On the assumption that steady-state aerodynamics applies, simple analytical expressions are derived for the average lift coefficient, Reynolds number, the aerodynamic power, the moment of inertia of the wing mass and the dynamic efficiency in animals which perform normal hovering with horizontally beating wings.
The majority of hovering animals, including large lamellicom beetles and sphin-gid moths, depend mainly on normal aerofoil action. However, in some groups with wing loading less than 10 N m−2 (1 kgf m−2), non-steady aerodynamics must play a major role, namely in very small insects at low Reynolds number, in true hover-flies (Syrphinae), in large dragonflies (Odonata) and in many butterflies (Lepidoptera Rhopalocera).
The specific aerodynamic power ranges between 1·3 and 4·7 WN−1 (11–40 cal h−1 gf−1) but power output does not vary systematically with size, inter alia because the lift/drag ratio deteriorates at low Reynolds number.
Comparisons between metabolic rate, aerodynamic power and dynamic efficiency show that the majority of insects require and depend upon an effective elastic system in the thorax which counteracts the bending moments caused by wing inertia.
The free flight of a very small chalcid wasp Encarsia formosa has been analysed by means of slow-motion films. At this low Reynolds number (10–20), the high lift coefficient of 2 or 3 is not possible with steady-state aerodynamics and the wasp must depend almost entirely on non-steady flow patterns.
The wings of Encarsia are moved almost horizontally during hovering, the body being vertical, and there are three unusual phases in the wing stroke: the clap, the fling and the flip. In the clap the wings are brought together at the top of the morphological upstroke. In the fling, which is a pronation at the beginning of the morphological downstroke, the opposed wings are flung open like a book, hinging about their posterior margins. In the flip, which is a supination at the beginning of the morphological upstroke, the wings are rapidly twisted through about 180°.
The fling is a hitherto undescribed mechanism for creating lift and for setting up the appropriate circulation over the wing in anticipation of the downstroke. In the case of Encarsia the calculated and observed wing velocities at which lift equals body weight are in agreement, and lift is produced almost instantaneously from the beginning of the downstroke and without any Wagner effect. The fling mechanism seems to be involved in the normal flight of butterflies and possibly of Drosophila and other small insects. Dimensional and other considerations show that it could be a useful mechanism in birds and bats during take-off and in emergencies.
The flip is also believed to be a means of setting up an appropriate circulation around the wing, which has hitherto escaped attention; but its operation is less well understood. It is not confined to Encarsia but operates in other insects, not only at the beginning of the upstroke (supination) but also at the beginning of the downstroke where a flip (pronation) replaces the clap and fling of Encarsia. A study of freely flying hover-flies strongly indicates that the Syrphinae (and Odonata) depend almost entirely upon the flip mechanism when hovering. In the case of these insects a transient circulation is presumed to be set up before the translation of the wing through the air, by the rapid pronation (or supination) which affects the stiff anterior margin before the soft posterior portions of the wing. In the flip mechanism vortices of opposite sense must be shed, and a Wagner effect must be present.
In some hovering insects the wing twistings occur so rapidly that the speed of propagation of the elastic torsional wave from base to tip plays a significant role and appears to introduce beneficial effects.
Non-steady periods, particularly flip effects, are present in all flapping animals and they will modify and become superimposed upon the steady-state pattern as described by the mathematical model presented here. However, the accumulated evidence indicates that the majority of hovering animals conform reasonably well with that model.
Many new types of analysis are indicated in the text and are now open for future theoretical and experimental research.
In a recent paper I have analysed the aerodynamics and energetics of hovering hummingbirds and Drosophila and have found that, in spite of non-steady periods, the main 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 which practise hovering or slow forward flight at similar Reynolds numbers (Re), 102 to 104. As discussed in that paper, there are of course non-steady flow situations at the start and stop of each half-stroke of the wings. Moreover, it does not follow that all hovering animals make use mainly of steady-state principles. It is therefore desirable to obtain as simple and as easily analytical expressions as possible which should make it feasible to estimate the forces on the wings and the work and power produced. In this way one may make use of the large number of observations on freely flying animals to be found in the scattered literature. It may then be possible to identify the deviating groups and to 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 since the true flight system is so complicated as to be unmanageable. However, when we confine 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 superficial, but rather that a procedure has been devised whereby the flight performance of a given animal can be evaluated quantitatively as well as qualitatively on the basis of only a few accurate observational data and a minimum of computation. In other words, given reliable information about bodily dimensions and wing-stroke parameters, the method enables one quickly to arrive at a first-order approximation so as to assess whether the animal makes use of well-established mechanisms or employs unusual or novel 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 wing stroke, the frequency of the wing beat and the weight of the animal which is sustained in hovering flight one can calculate, on the basis of steady-state aerodynamics, the minimum coefficient of lift which must be ascribed to the wings. From the same measurements one can also calculate the Reynolds number under which the wings operate, and from this figure one can obtain (using published data on wings) the maximum coefficient of lift which may be expected under these conditions. If the minimum coefficient of lift for steady-state aerodynamics does not exceed the maximum coefficient of lift obtainable at the Reynolds number in question one assumes that steadystate aerodynamics are adequate to explain hovering flight, and from then one can go on 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 is greater than the maximum obtainable, it is clear that hovering flight cannot be explained by steady-state aerodynamics and that a new approach must be made.
The main simplifications are that the animal is assumed to make use of steady-state flow patterns only, that lift is produced at right angles to the direction of the relative wind, that the stroke plane is horizontal with no tilt, that the induced wind can be disregarded because it is relatively small and, finally, that the wing movements are sinusoidal and of similar shape and the same duration for both the morphological upstroke and downstroke. As we shall see (p. 192), these assumptions lead to values for the average lift coefficients needed to remain airborne which are slightly higher than in the more complete treatment (Weis-Fogh, 1972). As to the specific aerodynamic power , the estimates are too small by a factor of about 2 but, as we shall also see (p. 195), it is possible in a simple manner to make appropriate corrections. The instantaneous aerodynamic force on a wing element F(t, r) of chord length c(r) and its vertical (lift) and horizontal (drag) components are shown in Fig. 1 A. In Fig. 1 and in the text, any quantity X which is a function of time t and distance r from the wing hinge, or fulcrum, is written as X(t, r). Appendix 1 gives a list of symbols and constants used.
The justification for the simplifications is that the majority of insects, birds and bats have been found to tilt the long axis of the body towards the vertical (Fig. 1B) when hovering, so that the wings beat in an almost horizontal plane, usually symmetrically about the average positional angle of 90° (Fig. 1C).
(a) Coefficient of lift
Fig. 2 A shows how the outline of a wing of total length R, and in particular the wing chord c(r), 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 function whether the moving wing is morphologically two-winged as in Lepidoptera and Hymenoptera or consists of only one part as in Diptera. It should be stressed here that there are hovering insects which make use of two pairs of wings which beat out of phase, as is the case in Odonata and Neuroptera, and these insects are not included in the present analysis.
(b) Reynolds number (Re)
(c) Drag, aerodynamic torque and power
WING INERTIA AND DYNAMIC EFFICIENCY
(a) Mass moment of inertia
The distribution of mass in fresh wings of Schistocerca gregaria is seen in Fig. 3 A and B. The insects were of different sizes but all results were recalculated to a locust of average size, a standard Schistocerca, by multiplying lengths with the ratio of the length indices, l/ls, and weights by the ratio of the volume indices, v/vs, where Is and vs are the standard indices for length and volume and I and v those of the individual insects used (Weis-Fogh, 1952). The standard deviations of the corrected figures 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 as closely as possible, it is seen that the mass tends to decrease linearly with distance r from the fulcrum both in the broad fan-shaped hindwings and in the stiff slender forewings. The deviations occur mainly towards the base, where r is small and the effect negligible. Consequently we can calculate I as being equal to that of a homogenous triangular plate of uniform thickness d and which is rotated about its base line c (Fig.3C).
It is often difficult to cut and weigh small insect wings, and for quick estimates it is obviously advantageous to use an expression like equation (27). Is it applicable to other insects? In order to answer this I have calculated the moment of inertia by means of equation (27) for the insects in which Sotavalta (1952, 1954) found I by weighings. His results are of course not as accurate as those for the locust because the wings are small and only a few strips could be cut. Nevertheless, the conformity between the calculated and the empirical results is good, as is seen from Table 3, in particular when the sources of error are taken into account. In fact, if accurate measurements are not readily available, the formula provides values which are entirely adequate for the present treatment. For flying insects considered as a group, the formula is accurate and it must reflect some structural principle of general significance, to be discussed 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 I alters when the wings are mutilated. We assume that . Let the moment of inertia of the mutilated wing be Im. It can 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 fulcrum. The moment of inertia of the cut section about the fulcrum is now
(b) Dynamic efficiency
Since we are dealing with ratios rather than absolute figures, let the angular movement represent i angular unit so that the wing moves from γ = 0 to γ = 1 during the upstroke in Fig. 5. Let the maximum aerodynamic moment in equation (22) be represented by +1 unit. The maximum inertial moment in equation (21) is then +N units at 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 Q(a+i) is represented by the thin curve from + N to E′. It is the area between the latter and the base line AE’ which represents the total positive work during one half-stroke.
