do thisduring the twisting phases. There is therefore a delayed effect in the aerodynamicsystem and the two vortices must become separated by translation in order to obtainfull lift. The similarity between the fling and the flip is that the transient vortices areproduced by the active wing twisting. Future observations and experiments are neededto verify this. At present we have none.

Let us now consider the entire wing stroke during hovering and, for the sake ofsimplicity, assume that pronation (J>) and supination (s) in Fig. 23 produce similareffects at the top and bottom of the stroke. The lift L is always at right angles to therelative wind, the drag D is in the direction of the wind, and the sense of the circula-


D

Dyingvortex

Dyingvortex

'Normallift

'Normal'lift

Fig. 23. The effect of the proposed flip mechanism when the wing-tip curve changes in ahorizontally hovering syrphine fly (A). The sense of the twist rotation is indicated duringpronation (p) and supination (»), the resulting bound vortex having the opposite sense (thinlines). In (B) and (C) the wings beat vertically up and down and a strong forward thrustresults but no vertical force. In (D) and (E) the fly would experience a steep climb, and in (F)it could hover on the spot, as discussed in the text. (G) is an attempt to combine the non-steadyflip mechanism with steady-state phases.


tion caused by the flip action is opposite to that of the original wing twist. The fljhovers with horizontal body as in (A). If the wings merely beat up and down in a ver-tical plane, there will be no net vertical force because the drag components will canceleach other. However, during both downstroke and upstroke there is a strong forwardshorizontal thrust caused by the ' flip' lift. If the stroke plane is tilted as in (D) and (E),the lift always points forwards and upwards and the fly will experience a steep climb.True hovering based exclusively on the flip mechanism can be achieved if the wing tiprotates anti-clockwise as in Fig. 23 F and so that the bound vortices die out duringthe vertical parts of the stroke. In Fig. 23 G I have indicated how the flip mechanism isprobably combined with normal aerofoil action in hover-flies and dragonflies duringhovering on the spot.

It should be noted that during truly non-steady periods the angle of attack is oflittle importance, and this may well be reflected in NachtingalTa (1966) results fromthe blow-fly Phormia regina reproduced as Fig. 20, particularly when one examines thelarge angles towards each end of the stroke. Another point is that even small altera-tions in wing-tip curve would have quick and drastic effects, as is needed for thesupreme control seen in hover-flies. Such changes are well known from Hollick's (1940)studies on the fly Muscma stabulans and are found in most medium-sized Hymeno-ptera and Diptera. It would be extremely interesting to study free unimpeded flight ofhover-flies by means of high-speed cinematography.

As far as I am aware, the solution deduced here for the flight of hover-flies (Syr-phinae) is new. It does not involve refined adjustments of the angle of attack andrequires basically two mechanisms to explain both the act of hovering and the extremeand rapid manoeuvres, namely that the basic flip mechanism dominates the aero-dynamic system and that the animal exerts rapid control over the wing-tip path, as is thecase in other Diptera (Hollick, 1940). This being fulfilled, the insect can do almostanything without altering the body axis, from hovering to a brisk forward dash, asteep climb or a fall. In a gust of wind the alula could be operated differentially on thetwo sides so that turning in the yawing plane could be added to the repertoire withoutinterfering with the basic control of flight.

(e) General discussion of hovering flight

At present it is not possible to provide a comprehensive picture but only to indicatethe interrelationships which have been established so far, or at least strongly suggested,and to indicate some problems which require a new theoretical or experimentalapproach.

