The local circulation method was applied to the free forward flight of the damselfly Ceriagrion melanurum Selys. The kinematic data used in the calculations were obtained by analyzing video-taped images of damselflies in free flight in a transparent container. The inclination of the stroke plane was smaller and the flapping amplitude was larger than those of dragonflies reported in other studies on odonate flight. However, the phase shift between the fore- and hindwings agreed with none of the previously reported patterns for damselflies: the forewings lead the hindwings by approximately a quarter-period. The calculated forces were within the expected range of error. The muscle-mass-specific power was between 40 and 80W kg−1. The vorticity distribution of trailing and shed vortices in the wake was also analyzed. Strong trailing vortices were observed at the wing tips, whereas shed vortices were concentrated near the wing root as the stroke switched direction.

The damselflies and dragonflies are the two major suborders of the extant Odonata. They have various morphological differences, of which those in the wings and thorax are the most interesting from an aerodynamic point of view. The fore- and hindwings of the damselfly are almost identical, whereas dragonfly hindwings are wider near the wing root than are their forewings. The pterothorax of the dragonfly is larger than that of the damselfly, and the angle between a line connecting the joints of the fore- and hindwings and the longitudinal body axis is generally large in damselflies, while this angle tends to be much smaller in dragonflies, especially in strong fliers (Fig. 1). These differences imply that there may also be differences in their flight mechanics. It is known that the damselfly is a less active flier than the dragonfly; however, its flight has not been studied as intensively as that of dragonflies.

Fig. 1.

Side views of a typical dragonfly (A) and damselfly (B) with the wings folded. Note that they are not shown to the same scale. (A) Anax parthenope julius Brauer, (B) Ceriagrion melanurum Selys.

Fig. 1.

Side views of a typical dragonfly (A) and damselfly (B) with the wings folded. Note that they are not shown to the same scale. (A) Anax parthenope julius Brauer, (B) Ceriagrion melanurum Selys.

The wing motion of Calopteryx splendens was studied by Rudolph (1976a,b), who reported that this species can perform a ‘clap and fling’, one of the novel mechanisms to generate lift proposed by Weis-Fogh (1973, 1975). A description of the flight of this species can also be found in Rüppell (1985). He analyzed films of Calopteryx splendens in free flight and concluded that this insect manoeuvres by changing the angle of inclination and the direction of the wingbeat. Kinematic parameters for several species of damselfly can be found in Rüppell (1989), but there are few descriptions available of the aerodynamics of damselfly flight. Marden (1987) measured the maximum lift production during take-off for various insects and birds by attaching increasingly heavy weights to them and found that the muscle-mass-specific lift of damselflies tended to be much higher than that of dragonflies. Ellington (1991) calculated the induced power requirements of these animals on the basis of Marden’s (1987) data.

A greater number of studies have been conducted on dragonfly flight. Weis-Fogh (1973), Norberg (1975) and Ellington (1984a) calculated the mean lift coefficient required during hovering flight using classical methods from rotary wing aerodynamics. As the calculated lift coefficient exceeded the range of lift obtained in steady-state flow, they concluded that dragonflies depend on novel unsteady mechanisms to augment lift. However, Azuma et al. (1985) and Azuma and Watanabe (1988) showed that dragonfly flight could be explained by steady-state aerodynamics if variations in the induced velocity with spanwise position and time were accurately evaluated. They used the local circulation method (LCM) to calculate aerodynamic forces and moments, a method that takes variations in the wake and interference between the wings into account by introducing ‘attenuation coefficients’ and is therefore based on a more realistic wake model than previous approaches. Two novel lift-generating mechanisms, the ‘clap and fling’ and the ‘flip’ were proposed by Weis-Fogh (1973, 1975). However, analysis of their wing kinematics has shown that dragonflies use neither mechanism.

An alternative mechanism, related to wing rotation between the downstroke and the upstroke, has been the subject of more recent work (Savage et al. 1979; Ellington, 1984c; Dickinson, 1994). Savage et al. (1979) estimated the aerodynamic forces during wing rotation from the visualized flow structure around a model wing, while Dickinson (1994) measured the instantaneous aerodynamic forces on the model directly. They concluded that this mechanism could be important in the production of aerodynamic forces.