EXPERIMENTAL METHODS AND MATERIAL
Most of the flight data derive from critical examination of the literature, but it was essential 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 of Syrphidae. So far, the best studied genera have been Volucella and Eristalis, which represent 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 with the body axis horizontal and with the wings oscillating in a non-horizontal plane and through a surprisingly small stroke angle. Obviously, they are exceptions to the general rule; the same applies to the Odonata (damsel flies and dragonflies). While I have been unable to study hovering dragonflies in nature, I have recently succeeded in recording the wing-stroke frequency of Syrphinae of the genera Syrphus, Sphaerophoria and Platychirus 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 no wind on calm summer days at Tibirke in Denmark. The flight tone and therefore the frequency was picked up by means of a Brüel & Kjaer 25 mm Condensor Microphone
Type 4145, with battery-driven Pre-amplifier Type 2619 and Power-supply Type 2804. The signals were recorded on a Nagra IV–S battery-driven tape-recorder supplied with a special noise-reduction system which ensured flat response characteristics between 100 and 970 Hz. The steep cut-off at either end corresponded to a reduction of 20 dB at 30 and 2000 Hz respectively. This was essential in order to prevent interference from other sound sources. The individuals were narcotized instantaneously while hovering by means of a strong jet from a Sparklets CO2-spray filled with chloroform and caught in a small net. The weights of the body and the wings were determined by means of an electronic microbalance immediately on return to the laboratory. 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 like Melolontha vulgaris (cockchafer), Rhizotrogus (now Amphimallon) and Cetonia 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 aerodynamic 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 substantial contributions by non-steady flow. Thus Osborne (1951) found an average lift coefficient of more than four. The beetle weighed 2·6 gf and had a ‘wing load’ of 3·2 kgf/ m2. I could not obtain this species but Professor J. W. S. Pringle, Oxford, brought some large elephant-dung beetles of the genus Heliocopris back from Kenya. He kindly measured the frequency acoustically during near-hovering, free flight and also gave me weights (8–13 gf) and wing dimensions; the wing load was 3–4 kgf/m2 and therefore comparable to that of the stag beetle. As in other lamellicorn beetles, the stroke angle ϕis about 180°.
Large hawk moths
Dr P. J. Wilkin, Brunel University, has recently shown me (personal communication) recordings of wind-tunnel experiments with the Florida tobacco homworm moth, Manduca sexta (Johannson), which was flying suspended almost horizontally from an aerodynamic balance. His results indicate that, under certain conditions, these insects can produce lift coefficients considerably in excess of steady-state values, and he seems to have confirmed this in model experiments. The question is how and to what extent freely flying sphingid moths make use of unusual aerodynamics – for instance, when feeding normally like hummingbirds in front of a flower. Caterpillars of Manduca sexta (Sphingidae, subfamily Acherontinae ; earlier name Protoparce sexta Johannson) were grown in the laboratory on a semi-synthetic diet (Hoffman, Lawson & Yamamoto, 1966). Adults were isolated and hovered spontaneously in front of an artificial flower but only shortly after ‘dusk’ and in subdued light. The flower was scented with the synthetic attractant isoamyl salicylate and the feeding solution contained 10% sucrose in water. The temperature was 26 °C and the relative humidity 90%. The flight chamber measured 71 × 71 × 62 cm3, and when the moth was hovering in the correct place near the centre of the almost dark cage two crossed light-beams of low intensity were interrupted by the insect. This blanked two C.D.S. phototransistors which triggered a time-delay circuit which then started a sound-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 override). It also triggered two 1000 W quartz-iodine lamps, and the timing was such that the lamps did not light up until the film had reached a sufficient speed, i.e. about 3000 frames/sec. This elaborate arrangement was essential because Manduca sexta is a nocturnal 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 as aerobatics 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 Encarsia formosa Gahan (family Eulophidae, subfamily Aphelinidae) were obtained from a commercial dealer; this species is used for biological control of the greenhouse white-fly, Trialeurodes vaporariorum Westwood (see Speyer, 1927; Burnett, 1949; Parr, 1968). The tiny adults (wing length 0·05–0·07 cm) both fly actively and make strong jumps. The procedure therefore was to 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 chamber was 5·4 cm high, 3·5 cm wide and 1·6 cm deep, so that the insects flying near the centre were within the range of focal depth (2 mm) and aerodynamically almost free of wall effects. The insects are diurnal and are attracted to fight. A 50 W fibre-optical vertical light beam was constantly on and tended to concentrate them in the vertical centre axis of the chamber, while three 150 W quartz-iodine lamps were triggered simultaneously with the camera (6–8000 frames/sec); they did not appear to interfere with flight. The lens was a Carl Zeiss 63 mm Luminar, f1:4·5, used at low focal opening, and the magnification on the film was one to four times. The temperature was 26–27 °C and the air almost saturated with water vapour. It was usually easy to distinguish between jumps, flight initiated by a jump, and free ‘unaided’ flight. Only the latter scenes are used here.
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 relevant SI 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 transport over long distances ; locusts and other migrating species are good examples. This type of flight is now reasonably well understood in birds and insects on the basis of ordinary aerofoil 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 practise slow 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 that this, the most strenuous form of flight, is less demanding in small animals than in large animals (because of the surface/weight ratio), but presumably also that the conquest by means of active flight of the small niches of the aerial biosphere has opened up an endless variety of ecologically different niches to which the response has been intense speciation, best exemplified by the 750000 known species of winged insects. Hovering flight is therefore of general significance, particularly in insects. The main problems are whether it is based on a common mechanism or not, and what the main principle is.
We have already seen that the hovering of hummingbirds and of Drosophila involves the same type of wing movements and can be understood on the basis of steady-state aerodynamics, although non-steady periods and effects undoubtedly exist and may play a significant role under some conditions (Weis-Fogh, 1972). Both types 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 a large stroke angle and (c) approximately in a horizontal plane, while (d) the long axis of the body is strongly inclined relative to the horizontal, sometimes almost to a vertical position. Fig. 7 shows some examples of normal hovering. It has been drawn from flash photographs of freely flying animals. The first question is how common normal hovering actually is.
All the animals listed in Tables 4 and 5 are known to hover, and if the name appears without an asterisk the species uses normal hovering. The examples have been greatly extended by direct observations of free flight in nature which I have made during the past few years. In the case of nocturnal species a high-pressure mercury-vapour lamp was used (Philips HP 125 W, 4000–7200 Å).
Normal hovering was observed in many species of Isoptera in East Africa (swarming termites), Plecoptera (stone-flies), Thysanoptera (thrips), Hemiptera (leaf-hoppers, aphids, backswimmers, may bugs), and Neuroptera (lacewings). The Coleóptera or beetles deserve more comment. Although the elytra (forewings) are usually spread out and 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 in a horizontal plane (cf. cockchafer in Fig. 7f) and sweeping through a large angle of about 180° on each side. However, because the body is broad, the two wings do not come into’close contact. Normal hovering is perhaps more easily studied in Coleóptera than in any other group except the large crane-flies (Tipulidae), and most species or groups make use of it, the tiger beetles (Cicindelidae) being possible exceptions.
Among Lepidoptera the moths generally appear to move their wings in accordance with the common pattern. This is indicated by my direct observations and also appears from flash photographs taken in nature (Fig. 7,b, Deilephila elpenor) and it was proved by the slow-motion films of the sphingid moth Manduca sexta hovering in the laboratory (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 often happens 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 wings is black. The sequence started when the left wings began the morphological downstroke (Fig. 8 a). The film speed was 3000 frames/sec, increasing to 3900 at the end of the stroke (Fig. 8p). Only every 10th frame was traced. From the apparent length of the body axis it is seen that the body is inclined about 45° relative to the horizontal. This scene, and films taken horizontally from the side, show that the wings beat almost horizontally like those of hovering hummingbirds, that they are twisted in a similar way and that upstroke and downstroke are of equal duration (cf. Weis-Fogh, 1972), the tilt β of the stroke plane being about 10°. It is apparent from the films that the fore- and hindwings move together and that a cleft between them does not normally open, although this happens during certain brisk manoevres. Furthermore, during normal hovering the acceleration of the wings does not lead to any significant elastic bending of the long axis of the wing (Figs. 8a–p; 7 c) but, again, substantial bending can occur during quick escape reactions (Fig. 7 e). We are not concerned with the latter situations here; the main point is that the elastic storage of energy is confined to 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 from normal hovering in front of the flower, but after the light and camera had been on for a short time it began to move backwards with vertical body axis (Fig. 9a; 3000 frames/ sec) and during the stroke the axis tilted backwards and the speed was also increased in the backwards direction. The morphological underside (black) of the wings is therefore seen during the main part of the stroke, whereas the upper side would predominate during fast forward flight, underlining the supreme manoeuvrability of the insect as well as the general principles of flight. On occasions like this the manoeuvre even 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 the top of the upstroke (a ‘clap’) and partly because it looks as if the angular movements deviate appreciably from the sinusoidal case. They have therefore been marked by a double asterisk (**) and need to be studied separately in another context.