Aerodynamics. A major conclusion from this study is that most insects performnormal hovering on the basis of the well-established principles of steady-state flow, i.e.normal aerofoil action. This implies that the analytical procedure and the mathematicalapparatus presented here provide a realistic approach both in qualitative and quanti-tative terms. However, one must also realize that any type of flapping flight also in-volves non-steady periods, particularly at the reversal points where active pronationand supination occur. The higher the forward speed and the lower the wing strokefrequency, the smaller is the required rate of wing twisting and the smaller the rela-tive importance of non-steady phenomena, as in a fast flying pigeon, and the empiricalobservations indicate that when the wing loading exceeds 1 kgf/ma, the steady-state


Principles seem to prevail even in hovering animals. The approach illustrated in Fig. 5and in Table 5 should then provide figures of the correct magnitude, although manymodifications can be expected in the light of more detailed knowledge. One of themost urgent requirements is to obtain reliable steady-state lift/drag diagrams of realanimal wings, particularly from insects, and measured at the right Reynolds numbersand in a wind field similar to that produced by a truly hovering animal.

In addition to the conventional type of hovering we have analysed two novel non-steady principles for generating lift, the fling and the flip mechanisms, which appear tohave become dominant in certain insects with small wing loadings. In both typescirculation is set up as a consequence ot the rapid twisting at the reversal pointe in thewing path and before the wing as a whole gains speed relative to the stationary air. Theessential difference between the steady and non-steady phases is therefore not thatflight depends on an aerodynamic cross-force, or lift, in the first case and on an entirelydifferent mechanism in the other cases; under the conditions considered here, liftdepends on the establishment of circulation in the form of a bound vortex round themoving wing. The difference is that a normal aerofoil or wing induces circulation andmaintains it as a consequence of viscous shearing forces, particularly in the air passingthe rear edge, whenever the wing is moved relative to the surrounding air. Once thecirculation has built up and the initial phase is over (Wagner effect) the steady-stateimplies that the circulation is maintained by the energy dissipated continuously in theshear system. In the case of the non-steady fling and flip mechanisms the vortex pat-terns which lead to circulations are created prior to and independent of the translationof the wing through the air. These vortex patterns would of course be useless unlessthey became superimposed upon the subsequent translational movement; but, onceestablished, the air speed of the wing can be reduced relative to what is needed in thesteady-state case and for as long as the initial bound vortex remains of significantstrength. After it has died out, circulation then has to be maintained by the conven-tional mechanism.

During slow forward flight and hovering the non-steady phases in birds, bats andinsects occupy a significant part of the wing-stroke cycle and, as we have seen, theycould be put to good use in some cases. Many new examples will undoubtedly cometo light in the future. The way in which the aerodynamic bending moment varies withthe positional angle in work diagrams like that in Fig. 5 will then be somewhat modi-fied in that the curve ABC will become steeper at the ends and more flat towards themiddle, even upwards concave in extreme cases. This will increase the inertial andelastic forces involved in starting a new half-stroke but it does not necessarily result inan increase of the enclosed area, i.e. in a larger aerodynamic power output, becausethe downwards momentum needed to hover remains the same, in accordance with theexperimental results of Wood (1970, 1972). As we have seen, the elastic system ofinsect wings tends to minimize vortex shedding so that the efficiency may remainreasonably high. If, then, the pronation and supination are used directly to createuseful circulation in addition to that obtained by normal aerofoil action, the averagecoefficient of lift CL as estimated from equation (9) will become reduced. The values inTable 5 are therefore likely to be too large, particularly at low wing loads, because thisfactor cannot be taken into account until we know more. Drosophila is a good exampleand has already been discussed (p. 217). What is needed now are new theoretical and

224 TOKKEL W E I S - F O G H

experimental studies on non-steady flow situations as exemplified by the fling and flijlmechanisms, but not necessarily confined to them.

It should be recalled that the dimensional analyses show that the fling and flip mecha-nisms are potentially useful in larger animals also and that they are not in conflictwith the conventional mode. It is therefore realistic to proceed with a general dimen-sional analysis of flapping flight to be presented in another context (paper in prepara-tion).