In the present study, kinematic parameters were obtained from video tapes of damselflies in free flight in the laboratory, since the available data (Rüppell, 1985, 1989) were neither complete nor accurate enough for calculating aerodynamic forces and moments accurately. Aerodynamic forces and moments were then calculated using the LCM to give a detailed picture of the aerodynamics of damselfly flight. Differences between the flight of the dragonfly and the damselfly are discussed.

Insects

Male damselflies, Ceriagrion melanurum Selys, were collected near a marsh in Shizuoka prefecture, Japan. Each damselfly was put into a separate paper envelope and transported to our laboratory, where it was placed in a cage (12.5 cm×18 cm×7 cm). Video-taping was performed on the day of capture.

Video-taping

A container constructed from transparent acrylic sheets was situated in a temperature-controlled laboratory (22 °C). The container was 15 cm×20 cm×45 cm (width × height × length) and was open at both ends. The damselfly flew along the length of the container towards a light situated at its far end. The flight was recorded using a high-speed video camera situated 67.5 cm from the long axis of the container and exactly half-way down its length. From the available literature on the boundary effect (Heyson, 1960, 1961, 1971; Heyson and Grunwald, 1966; Rayner and Thomas, 1991), the reduction in the vertical component of the induced velocity caused by the proximity of the container walls will be approximately 1 % for flight at the lowest speed recorded in the present experiments; such effects were therefore considered to be negligible.

Two thin stripes (0.2 mm wide) were painted across the width of the fore- and hindwings at approximately 75 % of the wing length from the wing joint (see Fig. 2). This allowed us to measure the feathering angle of the wing section (see below). The mass of the painted stripes (0.0051 mg) was 1.6 % of the mass of one wing (0.31 mg), and the moment of inertia about the wing joint was therefore increased by 2.2 %. This effect is less than those produced during previous manipulations on dragonflies (Azuma et al. 1985).

Fig. 2.

Side view of a damselfly showing the positions of the 14 digitized points (filled circles). The positions of the two thin stripes painted on the fore- and hindwing (at approximately 75 % of the wing length from the wing joint) nearest the camera are also shown.

Fig. 2.

Side view of a damselfly showing the positions of the 14 digitized points (filled circles). The positions of the two thin stripes painted on the fore- and hindwing (at approximately 75 % of the wing length from the wing joint) nearest the camera are also shown.

A high-speed video camera (Kodak Ektapro Motion Analyzer HS4540) with a 50 mm television lens at f=1.3 was used for all video recordings. The field of view covered the central region of the flight container, and the camera was operated at a speed of 1125 frames s−1. An area of approximately 139 mm×139 mm was recorded in the vertical plane containing the longitudinal axis of the container (object plane). For calibration of length in the recorded images, the back wall of the flight container was covered by a sheet of paper bearing 50 mm×50 mm grid lines. All recorded flights were examined separately by several members of our team, and flight sequences judged to be parallel to the image plane were noted. These sequences were repeatedly checked visually for the quality of the flight path. Flights not parallel to the image plane are distinguishable by apparent changes in size or in body orientation of the insects as they fly past the camera; any such sequences were rejected from further analysis. Body lengths were calculated from the points digitized from the video films during analysis (see below) and were compared with the expected values calculated for flights in the object plane. If the difference between these values exceeded 5 %, those sequences were rejected. Thus, we are confident that only flight sequences which were parallel with the longitudinal axis of the container were selected for further analysis.

Analysis of wing motion and morphology

The video system recorded each frame as 256 pixels × 256 pixels × 8 bits. For the kinematic analysis, an image-processing/analysis system was developed in the programming language C on an NEC PC-98 computer. The coordinates of 14 points on the damselfly (Fig. 2) were digitized from each frame. Because the flights were considered to be in the object plane (see above), digitized lengths (in image coordinates) can be converted to actual values with errors of only 1 % using perspective geometry and the background grid pattern. The resolution of the image was 0.54 mm in the object plane, which corresponds to 3.7 % of the wing length (semi-span). In all other analyses, orthographic projection was assumed (Ellington, 1984b).