The Mecoptera (scorpion flies), Trichoptera (caddis flies) and the large group of Hymenoptera (sawflies, swarming ants, wasps and bees) all appear to use normal hovering, at least as the main principle. It is well known that the honey-bee (Apis nullified) can change the inclination of the stroke plane and that this is almost horizontal when the insect is hovering in front of a flower (Neuhaus & Wohlgemuth, 1960 ; 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 observations and Fig. 7(d) indicate that they also use normal hovering. This was easy to observe with certainty in the large carpenter bees of the genus Xylocopa, which have deeply stained wings (X. violácea in Italy, 1972; and an unidentified species in Kenya, 1972). It may have been thought that the very small chalcid wasp Encarsia formosa would represent an exception but this is not the case, as is apparent from Fig. 14 and the description later. The reason for the triple asterisk (***) against its name in Tables 4 and 5 is that in every single wing stroke the two pairs of wings come close together in a distinct dorsal clap before the morphological downstroke starts. The significance of this will be analysed in the Discussion (p. 214), but the same phenomenon appears to be present in Drosophila virilis according to Vogel (1965, fig. III, I), who did not comment upon it however. It has therefore been similarly marked.
The large and diverse group of truly two-winged insects Diptera presents special problems. Whereas crane-flies, mosquitoes and other Nematocera as well as many large Brachycera and Cyclorrhapha undoubtedly use normal hovering in most cases (this applies for instance to EristaEs, Volucella and probably Calliphora), the dorsal clap seen in Encarsia and Drosophila probably occurs in many other small flies and insects from different orders. In addition, the true hover-flies, the Syrphinae, perform steady hovering flight according to principles which are unknown so far and which also seem to apply to large dragonflies, Odonata, as already pointed out. Their flight will therefore be treated separately.
Finally, some small bats make regular use of hovering, and although the main movements are consistent with the normal pattern the wings are somewhat flexed during the morphological upstroke (Eisentraut, 1936; Norberg, 1970), as is also the case in small passerine birds (Zimmer, 1943 ; Brown, 1951 ; Greenewalt, 1960). 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 exceptions 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 first glance it may seem difficult to find a simple geometrical function which describes the wing contour in sufficient detail for a realistic estimate of the second and third moment of the area, S and T. However, it is not difficult in practice, and in most cases the wing(3) can be described by means of a semi-ellipse where the wing length R corresponds to half the major axis and the minor axis is adjusted to give the best fit for the outer part of the wing ; this is the value for the adopted wing chord c in Table 4. The procedure is illustrated in Fig. 10 for both two-winged and four-winged animals. In some cases the major axis is tilted by a few degrees relative to the line from the fulcrum to the tip in order to give the best fit, as is seen in the case of Syrphus (Fig. 10c). This does not introduce any significant error. It may be thought that the hatched areas between the wing contour and the ellipse justify a substantial correction, but this is not the case. If, for example, we consider a wing in which the inner part is constricted, leaving a 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 lift coefficient and the aerodynamic power will then be underestimated by similarly small amounts and this is far too little to worry about in the present approach. The poorest fit is undoubtedly that of Pieris in Fig. 10(f) because the outer areas count most. However, the deviation is not large enough to be of real concern since much larger errors are introduced because our basic assumptions do not seem to apply to this group. In quantitative terms, the fit for the hummingbird Amazilia and for Manduca is almost perfect, taking into account that the contour does not remain really constant in 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 for free flight have been accepted and only if they belong to one set for each specimen included in the list. The data for Drosophila virilis are an exception since they derive mainly from Vogel’s wind-tunnel studies, as discussed elsewhere (Weis-Fogh, 1972). As to Encarsia formosa, the weight G is the average from some ten individuals and the first example refers to a particular flight of a particularly large specimen, whereas the second entry represents average values. Apart from body and wing dimensions, the most critical parameter is the wing-stroke frequency n because it varies greatly with size and it enters into the equations in the second or third power. However, in the selected examples the values can be considered to be accurate. The same does not apply to the stroke angle ϕ, but it cannot normally amount to more than 180° = n = 3·14 rad, and ϕ is usually about 120° = 2·09 rad in both large and small hovering animals.
(d) Gross results: Table 5
The weight of the animals ranges from 20 gf to 25 μgf and the wing length from 13 to 0·06 cm, the corresponding Reynolds numbers (Re) being 15000 and 15 respectively. It is possible to obtain lift coefficients approaching ro during normal steadystate flow down to (Re) between 10 and 100, and the square-law relationship between force and velocity holds down to 10 (Thom & Swart, 1940). It is therefore meaningful to 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 a factor 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 principles to be operative.
The major conclusion from Table 5 is that most hovering animals move their wings in accordance with steady-state principles and at Reynolds numbers well above the critical range, and so that the average lift coefficient is relatively small, seldom exceeding i-o. This applies to all the lamellicorn beetles hitherto considered exceptions (Osborne, 1951; Bennett, 1966, 1970) and to most wasps and bees. Apart from Encarsia the highest value among Hymenoptera is for Bombus terrestris, which many authors in the past have denied the ‘right’ to fly. Yet, the average lift coefficient of a particularly 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 principles. This does not exclude that during take-off or escape reactions non-steady situations may arise and be important, as will be discussed later, but it is hardly within the groups mentioned so far that we are likely to find the really interesting exceptions.
There are, however, some important deviations like Pieris napi (and probably P. brassicae) and, of course, the chalcid wasp Encarsia formosa. In both cases is much too 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 is hardly justified. As we shall see, the same applies to Syrphinae and Odonata. The results for the bat Plecotus auritus are also in parentheses but they are presumably not far from the true values in spite of the fact that the wings are somewhat flexed during the upstroke.
(e) Corrected results: a simple approach
In the simplified treatment there are two main variables which have been omitted, the tilt ft of the stroke plane relative to the horizontal and the induced wind w. As we shall now see, the tilt is of relatively small significance whereas the induced wind is a major factor when estimating the aerodynamic bending moment Qa and the power Pa.
In Table 6 the lift/drag ratio is also given, and the correction procedure is best illustrated with reference to Fig. 11 for Drosophila virilis, where = 0·2, L/D = 7:4 and’β = 20°. 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, − vg is actually tilted by 20° and the vertical w must be added to give the resulting relative wind velocity Vg. The true resulting wind force F then has a vertical component which is 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 time Vg is 5 % smaller than − vg, and the velocity squared is therefore 10 % smaller than the assumed value. The combined result is that has been estimated almost correctly whereas the aerodynamic power Pa has been underestimated by 30%. The exact corrections vary with β, L[D ratio, and w/vg ratio, but for the material as a whole there is no justification for correcting the values, which appear to be valid within ±15%. However, the power is always systematically underestimated, and some representative values are compared in Table 6. In this and in the previous table, we use the specific aerodynamic power, or power per unit body weight . It is seen that the correction is substantial and that the corrected figures compare reasonably well with the results obtained in a more comprehensive treatment of Amazilia fimbriata and Drosophila virilis (Weis-Fogh, 1972). It should be noted that the corrections are largest when the L/D ratios are high.
It is helpful to consider the more general diagram in Fig. 12, which applies to an L/D ratio of 6:1 and a w/vg ratio of 0·2 for semi-elliptical wings. It is seen that the optimum tilt of the stroke plane with respect to the horizontal is about + 20° and that this does not alter the calculated significantly. However, the true drag in the stroke plane is about double that estimated from the simple theory. Consequently, in the following we shall use the corrected values for , N and η in Table 5. They hardly differ by more than 20% from the true values. It should be noted that quite large variations in β have relatively little effect and we are therefore justified in disregarding this parameter in a treatment whose aim is a first approximation based on simple expressions.
(f) Specific aerodynamic power
Of the 30 examples from 28 species in Table 5 the available information was incomplete for the bat Plecotus auritus and the moth Amathes bicolorago, which also had an unusually high lift coefficient (f6). For the two Pieris butterflies and the chalcid wasp Encarsia 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 them separately.