Wing inertia and elastic forces. The same applies to the bending moments caused bythe acceleration of the wing mass and by the elastic deformations. So far, the details ofthe elastic system only seems to have been analysed in the desert locust Schistocercagregaria (Weis-Fogh, unpublished) and to some extent in Diptera and Hymenoptera(Pringle, 1957, 1968) but it is now clear that similar principles apply to insects ingeneral. However, it is necessary to consider the possible effects of non-steady aero-dynamics in this context.

rt will be remembered that the inertial torque Q* is relatively insignificant inDrosophila (Weis-Fogh, 1972, and present study). The same applies to the truehover-flies. In typical Syrphinae the wings are relatively very light (0-7% of bodyweight) and the stroke angles and frequencies are low. When the moment of inertia Iwas calculated from equation (27) and the ratio iV from equation (23), it was found thatthe ratio between maximum inertial and aerodynamic bending moments is consider-ably smaller than unity. This means that in Drosophila and in the Syrphinae the phaselag between tension and shortening essential for the operation of fibrillar muscles(Pringle, 1957, 1967) cannot be provided by wing inertia. At least in the case of Syr-phinae it is unlikely that the effective lever arm of the wings changes much with posi-tion because of the small stroke angle, and the problem is how a phase lag is introduced.It is possible, however, that a substantial muscle tension is needed to twist the wingsagainst both elastic and aerodynamic forces so that pronation and supination couldprovide the necessary lag as well as producing useful aerodynamic work. During theevolution of conventional into non-conventional types, or vice versa, there need there-fore not have been any real change with respect to myofibrillar mechanism.

In Hymenoptera and Diptera the wing twisting is an automatic consequence of thedeformation of the elastic thoracic box caused by the strong indirect wing muscles(Pringle, 1957, 1968). In Odonata, Lepidoptera and Coleoptera wing twisting isdirectly controlled by the basalar and subalar muscles which are well developed in allthree orders. We have seen that the anisopterous dragonflies and the LepidopteraRhopalocera (butterflies) actually make use of non-steady principles in spite of theirrelatively large size. The fling phase during the start of flight in Venessa to has alreadybeen mentioned, but it may also be significant that Magnan (1934, fig. 174) observedan open wing-tip loop in Aeshna mixta very similar to the one indicated in Fig. 23 G asbeing optimal for hovering flight based on the flip mechanism. Apart from Hymeno-ptera the other very small flying insects belong to the Coleoptera and the Thysanopteraand in both orders the basalar muscles are well developed (Matsuda, 1970). There ishardly any doubt that further studies on the dynamics of the flight system in insectswill lead to many reinterpretations of the functional morphology of winged insects.The same applies in principle to birds and bats.

Apart from propagation of the heart pulse in vertebrate arteries, one does not norm-

. Flight fitness in hovering animals 225

Illy have to take the speed of propagation of elastic deformations into account whendealing with animal dynamics. However, this factor is important for the non-steadyaerodynamics in insects and probably also in the wings of birds and bats.

The wing system of insects is often referred to as a mechanical resonant systemwithout further qualifications. This is obviously not true in species with synchronousor non-fibriilar muscles where the rhythm is determined ultimately by the centralnervous system, although natural selection would tend to optimize inertial and elasticforces one way or the other. In species with myogenic or fibrillar wing muscles there isno direct nervous control of the fundamental frequency n, and one would expect thatn oc J~i provided that the stiffness of the elastic system remains independent of n. Atleast, this relationship should hold in species with large inertial bending moments suchas EristaUs and Calliphora (but not necessarily in Syrphinae). It is therefore dis-concerting that Danzer (1956) found that n oc /-°-22 in Calliphora erythrocephala inexperiments where the moment of inertia was reduced by shortening the wings.However, he calculated Im from the wrong assumption that the mass was distributedelliptically. For some years Dr Machin in this Department has conducted class experi-ments with this species and confirmed the predicted relationship, whether Im wasfound by means of equation (30) or directly by weighing, a minor deviation beingcaused by the fact that the stiffness of the active flight muscles decreases with in-creasing frequency (Machin & Pringle, i960). There is therefore good reason toassume that the elastic system is both necessary and efficient in insects whetherpowered by neurogenic or myogenic muscles.