The wing motion can be represented by the flapping of the axis connecting the wing root to the tip and the feathering motion around this axis (the feathering axis). As the flapping motion is almost completely confined to a conical region whose apex is located at the wing root (joint), it can be described by the inclination of the stroke plane (making up the base of the cone), the coning angle and the azimuth angle (Fig. 3). The azimuth angle is the angle between the projection of the feathering axis of the wing on the stroke plane and the intersection of the horizontal plane and the stroke plane. The calculation of these angles from the digitized points was relatively simple by trigonometry because the flights analysed did not involve rolling, yawing or banking. The derivation of the equations used is therefore omitted from this paper.

Fig. 3.

Definitions of the kinematic parameters. (A) Side view. The damselfly body is oriented at angle Θ to the horizontal and the flight path is at angle Ε to the horizontal. The flapping motion of each wing is confined to a conical region (shaded) whose apex is located at the wing root. The base of the cone is in the same plane as the stroke plane angle ε. βf and βh are the coning angles of the fore-and hindwings; εf and εh are the stroke plane angles of the fore- and hindwings. (B) Stroke plane and wing motion projected onto it (viewed from the direction normal to the stroke plane, view A). ψ is azimuth angle. ψ0 and ψ1 are the steady-state and the first harmonic components of flapping motion.

Fig. 3.

Definitions of the kinematic parameters. (A) Side view. The damselfly body is oriented at angle Θ to the horizontal and the flight path is at angle Ε to the horizontal. The flapping motion of each wing is confined to a conical region (shaded) whose apex is located at the wing root. The base of the cone is in the same plane as the stroke plane angle ε. βf and βh are the coning angles of the fore-and hindwings; εf and εh are the stroke plane angles of the fore- and hindwings. (B) Stroke plane and wing motion projected onto it (viewed from the direction normal to the stroke plane, view A). ψ is azimuth angle. ψ0 and ψ1 are the steady-state and the first harmonic components of flapping motion.

The feathering motion is described by the feathering angle, which is the angle between the chord of a wing section and the stroke plane in the plane perpendicular to the feathering axis. The feathering angles were estimated by the following method. The feathering angle (θ) is related to the angle χ between a painted stripe and the horizontal axis in an image such that:
formula
where a11=−cosεsinψsinβ+sinεcosβ, a12=cosεcosψ, a21= sinεsinψsinβ+cosεcosβ, a22=−sinεcosψ, ε is the inclination of the stroke plane, ψ is the azimuth angle in the plane parallel to the stroke plane and β is the coning angle of the cone whose base is parallel to the stroke plane and whose surface contains the flapping axis. The angles ε, ψ and β are determined from digitized points other than those on the painted stripe (see Fig. 2); consequently, the coefficients a11, a12, a21 and a22 are independent of θ, allowing θ to be obtained by solving equation 1. As can be seen from equation 1, θ cannot be determined when ψ is ±π/2 rad. The maximum error expected using this method is 25 % according to our calculations.

The morphological measurements required in the analysis (see Table 1) were obtained after filming was completed.

Table 1.

Morphological characteristics of the damselflies Ceriagrion melanurum used in this study

Morphological characteristics of the damselflies Ceriagrion melanurum used in this study
Morphological characteristics of the damselflies Ceriagrion melanurum used in this study

Kinematics

The morphological characteristics of the four damselflies for which flights were observed that met the selection criteria are shown in Table 1. Kinematic variables for these flights are summarized in Table 2.

Table 2.

Kinematic variables measured for flights analysed from four damselflies, Ceriagrion melanurum

Kinematic variables measured for flights analysed from four damselflies, Ceriagrion melanurum
Kinematic variables measured for flights analysed from four damselflies, Ceriagrion melanurum

The wing span and body mass of the damselfly C. melanurum are approximately 40 % and 5 %, respectively, of the values of these variables in the dragonfly Anax parthenape julius (Azuma and Watanabe, 1988). Wing loading [mg/(Sf+Sh)], where m is body mass, g is the acceleration due to gravity and Sf and Sh are the areas of the fore- and hindwings, is approximately 60 % and the aspect ratio (b2/S, where b is wing span) of the fore- or hindwings is approximately 150 % of these values in the dragonfly. The flapping amplitude of the wings is relatively large, whereas the inclination of the stroke planes is smaller than that of the dragonfly. Damselflies beat their wings at almost the same frequency as dragonflies, and wingbeat frequency seems to be independent of forward velocity, as for dragonfly flight. These kinematic characteristics agree with previous results for damselflies (Rüppell, 1985, 1989); however, we also found a novel phase shift between the fore- and hindwings. In the present study, the forewing leads the hindwing by approximately a quarter-period, while Rüppell reported both parallel stroking (0 ° phase shift) and counter-stroking (180 ° phase shift). For dragonfly flight, Azuma and Watanabe (1988) found that the hindwing led the forewing by approximately a quarter-period.