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 average being 21 cal h−1 gf−1. The upper limit is 40 for the heavy lamellicorn beetle Heliocopris and the lower limit is 11 for the large sphingid moth Manduca. This must be accepted as an empirical fact and both the absolute magnitude and the range are consistent 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 must also consider the effect of Reynolds number (Re) on the drag coefficient , as will be discussed elsewhere (Weis-Fogh, in preparation). It is also consistent with the fact that the specific chemical power consumption, the metabolic rate, is of the same magnitude in small and large flying insects (Weis-Fogh, 1964), and probably in small birds and bats as well. This obviously requires more discussion.
If we accept that the small insect Encarsia formosa could produce the necessary lift coefficients 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) and therefore be of the same magnitude as in other hovering animals. This came as a surprise.
(g) Dynamic efficiency and elastic forces
Table 5 -clearly shows that the ratio N = |Qi| max l|Qa| max is high in almost all insects analysed, with the exception of butterflies and Drosophila. This means that the dynamic efficiency η is low and that most species must spend 2–3 times more mechanical energy than needed for flight alone, provided that there is no elastic system present in 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. If we consider the metabolic rates of the six species for which the data are sufficiently good, Table 7 shows that, as to energetics, flight would not be feasible in Vespa crabro, Aëdes aegypti and Eristalis tenax unless they could make use of elastic bending moments in the thorax or wings similar to those in the locust Schistocerca gregaria, the moth Sphinx ligustri and the dragonfly Aeshna grandis (Weis-Fogh, 1961, 1972). As to the honey-bee, Apis mellifica has the highest continuous metabolic rate measured in any insect 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 available evidence amounts to a circumstantial proof that as a group, flying insects possess and depend upon elastic forces in order to store and release the kinetic energy of the oscillating wings. Under conditions of normal continuous flight the elastic system resides in the thorax and not in the wings because the wings are only bent elastically and to a significant extent during exceptional circumstances. This does not apply to Encarsia formosa.
(h) Hovering flight of a very small insect
The calculated average lift coefficient of about 3 for Encarsia formosa in Table 5 is much too high to be compatible with steady-state aerodynamics. It may be argued that the flight data in Table 4 are incorrect or misleading. To some extent this may be true because the morphological upstroke is of shorter duration than predicted from an harmonic movement of the wings due to the ‘clap’ period towards the end of each upstroke (cf. Figs. 14, 15), but the effect could not reduce the lift coefficient to less than about 2. We are therefore faced with the problem that these tiny insects do in fact hover and produce an average lift coefficient of 2–3 at a Reynolds number of between 10 and 20. It should be emphasized that we have not taken into account the brim of hairs and bristles seen in Fig. 13 but only the size and shape of the membranous parts of the wings. At present, we can hardly do better since the hairs are invisible on the available films, and it is doubtful to what extent they play a direct role in the process of lift production, as will be discussed when a new mechanism is proposed in the Discussion (p. 217).
As to morphology, Fig. 13 shows that Encarsia formosa Gahan resembles an ordinary four-winged wasp. The most outstanding deviations are shared with other very small insects among Hymenoptera and Coleóptera, in particular the brim of long marginal hairs, the ‘stalked.’ wings, and the lack of any significant wing veins apart from a strong reinforcement of the leading edge (costal veins). In addition, the wing hinge is so constructed that the ‘stalked’ elastic wing bases can be swung above the dorsum and attain a positional angle of more than 180° (cf. Fig. 15). The effect is that the two pairs of wings can be ‘clapped’ closely together above the dorsum in spite of the relatively large distance between the wing fulcra of opposite sides. I have often seen this dorsally ‘clapped’ posture of the wings in dead or preserved specimens, but it is always present towards the end of the upstroke during active flight, whether the insect moves horizontally forwards or backwards in the air, hovers or is engaged in refined manoeuvres.
Some details deserve attention since they may be of functional significance. Fig. 13 was constructed from a set of photomicrographs of specimens embedded in glycerol jelly after mild treatment in lactic acid. Staining with methylene blue revealed that resilia hardly plays any significant role in the wing system so that the main elasticity must reside in the flight muscles and in the hard cuticle of thorax and wings, as in larger Hymenoptera. The inner part of the forewing, from the axillary sclerites and the tegula (a) to point (b) where the marginal vein (c) starts, is not ordinary wing membrane but a rather solid plate of hardened cuticle. I have not been able directly to compare 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 costal cell and the submarginal vein (d) is the strongly reinforced leading edge itself. It is hollow and carries a row of what appear to be campaniform sensilla at its posterior margin, but the most important characteristics are (1) that it can be twisted about its long axis as a thick-walled tube (pronation and supination) and (2) that it is connected with 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 basically the same way as the whole wing is in large insects, but the twisting does not necessarily result 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, elastic bending zone from (b) to the tip of the anal vein (e). This means that the outer, main part 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 with small bristles, about 20 μm long. They may not have any aerodynamic importance but they would prevent the wings from adhering to other surfaces, including the opposing wing surfaces, during the ‘clap’ period. Finally, the distance between the wing bases of the stalked hindwings is smaller than that of the forewings. This and the fact that the forewings are moved both up and down and forwards and backwards relative to the body axis require a sliding coupling between the two wing pairs. The coupling consists of the rolled-in hind margin of the forewings (called the retinaculum by Compere, 1931, broken line) and the two upwardly directed hooklets at the tip of the marginal vein of the hindwing. In preserved specimens the hindwings are usually unhooked and held at a different positional angle from that of the forewings, indicating that the hindwings do not merely follow the forewings passively although they move together during flight.
On the left-hand wings in Fig. 13 A the triangle drawn in broken lines corresponds to the contour function used in the first example for Encarsia in Tables 4 and 5. It should take care of the possibility that the marginal hairs may be considered as part of the lift-producing surface. More realistically, the hatched rectangle represents the best simple fit for the case that only the membranous parts matter significantly. It should be noted that only the outer two-thirds of the rectangle is used in the calculations because the wings are ‘stalked’. In fact, this refinement makes little difference as to but is important for the further treatment of lift production.
About 50 slow-motion films were obtained, which assisted in building up a picture of free flight, but only ten gave quantitative information about true hovering.
It was a surprise to find that the movements of Encarsia wings basically resemble those of other insects, small or large. During fast forward flight the body angle is small and the wings beat obliquely up and down. During hovering on the spot, or when flying slowly backwards, Fig. 14 shows that the body is almost vertical and the wings beat almost horizontally. A detailed description is necessary but it must be emphasized that the film speed (7150 frames/sec) is insufficient for accurate tracings of the outlines 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 the left to the right and so that it also climbed at an angle of about 45° (Fig. 14). Over four complete strokes the frequency was 403 sec−1 and the translational speed of the body was 9 cm sec−1 in the backwards-upwards direction, or sufficiently small to be ignored in relation to the speed of the wings themselves relative to the air, which exceeds 150 cm sec−1 at the radius of gyration during the downstroke and is significantly higher during the upstroke. However, the fact that the insect climbed not only showed that it could produce lift in excess of what is needed for hovering but it also made it possible to estimate during which part of the stroke lift is produced, by means of measuring the vertical oscillations of the body. The tracings in Fig. 14 are from wingstroke no. 3 and relate to the movements of the wings relative to the body. The vertical and horizontal axes are indicated; and frames are numbered so that no. o is the beginning of the morphological downstroke. The animal is seen at a somewhat oblique angle with its long axis slightly tilted by about 20° towards the reader and so that the left-hand side of the head is above the plane of the paper. The right pair of wings (on the far side of the insect) then appears at or slightly above the level of the head, 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 to outline the wings. The anterior or costal edge of the forewings is drawn in heavy line.
The wing stroke
Although forewing and hindwing on each side can separate and sometimes do so, basically they move as a single unit, the pair of booklets ensuring a sliding anchorage to the retinaculum of the forewing. The stroke in Fig. 14 is covered by 18 frames and starts in no. o with the wings held closely together as two plane plates above the dorsum and with the leading edge at about the same level as the forehead. During the next 0·1 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 then separate and move almost horizontally during the morphological downstroke as if they were ordinary aerofoils working at moderate angles of attack (nos. 3–5), until they pass the midpoint where γ = 90°. The angle of attack is then increased and the wings continue to move for another 30° but are also swung upwards until the end of the downstroke is reached somewhere between nos. 7 and 8. The morphological upstroke starts partly with a rapid reversal of angular movement which actually leads to an elastic bending of the wings at the transverse bending zone (nos. 9 and 10) and partly with a quick ‘flip’ β the main wing surface whereby it is rotated as a whole through a large angle so that the costal edge again leads (nos. 7–9). It resembles the flipping of a pancake in the air, hence the term. The rapid morphological upstroke then follows with a tilt of the stroke-plane angle of about 30° and elastic straightening of 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 cannot produce any useful aerodynamic forces, because the two pairs are held closely together, their flat surfaces are vertical, and they are moved vertically upwards until the tips reach the level of the head again (no. 17).