SUMMARY

1. On the assumption that steady-state aerodynamics applies, simple analyticalexpressions are derived for the average lift coefficient, Reynolds number, the aero-dynamic power, the moment of inertia of the wing mass and the dynamic efficiency inanimals which perform normal hovering with horizontally beating wings.

2. The majority of hovering animals, including large lamellicorn beetles and sphin-gid moths, depend mainly on normal aerofoil action. However, in some groups withwing loading less than 10 N m~2 (1 kgf m~2), non-steady aerodynamics must play amajor role, namely in very small insects at low Reynolds number, in true hover-flies(Syrphinae), in large dragonflies (Odonata) and in many butterflies (LepidopteraRhopalocera).

3. The specific aerodynamic power ranges between 1-3 and 4 7 W N " 1 (11-40 cal h"1 gf"1) but power output does not vary systematically with size, inter aliabecause the lift/drag ratio deteriorates at low Reynolds number.

4. Comparisons between metabolic rate, aerodynamic power and dynamic effi-ciency show that the majority of insects require and depend upon an effective elasticsystem in the thorax which counteracts the bending moments caused by winginertia.

5. The free flight of a very small chalcid wasp Encarsiaformosa has been analysed bymeans of slow-motion films. At this low Reynolds number (10-20), the high lift co-efficient of 2 or 3 is not possible with steady-state aerodynamics and the wasp mustdepend almost entirely on non-steady flow patterns.

15 BIB 59


6. The wings of Encarsia are moved almost horizontally during hovering, the bod'being vertical, and there are three unusual phases in the wing stroke: the clap, thefling and the flip. In the clap the wings are brought together at the top of the morpho-logical upstroke. In the fling, which is a pronation at the beginning of the morpho-logical downstroke, the opposed wings are flung open like a book, hinging about theirposterior margins. In the flip, which is a supination at the beginning of the morpho-logical upstroke, the wings are rapidly twisted through about 1800.

7. The fling is a hitherto undescribed mechanism for creating lift and for setting upthe appropriate circulation over the wing in anticipation of the downstroke. In the caseof Encarsia the calculated and observed wing velocities at which lift equals body weightare in agreement, and lift is produced almost instantaneously from the beginning ofthe downstroke and without any Wagner effect. The fling mechanism seems to be in-volved in the normal flight of butterflies and possibly of Drosophila and other smallinsects. Dimensional and other considerations show that it could be a useful mecha-nism in birds and bats during take-off and in emergencies.

8. The flip is also believed to be a means of setting up an appropriate circulationaround the wing, which has hitherto escaped attention; but its operation is less wellunderstood. It is not confined to Encarsia but operates in other insects, not only at thebeginning of the upstroke (supination) but also at the beginning of the downstrokewhere a flip (pronation) replaces the clap and fling of Encarsia. A study of freelyflying hover-flies strongly indicates that the Syrphinae (and Odonata) depend almostentirely upon the flip mechanism when hovering. In the case of these insects a tran-sient circulation is presumed to be set up before the translation of the wing through theair, by the rapid pronation (or supination) which affects the stiff anterior margin beforethe soft posterior portions of the wing. In the flip mechanism vortices of oppositesense must be shed, and a Wagner effect must be present.

9. In some hovering insects the wing twistings occur so rapidly that the speed ofpropagation of the elastic torsional wave from base to tip plays a significant role andappears to introduce beneficial effects.

10. Non-steady periods, particularly flip effects, are present in all flapping animalsand they will modify and become superimposed upon the steady-state pattern asdescribed by the mathematical model presented here. However, the accumulatedevidence indicates that the majority of hovering animals conform reasonably well withthat model.

11. Many new types of analysis are indicated in the text and are now open forfuture theoretical and experimental research.

This study is dedicated to the memory of my late wife, Hanne Weis-Fogh, n6eHeckscher, who helped, encouraged and inspired me throughout twenty-five yearsof research, until she died suddenly in a car accident on 17 April 1971.