Examples of the paths traced by the tips of the fore- and hindwings during an entire wingbeat cycle are shown in Fig. 4 for damselfly C flying at 0.71 m s−1. The conical regions along which the flapping motions are performed have smaller coning angles than those of a dragonfly. The conical region is open upwards for the forewing and downwards for the hindwing, because the stroke plane of the forewing is located above the wing root, while that of the hindwing is below the wing root. The stroke plane angle of the hindwing is larger than that of the forewing.

Fig. 4.

Positions of the wing tips during a single wingbeat. The digitized points are from separate video frames. The stroke plane angles ε are also shown. (A) Forewing; (B) hindwing. Data are for damselfly C. V=0.71 m s−1. See Table 2 for further details of this sequence. The conical region traced by the wings during the wingbeat is shown shaded (see also Fig. 3).

Fig. 4.

Positions of the wing tips during a single wingbeat. The digitized points are from separate video frames. The stroke plane angles ε are also shown. (A) Forewing; (B) hindwing. Data are for damselfly C. V=0.71 m s−1. See Table 2 for further details of this sequence. The conical region traced by the wings during the wingbeat is shown shaded (see also Fig. 3).

The azimuth angles ψ in the stroke plane and the feathering angles θ with respect to the stroke plane are shown in Fig. 5A,B. Table 3 shows the Fourier coefficients of these angles defined by:
formula
formula
for both wings, where k is the kth harmonic component, ω is the angular velocity of the beating motion (2πf) and t is time. The first harmonic is predominant in the flapping motion, whereas higher harmonics are included in the feathering motion.
Table 3.

Fourier coefficients calculated for the azimuth angle and feathering angle with respect to the stroke plane for the forewing and hindwing of four damselflies (A–D)

Fourier coefficients calculated for the azimuth angle and feathering angle with respect to the stroke plane for the forewing and hindwing of four damselflies (A–D)
Fourier coefficients calculated for the azimuth angle and feathering angle with respect to the stroke plane for the forewing and hindwing of four damselflies (A–D)
Fig. 5.

Azimuth angles (A) and feathering angles (B) of damselfly C shown with respect to the stroke plane for a complete wingbeat cycle. The points are plotted relative to an entire wingbeat period T=29 ms. V=0.71 m s−1.

Fig. 5.

Azimuth angles (A) and feathering angles (B) of damselfly C shown with respect to the stroke plane for a complete wingbeat cycle. The points are plotted relative to an entire wingbeat period T=29 ms. V=0.71 m s−1.

Although previous reports suggest that some damselflies perform the ‘clap and fling’ (Rudolph, 1976a,b), this was not observed during the present experiments.

Aerodynamic analysis of flight performance

In most previous studies on the flight performance of insects (Weis-Fogh, 1973; Norberg, 1975; Ellington, 1984a), the lift coefficient and induced velocity are assumed implicitly to be constant along the span and azimuthal position of the wing. These assumptions do not result in significant errors at high speeds when the variance in the inflow (the resultant flow incident on a wing element or thin wing section) velocity is not large. However, for insects flying at low speeds, analyses based on these assumptions could lead to substantial errors.

In the present study, the aerodynamic forces and the hinge moment or torque acting on the wings are calculated using the LCM (Azuma et al. 1985) because it is not necessary to make assumptions regarding the lift coefficient and induced velocity when using this method. The LCM calculates the distribution of induced velocity and aerodynamic forces as a function of time and spanwise position. The effect of unsteadiness is incorporated into the lift curve and the wake model, which consists of periodically changing shed and trailing vortices, used to calculate the attenuation coefficient. We consider the unsteady effects of wing rotation simply by introducing a reduction in the lift slope caused by the Theodorsen function (Theodorsen, 1934) and a dynamic stall accompanied by periodic changes in the translational velocity and feathering angle (see below). Thus, the results of Dickinson (1994) on the effects of wing rotation are not directly applicable to the specific conditions of the present study.