Detailed analysis of the movements of the body relative to a co-ordinate system fixed in space showed that in three successive wing strokes in this sequence (strokes 1–3), the vertical movements followed the same pattern, as listed in Table 8. The horizontal movements of the body are of less significance since the speed in this direction is small, the wind resistance is therefore small, and the kinetic energy would tend to blur any variations. The division between ‘upstroke’ and ‘downstroke’ cannot be made very accurately, but comparison with Fig. 14 should clarify doubts as to the interpretation. It is seen that both the horizontal downstroke and the tilted upstroke produce lift in excess of the body weight in all three strokes, and in approximately equal amounts. Also, the ‘clap’ period does not appear to contribute to the vertical force ; the apparent exception in stroke 2 is probably caused by the ‘up shoot’ following the exceptionally strong lift during the preceding two phases. In any case the resolution of the photographs is too small to permit any further conclusions about the ‘clap’ period. However, another important observation is that a substantial amount of climb and therefore of lift is produced between frames o and 3, i.e. well before the wings reach maximum angular velocity during the ‘downstroke’ and immediately following the ‘flinging open’ phase. Similarly, it looks as if lift builds up immediately after the ‘flip’, which initiates the ‘upstroke’, but the rapid angular movements and the elastic deformations 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 forewing are summarized in Fig. 15, which derives from another specimen in film 47 A. The insect was photographed from in front and almost along the long axis of the body, the wing-stroke frequency being 374 sec−1. During the period of ‘clap’ the positional angle γ is larger than 180°, as already explained, and the ‘downstroke’ follows a course not far from that of an harmonic movement at the overall frequency of oscillation. On the other hand the ‘upstroke’ is considerably faster since the half-cycle includes both the upwards movement and the ‘clap’ period.
Both these and other films were used to estimate the rate of rotation of the wing plane during the periods of ‘flinging open’ and of ‘flip’, but these figures are more relevant when we test the new proposed mechanism for lift production in the Discussion to which these observations lead (p. 216). What should be kept in mind at this stage is that the wings are moved essentially as if ordinary aerodynamic principles apply; the useful force is produced almost at right angles to the movement, in contrast to a principle based upon pushing or rowing, which have often been invoked to explain the flight of very small insects. Our problem can then be re-defined as follows: how is an apparent conventional aerodynamic cross-force, i.e. aerodynamic lift, produced by means of flapping wings under conditions which prevent the building up of sufficient circulation T in the usual way?
(i) Hovering flight of syrphid flies and dragonflies
The true hover-flies belong to the subfamily Syrphinae. Apart from the frequency 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 known about their flight. They are of course easily observed ; but, in contrast to the distinct hum produced by flying species of the subfamily Eristalinae (Eristalis, Helophilus, Myiairopa, etc.) and Volucellinae (Volucella), whose wing-stroke frequencies have often been recorded, the flight sounds emitted by the Syrphinae are very weak. It is also characteristic that they retain an almost horizontal body axis whether they hover or dash off in fast forward flight when disturbed, as mentioned earlier (p. 181). Both in Syrphinae and in Odonata this is probably an adaptation to an adult life entirely dependent upon flight for feeding and sexual display. In both groups no time is wasted by 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 the body axis tilted head downwards. During typical hovering in still air the stroke angle, ϕ, 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, λ, in equation (35) is only about 1·4 in Syrphinae. The wings appear to beat up and down with a tilt of about 45° to the horizontal and the tips definitely do not follow the usual horizontal figure-of-eight; but, again, future slow-motion films or stroboscopic observations of freely flying insects are needed to elucidate the details. In contrast, my unaided observations on foraging Eristalis, Helophilus and VoluceUa indicate that representatives of these subfamilies make use of normal hovering. In fact, Eristalis can be quite difficult to distinguish by sight or sound from a foraging bee.
The sounds produced by hovering Syrphinae are barely audible to the human ear and they are complex, as is seen from Fig. 16. The upper three recordings (a–c) are from the expert hoverer Syrphus balteatus, which can vary its frequency appreciably. It is 177 sec−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 frequencies the pattern is very complex, and this applies to all the recordings including those of Syrphus ribesü (d), 5. coroUae (g) and Sphaerophoria scripta (e, f). The latter clearly demonstrates that the true wing-stroke frequency of 315 sec−1 in (f) would be estimated to be twice as high if a purely auditory method was used, the so-called tenor error discussed by Sotavalta (1947).
No attempt is made in this study to interpret the sound tracings. They are merely used to determine the frequency of the beating wings during true hovering of each individual included in Table 9, but they may offer important clues about the flight mechanism at a later stage.
The aerodynamic problem
If an animal were restricted to the use of conventional aerofoil action for hovering, the best possible solution would be (a) to make equal use of upstroke and downstroke and (b) to beat the wings almost horizontally, as during normal hovering. Any substantial deviation would require higher lift coefficients. If therefore we calculate the average for the syrphid flies on this assumption, known to 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 best fit for each individual insect. The figures in parentheses are the results when the stroke angles p differ appreciably from the true values. It is seen that in all Syrphinae of the three morphological types should be as large as 2–3, i.e. incompatible with steadystate aerodynamics by a factor of at least 2 or 3. In fact the real discrepancy is larger partly because λ is so small (1·4) and partly because the wings do not beat in a horizontal plane. Clearly, the non-steady flow patterns present at each end of the stroke in any hovering animal (Weis-Fogh, 1972) must be utilized and must have become dominant in Syrphinae, rather than the steady-state phases which play a major role in most other groups.
This extreme situation is probably correlated with the relatively low figures for the wing loading of Syrphinae, 0· kgf m−2, which contrasts with those for Eristalinae and Volucellinae, 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 to rely mainly on conventional principles. However, within flying insects, and in particular within Diptera, there must clearly be a whole spectrum ranging from the ordinary to the exceptional mode of lift production.
As to Odonata, calculations similar to those in Table 9 for Aeshna júncea (based on Sotavalta, 1947) and for Aeshna grandis (Weis-Fogh, 1967) show that hovering dragonflies must rely on non-steady aerodynamics because in both species each of the four wings should operate with a minimum of 2·3. Again, the wing-load figure for Odonata is low, 0·1–0·6 kgf m−2 (Müllenhoff, 1885).
In hovering Syrphinae and Odonata there is no ‘flinging open’ phase, but at each end of the wing-stroke there must be a rapid rotation of the wing surface about its long axis, or a phase corresponding in principle to the ‘flip* of the main part of the wing surface in the chalcid wasp.
This investigation represents a first-order approximation so that refinements and supposedly second-order effects have been avoided deliberately in order to make the survey manageable in terms of labour and also to discover new principles and solution which may have been overlooked in previous studies. In fact, the conclusions arrived at here and in the previous sections should make it clear that we are now beginning to understand the major mechanisms in hovering flight in terms not only of conventional aerodynamics 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 of mechanisms, as is so often the case in biological systems where countless generations have been selected for fitness in the course of evolution. Since late Devonian, about 350 million years ago, nature has explored aerial locomotion in all its aspects and over a 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 the gliding and soaring flight of birds (Pennycuick, 1968b, 1972a ; Tucker & Parrot, 1970), bats (Pennycuick, 1971), of the extinct pterosaurs (Bramwell, 1971) and of some insects (Jensen, 1956; Nachtigall, 1967; R. Å. Norberg, 1972). Similarly, powered flight has been 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 are either borne out or inherent in most results and arguments presented so far: (1) that the basic mechanism is similar to that first proposed by Lilienthal (1889) for large birds and 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 steadystate aerodynamics (Brown, 1948, 1951, 1953; Weis-Fogh & Jensen, 1956; Jensen, 1956; Pennycuick, 1968a, b), and (3) that many refinements are present within the framework stated under (1) and (2) which can be considered to be consequences of refined kinematics seen in birds, bats and insects (cf. Nachtigall, 1966; Nachtigall & Kempf, 1971; Rüppel, 1971; Bilo, 1972). In a recent book Pennycuick (1972b) has provided a synthesis of several aspects of vertebrate flight. Although some of his points are arguable, the general conclusions and line of thoughts provide an admirably clear picture of the present state.
In contrast to this there have been claims in the literature that some insects with high wing loadings, in particular lamellicorn beetles, must make use of non-steady principles to a major extent (Osborne, 1951,Bennett, 1966,1970). We have seen that, although no new principles have actually been proposed but must be related to a high rate of wing twist in some way (Bennett, 1970), the relatively heavy beetles and bees need not necessarily use non-steady aerodynamics, not even when they hover. One of the results of the present study is that exceptions to this rule (i.e. steady-state aerodynamics) are to be found among the insects (and birds?) with a low rather than a high wing loading, as is clearly demonstrated in Table 10.