I am greatly indebted to Professor Sir James Lighthill, F.R.S., not only for permis-sion to quote his unpublished caclulations, but also for the open mind with which hereceived me. I thank Professor J. W. S. Pringle, F.R.S., for information on largebeetles, Dr R. H. J. Brown for showing me unpublished slow-motion exposures offlying pigeons, Professor W. D. Biggs for information about wave propagation insolids, Dr Martin Jensen for permission to reproduce Fig. 19, Professor W. NachtigalJ


lor permission to reproduce Fig. 20, Dr K. E. Machin for numerous discussions, andDr J. Smart for expert advice on insect wings.

It would have been impossible to provide the slow-motion films and the soundrecordings without the enthusiastic assistance of Messrs B. J. Fuller, G. G. Runnallsand D. M. Unwin. I also thank Mr J. W. Rodford for drawing most of the figuresand Mr R. T. Hughes for checking the references.

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APPENDIX I

Main symbols and some constants and conversion factors between the technicalgram force - centimetre - second system (gf - cm - s) used here and the SI system ofmeasurement.

Superscriptm1 Specific, i.e. per unit weight.

Subscript

aDe

aerodynamicdragelastic

Symbols and constants

cDcLcDGILIMNnPQRr

coefficient of dragcoefficient of liftwing chord, as definedaerodynamic dragweight of animalmoment of inertiaaerodynamic liftrepresentative lengthmassratio, |j5<|niax/|Qalmaxwing-stroke frequencypowerbending momentwing lengthdistance, radius

g gyrationi inertialL lift

(Re) Reynolds number5 acR3, second moment of wing areas

about fulcraT TCR*, third moment of wing areas

about fulcrat timeV,v,u velocity (volume)W workw induced windP angle of tiltF circulation7 positional angle7] dynamic efficiency6 angle of twistA ratio, stroke-arc/wing-chord

230

kinematic viscosity: for air at15 °C and 760 mmHg,v = 0-14 cm2 s"1

= 0-14 x io~4m2s~1.p mass density; for air at 15 °C and

760 mmHg,p = 1-25 x io~* gf cm"4 sa

= 1*23 x io~3 g cm—"= 1 -23 kg m~s.

Conversion factors

force 1 gf equals 9-81 x io"3 N.work 1 gf cm equals 9-81 x io"6 J.power 1 gf cm s"1 equals

9 - 8 I X I O - « W

= 8-44 x io~2 cal h"1

= 1-3 x io~7Hp (int.).specific 1 cal h"1 gf"1 = o-1186 W N"1.power

TORKEL WEIS-FOGH

o" shape factor for second moment ofwing area.

T shape factor for third moment ofwing area.

<f> wing-stroke angle; 1 rad = 57*3°.(o angular velocity; rad a"1.

1 gf cm"1 sa equals 981 g.1 gf cm"4 s2 equals

-3

massdensity

981 g cmmoment of 1 gf cm1 sa equals

inertia si g cma

APPENDIX 2

Second and third moments of the wing areaSecond moment S

The general expression is 5 = crcR3, where cr is the shape factor. It can be found inmost engineering handbooks but is derived here for the case of a semi-elliptical wing inwhich c is the minor axis and R is the major semi-axis. The semi-chord then varieswith the distance r from the fulcrum as

Wr) = {cl2R){R*-r*)i. (48)

If we calculate the moment about c for two entire wings (or for two whole pairs ofwings),

and therefore cr = \ny as in Table 1.

Third moment T

For any wing T is defined as

J r - 0

where T is the shape factor. In the case of a semi-elliptical wing as before, and if weconsider two entire wings,

•r-B= TCR*, (So)

where ^ is the shape factor. It was calculated in a similar way for the other wingshapes in Table 1.

Download - QUICK ESTIMATES OF FLIGHT FITNESS IN HOVERING …Flight fitness in hovering animals 173 Table 1. Shape factor a for the second moment of the area of two whole functional wings (or

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