In spite of considerable progress in recent years it still remains true that the flying desert locust Schistocerca gregaria is the best understood and analysed example of fast forward flight, and we should therefore briefly consider Martin Jensen’s results from 1956 on the basis of our analysis of the wing movements seen in Fig. 17 (not previously published). It represents traces of the sequence of 18 equally spaced flash photographs (10−6 sec) from stroboscopic film I. The wing-stroke frequency n was 17·5 sec−1 and all other flight parameters also appeared to be normal for steady horizontal flight at 3·5 m sec−1. The average vertical lifting force was 2·23 gf as measured on the aerodynamic balance, or 97% of the body weight, while the detailed analysis based upon steady-state aerodynamics resulted in 2·17 gf. This discrepancy (3%) is well within the accuracy of measurements and the error arising from predictable non-steady effects (Jensen, 1956). The same applies to the measured and calculated average horizontal thrust which is much smaller (0·10 gf and 0·14 gf respectively). We can therefore express 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 by means of two very fine white hairs placed normal to the long axis of the wing, one near the tip and the other further towards the fulcrum, where as distinct rear flap is present in the form of the stiff vannal area of the forewings (corresponding to the flexible fan-shaped vannal membrane of the hind wing). The hair was cut at the natural line of bending to permit free operation of the flap. Apart from the downward angular movements of the whole wing the downstroke is characterized by a nose-down twist of the wing, a pronation. This deformation is the result of active movements at the wing base and results in an almost linear twist θ with distance from the fulcrum. Furthermore, when the forewing approaches the horizontal position and the angular velocity has reached its maximum and begins to decline again, the vannal flap is automatically tilted downwards, altering the wing profile and increasing the lift coefficient (Jensen 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 the flap (nos. 14–16). This gives rise to the so-called Z-profile which offers little lift and drag (Jensen, 1956). These deformations are always present whenever the locust flies actively, but may of course become exaggerated in still air and under other adverse conditions, as seen in Fig. 18. In the latter case the large stroke angle and the air resistance which meets the wing of a struggling tethered animal can result in visible elastic bending, but this is not the case during normal forward flight (cf. Fig. 17 (14) and Fig. 18f).
In principle, the hind wings are moved in a similar way although the flexible vannus tends to obscure the basic similarity. In an extremely detailed study of the movements of the forewings of the migratory locust (Locusta migratoria) by means of stereo-photographs taken from above, Zamack (1972) has recently claimed that Jensen’s kinematic analysis is invalid. What he means is presumably that there were second-order effects not taken into account, such as various elastic bendings, and this is undoubltedly so. However, Zamack failed to demonstrate the much more important flap and the Z-profile, both of considerable aerodynamic importance, and my own conclusion is that his method cannot be as accurate as he himself claims since innumerable direct observations on three species of locusts (Schistocerca gregaria, Locusta migratoria and Noma-dacris septemfasciata), stroboscopic films from the side (Fig. 17) and flash photographs of Schistocerca gregaria (Fig. 18) never fail to show these features. There are next to no differences in morphology and flight between the three species.
The net result of the movements relative to the air is illustrated in Fig. 19 (reproduced from Jensen, 1956, by combining his figs. Ill, 6 and III, 8). The tracings refer to the tip and to the middle section of a forewing as seen on the unfolded elliptical cylinder, 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 carefully adjusted to the path. It should be noted that Jensen’s calculations are based on the true three-dimensional movements and that the mutual interference between the wings was taken into account in estimating the aerodynamic angles of attack. The diagram is presumably valid in principle for most types of fast forward flight. It is also reminiscent of a similar diagram in Fig. 20 for the two-winged fly Phormia regina flying forwards at 2·8 m sec−1 and analysed by Nachtigall (1966), the main differences being (1) a much higher rate of wing twist dθ/dt of about 6000 rad sec−1 as against maximally 500 rad sec−1 in Schistocerca, (2) high geometrical angles of attack towards the extremes of each half-stroke where the high twist rates also occur, and (3) a larger ratio of flapping speed to forward speed. These points are all relevant in the discussion of hovering flight but at this point it is obvious that the flight of these two very different insects is based essentially on the same type of movement, and that one can envisage a continuous evolution from the one to the other. This also means that the same individual can utilize basically identical mechanisms for fast and slow forward flight as well as for true hovering.
(b) ‘Delayed,’ elasticity
How does this influence flight? In the case of a locust with wing length 5 cm and wing-stroke frequency 20 sec−1, each stroke lasts 50 × 10−3 sec and the propagation time is only 2% of the period. The effect will be insignificant. In the fly Phormia regina the duration of one stroke is 8 × 10−3 sec, of which the twist phases at top and bottom last about 0·5 × 0−3 sec each. With a wing length of 1 cm the propagation time is at least 0·2 × 10−3 sec. The same applies to syrphid flies, and it means that if the wing twisting itself is instrumental in creating the aerodynamic flow upon which lift depends in these insects, i.e. if the flip is of importance, the action must be propagated from base to tip with a delay which is appreciable or even of the same magnitude as the duration of the pronation or supination. This may have an important and beneficial consequence for hovering flight in Syrphinae and other insects, as will be discussed later.
In the case of the small Chalcid wasp Encarsia formosa, the marginal veins are 0-035 cm long and the total wing length is 0·062 cm. The minimum propagation times are then 0·7 × 10−5 and 1·2 × 10−6 sec respectively. During the ‘flinging open’ phase, the two wings open by at least 6° in io-8 sec and probably by 0° on the average (see later). We therefore cannot neglect the propagation time of the torsional movement when dealing with the rapid changes at either end of the wing stroke, but it is unlikely to be of importance during the middle part.
It should be noted that in geometrically similar animals the relative importance of the propagation time is independent of size (t ∝ l0) when n ∝ l−1, as is usually but not always the case. The significance of this factor is therefore potentially the same it small and large flapping animals.
(c) A novel mechanism for lift production in Encarsia: the fling
The low Reynolds number makes high lift coefficients impossible and yet was found to be about 3. It should also be recalled that lift in excess of the body weight is produced early after the ‘flinging open’ phase seen in Figs. 14 and 15, hereafter called the fling, namely between frames 3 and 4 or considerably earlier than the attainment of corresponding maximum angular velocity by the horizontally beating wings. This is not consistent with ordinary aerofoil theory and practice. Could it be that the fling is a mechanism for creating circulation round the wings before the right and left pair separate and start to move in a horizontal plane during the morphological downstroke in Fig 21C? At first sight this seems improbable because what is needed is the setting up of a circulation in the sense indicated in Fig. 21 A, and the twisting movements (pronation) are of the opposite sense. We are therefore not dealing with an ordinary Magnus effect. However, the possibility exists that a transient circulation of the desired sense could be induced. Once established it would immediately produce lift normal to the direction of the horizontal translational velocity vt and in an upwards direction. It could then be reinforced to some extent during the stroke, but there would always be a tendency for it to approach the low value corresponding to the small steady-state lift coefficients at the relevant Reynolds number. The mechanism then depends 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 the rear edges which can be represented as a ‘hinge’ (Fig. 21 A) so that air must flow into the space created between the wings from in front. The opposite flow is prevented by the adjoining rear edges forming the ‘hinge’. Immediately after this fling, which corresponds to the pronation in normal flight, the wings on either side separate, each carrying a bound vortex with it. The question is, will this movement result in a circulation of the necessary strength T and the correct sense? It is a problem of flow relative to a solid body which suddenly starts to move (impulsive start) in a viscous incompressible fluid ; the resulting motion of the fluid must then be irrotational and without circulation. The point is that the body then breaks into two, each of which carries its own circulation of opposite sign but of equal strength.
Examination of the films show that the fling lasts about 3 × 10− 4 sec as a maximum.. This corresponds to a thickness of 65 μm. Although the hairs (20 μm long) may increase the effective thickness, the boundary layer should not constitute a serious obstacle.
At this stage I consulted Professor Sir James Lighthill, who most kindly discussed these ideas with me and wrote (io December 1972):
I have made calculation of the circulation at the end of the fling phase on the following basis: [cf. Fig. 21B], The angular velocity (radians per second) with which the two wings ap and aq are opening about the point a is taken as ω. Thus I first calculate the circulation around the contours shown [B 3], neglecting shedding of vorticity during the fling. This is a classical calculation for vortex-free (‘irrotational’) flow. The answer is exact: for the geometry shown the circulation is 0·69 ωc2 round both the pencilled contours.
If for the moment this value be acepted, then we may consider what happens when a small gap appears between the two wings. This should not make any significant difference to the flow because that flow has zero velocity at the point a on both sides. The circulation round the pencilled contours [thin lines in Fig. 21B] should be unchanged, therefore; and should remain 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 going to diminish the circulation around the contours shown in the previous figure [B3]. Since the calculation has to be made while the wings are still connected at a, however, those circulations are not around truly closed circuits and so we cannot say for certain whether they are diminished 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 if the vorticity 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] are diminished 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 remain at the end of the ‘fling’.
According to equation (45) and the values for R and vt used there, the flapping speed vt at which the total lift equals the body weight is then 93 cm sec-1. This speed is achieved at about frame 3 in Figs. 14 and 15, i.e. at the time when the films show that the body begins to gain height (cf. Table 8). There is therefore hardly any doubt that this novel method of creating circulation is sufficient and essential for making a small insect like Encarsia formosa airborne.
Details of Professor Lighthill’s analysis and calculations, which include some considerations 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 principle of which should remain true, the initial period of the fling is crucial as far as energy transfer is concerned and also with respect to Reynolds number (equation 44). We have already seen that the minimum time needed for a torsion of the anterior submarginal and marginal veins (cf. Fig. 13) to travel from the wing base to the distal end is 0·7 × 10−8 sec; and if the wing surface is made of a similar material it will take 1·2×10−8 sec to reach the wing tip. During the initial 10−4 sec the average value of w is 11 × 103 rad sec−1 so that 10−6 sec corresponds to an angular opening of 6°. During the initial and energetically most important phase there is therefore no doubt that the cleft 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 has reached the tip.
In this context it may well be that the marginal hairs have a specific function - that of preventing or delaying tip-vortex formation. It is instructive to note the difference between 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 larger and 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 such wings 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 from those of ordinary aerofoils, the actual shape of the wing is of little importance. What is needed is an elongated solid body which carries a bound vortex and which is moved relative to the stationary air.
The hind wings and the prominent hairs at their rear edge in Encarsia may improve the mechanism because it should be possible to fling open the forewings not as one opens a book, but by pulling the two parallel wing surfaces apart while the hind wings seal the rear entrance. This could be the case during the initial phase and should also make it possible for the animal to exert more control than is otherwise possible. In addition, the long hairs on the hindwings could counteract the unavoidable inflow of air at the proximal wing margin.
Other flying animals
It was noted that Drosophila virilis appears to have a clap and a fling 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 the basis of steady-state principles (Weis-Fogh, 1972), the usual mechanism does not leave any safety margin, and it appears highly likely that this and other small insects make use of the fling principle, thereby gaining extra lift and manoeuvrability. The cost involved is unknown at present.
As to butterflies (Lepidoptera, Rhopalocera) they take off with the wings often held in 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 too high (2·2) for wings of very low aspect ratio strongly indicates that this group of insects regularly utilize the fling method.
As to birds, it is easy to hear one or two distinct claps when a pigeon suddenly takes off from a perch when disturbed. Dr R. H. J. Brown has kindly shown me unpublished slow-motion films of flying pigeons from which it appears that at very low air speeds the wings touch each other at the top, when the sound is produced. They are then flung open in a way similar to the fling in Encarsia. It would be interesting to study this phenomenon in large insects also, and in other birds where it might assist flight particularly during take-off in emergencies. Could it explain some of Dr Wilkin’s results referred to on page 182 above?
(d) The flip mechanism in hover flies
Obviously the fling mechanism whereby two sets of wings start by forming one structure which is then suddenly deformed before the sets rapidly move apart cannot explain the flight of the true hover-flies, the Syrphinae. However, there are some similarities, and although the following account is much more speculative than the preceding one it is concluded that we are confronted with a special mechanism which I believe to be novel and which I shall call the flip mechanism.
Consider an ordinary insect wing which is stretched out from the body. During a period of twist at the extreme positions, be it pronation or supination, the leading edge is suddenly swung actively, nose-down during pronation and nose-up during supination. In many insects the posterior part of the wing is often flexible and does not follow the rapid movement of the front part at all closely. In Encarsia each effective wing 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 the flexible 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. 22A, B) are extremely interesting in this context and differ from other Dipteran wings in several respects. (1) The wing itself has a concentration of strong veins going from base to tip in the anterior part while the posterior part is poorly supplied; this is the ‘aderarme’ wing of Rohdendorf (1958/9). (2) There is a special reinforcement known only in the family Syrphidae, the so-called vena spuria (v.s.). It tends to reinforce the anterior almost triangular part and to provide linear reinforcement parallel to which the wing must bend during pronation and supination, separating the anterior, and the posterior parts funtionally. (3) I have observed that the cuticle of the posterior part is extremely thin, soft and pliable in the living insect, 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 contrasts sharply with the common observation that Hymenoptera, Lepidoptera and many 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 (aZ. in Fig. 22A), is unusually large and can be bent upwards or downwards at right angles to the remaining wing surface, presumably by means of a pleuro-axillary muscle. It is probably used for control and manoeuvres, particularly in gusty wind, but since it cannot be of importance for lift generation we shall not analyse it further.
As a first approximation the wing can be considered as a plate with a stiff anterior part of roughly triangular shape which can be twisted actively from the base and another posterior part which is pliable and is indicated by transverse hatching in Fig. 22 C. The wing length R in a typical hover-fly is 1 cm, and the propagation of a torsional movement from base to tip therefore lasts at least 2 × 10−4 sec. In fact, it is likely 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, at least in Odonata (R. Å. Norberg, 1972). At the typical frequency of 150 sec−1, the wing-stroke period is 6·7 × 10−3 sec of which 7% is used for twisting at either end (Nachtigall, 1966, in Phormia), or 5 × 10−4 sec for pronation and supination respectively. There must therefore be a significant delay between the onset of twisting at base and tip. Also, the soft posterior membrane has a much lower elastic modulus so that the deformation cannot reach this region until after the active twist is completed, i.e. 6–20 × 0−4 sec after the onset. The air at the hind margin is then at rest during the twisting phases, the margin representing a stagnation line.
On this basis I have attempted to illustrate the probable deformations in Fig. 22 C at the onset of pronation (1) when t = 0, (2) when the wave has reached the middle part of the wing at t = 10−4sec, and (3) when it arrives at the tip at about t = 2 × 10−4 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 Encarsia but a circulation in the form of a bound vortex cannot be established unless an opposite posterior vortex of opposite sense is created. It will be noted that at this stage, the tip is not moved so that any shedding of vorticity at the tip will be insignificant. The propagation of the twist therefore introduces economy. In (3) the entire wing has undergone twist and has acquired what corresponds to a bound vortex although some useful circulation must now be lost due to the unavoidable tip vortex. When the wing then begins to swing down, leaving the system of posterior vortices behind as if they were ordinary starting vortices, it has already acquired circulation of a strength unrelated to the flapping speed and may produce lift at much lower speeds than envisaged in a steady-state system. It should be mentioned that the vertical induced wind is too small to remove the trailing vortex significantly during the time occupied by the twisting movements.
In this mechanism the soft posterior membrane is essential for the initial distribution of the two vortices relative to the wing, the anterior one bound to the wing and the posterior one free. If the posterior part were stiff and the wing a simple torsion plate, the rear part would swing up as the leading part swings down, the system would be symmetrical about the neutral axis and no useful circulation could build up before the translation starts.’ Once the real wing is in movement the soft rear flap will also increase the downwash and prevent separation or stall.
Without quantitative proof I propose that this is the mechanism which enables Syrphinae and Odonata to obtain the high lift coefficients of 3 or more during hovering flight. The principle could be used by any animal with similar wing characteristics. It differs in principle from the fling mechanism in the way that the posterior vortex cannot be used and tends to counteract the useful bound vortex until the two have become separated in space, and the induced wind is too small (about 50 cm sec−1) to do this during the twisting phases. There is therefore a delayed effect in the aerodynamic system and the two vortices must become separated by translation in order to obtain full lift. The similarity between the fling and the flip is that the transient vortices are produced by the active wing twisting. Future observations and experiments are needed to verify this. At present we have none.
Let us now consider the entire wing stroke during hovering and, for the sake of simplicity, assume that pronation (p) and supination (s) in Fig. 23 produce similar effects at the top and bottom of the stroke. The lift L is always at right angles to the relative wind, the drag D is in the direction of the wind, and the sense of the circulation caused by the flip action is opposite to that of the original wing twist. The fly hovers with horizontal body as in (A). If the wings merely beat up and down in a vertical plane, there will be no net vertical force because the drag components will cancel each other. However, during both downstroke and upstroke there is a strong forwards horizontal 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 tip rotates anti-clockwise as in Fig. 23 F and so that the bound vortices die out during the vertical parts of the stroke. In Fig. 23 G I have indicated how the flip mechanism is probably combined with normal aerofoil action in hover-flies and dragonflies during hovering on the spot.
It should be noted that during truly non-steady periods the angle of attack is of little importance, and this may well be reflected in Nachtingall’s (1966) results from the blow-fly Phormia regina reproduced as Fig. 20, particularly when one examines the large angles towards each end of the stroke. Another point is that even small alterations in wing-tip curve would have quick and drastic effects, as is needed for the supreme control seen in hover-flies. Such changes are well known from Hollick’s (1940) studies on the fly Muscina tabulons and are found in most medium-sized Hymeno-ptera and Diptera. It would be extremely interesting to study free unimpeded flight of hover-flies by means of high-speed cinematography.
As far as I am aware, the solution deduced here for the flight of hover-flies (Syrphinae) is new. It does not involve refined adjustments of the angle of attack and requires basically two mechanisms to explain both the act of hovering and the extreme and rapid manoeuvres, namely that the basic flip mechanism dominates the aerodynamic system and that the animal exerts rapid control over the wing-tip path, as is the case in other Diptera (Hollick, 1940). This being fulfilled, the insect can do almost anything without altering the body axis, from hovering to a brisk forward dash, a steep climb or a fall. In a gust of wind the alula could be operated differentially on the two sides so that turning in the yawing plane could be added to the repertoire without interfering 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 indicate the interrelationships which have been established so far, or at least strongly suggested, and to indicate some problems which require a new theoretical or experimental approach.
A major conclusion from this study is that most insects perform normal 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 mathematical apparatus presented here provide a realistic approach both in qualitative and quantitative terms. However, one must also realize that any type of flapping flight also involves non-steady periods, particularly at the reversal points where active pronation and supination occur. The higher the forward speed and the lower the wing stroke frequency, the smaller is the required rate of wing twisting and the smaller the relative importance of non-steady phenomena, as in a fast flying pigeon, and the empirical observations indicate that when the wing loading exceeds 1 kgf/m2, the steady-state principles seem to prevail even in hovering animals. The approach illustrated in Fig. 5 and in Table 5 should then provide figures of the correct magnitude, although many modifications can be expected in the light of more detailed knowledge. One of the most urgent requirements is to obtain reliable steady-state lift/drag diagrams of real animal wings, particularly from insects, and measured at the right Reynolds numbers and 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 nonsteady principles for generating lift, the fling and the flip mechanisms, which appear to have become dominant in certain insects with small wing loadings. In both types circulation is set up as a consequence of the rapid twisting at the reversal points in the wing path and before the wing as a whole gains speed relative to the stationary air. The essential difference between the steady and non-steady phases is therefore not that flight depends on an aerodynamic cross-force, or lift, in the first case and on an entirely different mechanism in the other cases; under the conditions considered here, lift depends on the establishment of circulation in the form of a bound vortex round the moving wing. The difference is that a normal aerofoil or wing induces circulation and maintains it as a consequence of viscous shearing forces, particularly in the air passing the rear edge, whenever the wing is moved relative to the surrounding air. Once the circulation has built up and the initial phase is over (Wagner effect) the steady-state implies that the circulation is maintained by the energy dissipated continuously in the shear system. In the case of the non-steady fling and flip mechanisms the vortex patterns which lead to circulations are created prior to and independent of the translation of the wing through the air. These vortex patterns would of course be useless unless they became superimposed upon the subsequent translational movement; but, once established, the air speed of the wing can be reduced relative to what is needed in the steady-state case and for as long as the initial bound vortex remains of significant strength. After it has died out, circulation then has to be maintained by the conventional mechanism.
During slow forward flight and hovering the non-steady phases in birds, bats and insects occupy a significant part of the wing-stroke cycle and, as we have seen, they could be put to good use in some cases. Many new examples will undoubtedly come to light in the future. The way in which the aerodynamic bending moment varies with the positional angle in work diagrams like that in Fig. 5 will then be somewhat modified in that the curve ABC will become steeper at the ends and more flat towards the middle, even upwards concave in extreme cases. This will increase the inertial and elastic forces involved in starting a new half-stroke but it does not necessarily result in an increase of the enclosed area, i.e. in a larger aerodynamic power output, because the downwards momentum needed to hover remains the same, in accordance with the experimental results of Wood (1970, 1972). As we have seen, the elastic system of insect wings tends to minimize vortex shedding so that the efficiency may remain reasonably high. If, then, the pronation and supination are used directly to create useful circulation in addition to that obtained by normal aerofoil action, the average coefficient of lift as estimated from equation (9) will become reduced. The values in Table 5 are therefore likely to be too large, particularly at low wing loads, because this factor cannot be taken into account until we know more. Drosophila is a good example and has already been discussed (p. 217). What is needed now are new theoretical and experimental studies on non-steady flow situations as exemplified by the fling and fling mechanisms, but not necessarily confined to them.
It should be recalled that the dimensional analyses show that the fling and flip mechanisms are potentially useful in larger animals also and that they are not in conflict with the conventional mode. It is therefore realistic to proceed with a general dimensional analysis of flapping flight to be presented in another context (paper in preparation).
Wing inertia and elastic forces
The same applies to the bending moments caused by the acceleration of the wing mass and by the elastic deformations. So far, the details of the elastic system only seems to have been analysed in the desert locust Schistocerca gregaria (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 in general. However, it is necessary to consider the possible effects of non-steady aerodynamics in this context.
It will be remembered that the inertial torque is relatively insignificant in Drosophila (Weis-Fogh, 1972, and present study). The same applies to the true hover-flies. In typical Syrphinae the wings are relatively very fight (0·7% of body weight) and the stroke angles and frequencies are low. When the moment of inertia I was calculated from equation (27) and the ratio N from equation (23), it was found that the ratio between maximum inertial and aerodynamic bending moments is considerably smaller than unity. This means that in Drosophila and in the Syrphinae the phase lag 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 Syrphinae it is unlikely that the effective lever arm of the wings changes much with position 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 wings against both elastic and aerodynamic forces so that pronation and supination could provide the necessary lag as well as producing useful aerodynamic work. During the evolution of conventional into non-conventional types, or vice versa, there need therefore not have been any real change with respect to myofibrillar mechanism.
In Hymenoptera and Diptera the wing twisting is an automatic consequence of the deformation of the elastic thoracic box caused by the strong indirect wing muscles (Pringle, 1957, 1968). In Odonata, Lepidoptera and Coleoptera wing twisting is directly controlled by the basalar and subalar muscles which are well developed in all three orders. We have seen that the anisopterous dragonflies and the Lepidoptera Rhopalocera (butterflies) actually make use of non-steady principles in spite of their relatively large size. The fling phase during the start of flight in Venessa io has already been mentioned, but it may also be significant that Magnan (1934, fig. 174) observed an open wing-tip loop in Aeshna mixta very similar to the one indicated in Fig. 23 G as being optimal for hovering flight based on the flip mechanism. Apart from Hymenoptera the other very small flying insects belong to the Coleóptera and the Thysanoptera and in both orders the basalar muscles are well developed (Matsuda, 1970). There is hardly any doubt that further studies on the dynamics of the flight system in insects will 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 normally have to take the speed of propagation of elastic deformations into account when dealing with animal dynamics. However, this factor is important for the non-steady aerodynamics 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 system without further qualifications. This is obviously not true in species with synchronous or non-fibrillar muscles where the rhythm is determined ultimately by the central nervous system, although natural selection would tend to optimize inertial and elastic forces one way or the other. In species with myogenic or fibrillar wing muscles there is no direct nervous control of the fundamental frequency n, and one would expect that provided that the stiffness of the elastic system remains independent of n. At least, this relationship should hold in species with large inertial bending moments such as Eristalis and Calliphora (but not necessarily in Syrphinae). It is therefore disconcerting that Danzer (1956) found that n ∝ I−0·22 in Calliphora erythrocephala in experiments where the moment of inertia was reduced by shortening the wings. However, he calculated Im from the wrong assumption that the mass was distributed eUiptically. For some years Dr Machin in this Department has conducted class experiments with this species and confirmed the predicted relationship, whether Im was found by means of equation (30) or directly by weighing, a minor deviation being caused by the fact that the stiffness of the active flight muscles decreases with increasing frequency (Machin & Pringle, 1960). There is therefore good reason to assume that the elastic system is both necessary and efficient in insects whether powered by neurogenic or myogenic muscles.
This study is dedicated to the memory of my late wife, Hanne Weis-Fogh, née Heckscher, who helped, encouraged and inspired me throughout twenty-five years of 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 permission to quote his unpublished caclulations, but also for the open mind with which he received me. I thank Professor J. W. S. Pringle, F.R.S., for information on large beetles, Dr R. H. J. Brown for showing me unpublished slow-motion exposures of flying pigeons, Professor W. D. Biggs for information about wave propagation in solids, Dr Martin Jensen for permission to reproduce Fig. 19, Professor W. Nachtigall for permission to reproduce Fig. 20, Dr K. E. Machin for numerous discussions, and Dr J. Smart for expert advice on insect wings.
It would have been impossible to provide the slow-motion films and the sound recordings without the enthusiastic assistance of Messrs B. J. Fuller, G. G. Runnails and D. M. Unwin. I also thank Mr J. W. Rodford for drawing most of the figures and Mr R. T. Hughes for checking the references.
Main symbols and some constants and conversion factors between the technical gram force – centimetre – second system (gf – cm – s) used here and the SI system of measurement.
* Specific, i.e. per unit weight.