The flapping and feathering motions of the wings were modelled using the coefficients from the Fourier series up to the fourth harmonic shown in Table 3. The first harmonic of the feathering angle, θ1 is assumed to have a linear twist θt≅0.2 rad along the non-dimensional spanwise position η=y/(b/2), where y is the spanwise position along the feathering axis of the wing and b is wing span, as follows:
formula

Lift

Fig. 6 shows the predicted aerofoil characteristics of a damselfly wing. These curves were obtained by modifying values from dragonfly wings (Okamoto et al. 1996), taking into account the difference in Reynolds number Re (dragonfly, 3×103, damselfly, 103). Although the three-dimensional shape of the wing differs considerably between these two suborders, the difference in aerofoil (two-dimensional wing or the wing section with a plane perpendicular to the feathering axis) shape is not large; therefore, the aerofoil characteristics of a dragonfly can be used to predict those of a damselfly. Because of the periodic changes in flapping velocity and feathering angle, dynamic stall (Ham, 1968) was expected to occur. Therefore, the lift curve in Fig. 6 was modified to incorporate the effects of dynamic stall to give delayed stall at high angles of attack and an increase in the maximum lift coefficient. The increase in the maximum lift coefficient was estimated to be approximately 20 % in previous studies (Carr et al. 1977; Francis and Keesee, 1985); therefore, we delayed the stall until the lift coefficient was 120 % of the maximum predicted steady-state lift coefficient.

Fig. 6.

Predicted aerofoil characteristics of a damselfly wing. Values are derived from those measured for dragonfly wings (Okamato et al. 1996) corrected for the difference in Reynolds number. Cl, lift coefficient; Cd, drag coefficient. See text for further details.

Fig. 6.

Predicted aerofoil characteristics of a damselfly wing. Values are derived from those measured for dragonfly wings (Okamato et al. 1996) corrected for the difference in Reynolds number. Cl, lift coefficient; Cd, drag coefficient. See text for further details.

Fig. 7 shows the variation in the angle of attack α with spanwise and azimuthal position of the wings calculated using the LCM. The angle of attack is defined as the angle between the chord of a wing section and the resultant flow incident on it; therefore, it varies with the spanwise and azimuthal positionof the wings (for details, see Azuma and Watanabe, 1988). The damselfly uses an angle of attack below the stall (i.e. below 15 °, see Fig. 6) for approximately 70 % of the area covered by the wing motion during the downstroke; during the upstroke, stalling occurs over a larger proportion of the covered area. It is clear that the angle of attack is far from constant during the wingbeat and along the wingspan. Therefore, the assumption of a constant lift coefficient by some aerodynamic models is not realistic.

Fig. 7.

Changes in the angle of attack α during a wingbeat cycle calculated using the local circulation method. All values are given in degrees.

Fig. 7.

Changes in the angle of attack α during a wingbeat cycle calculated using the local circulation method. All values are given in degrees.

Fig. 8A,B shows the variation in the calculated aerodynamicforces with time for damselfly C flying at a speed of 0.71 m s−1. It is clear (1) that the hindwing is responsible for generating much of the vertical force, avoiding any unfavourable effects caused by the forewing wake due to the phase shift between the fore- and hindwings, (2) that the vertical force is almost always positive (upwards) and the greater part of this force is generated by both wings during their respective downstrokes, (3) that most of the positive horizontal force (forward thrust) is generated during the upstroke, whereas negative horizontal force (backward thrust) is generated during the downstroke, and (4) that the mean vertical force represents 93 % of the body weight for damselfly C, and 107, 84 and 80 % for damselflies A, B and D, respectively. A value of 100 % should be obtained if the flights are perfectly horizontal and completely trimmed; however, we did not impose this condition on the calculation process. The forces were calculated using the measured kinematic variables, so that the accuracy of these calculated forces depends directly on that of the kinematic data. Considering that the kinematic data were obtained by analyzing video-taped images, deviations of this magnitude in the calculated vertical force from the expected values (equal to body weight) can be attributed to measurement errors in the kinematic data.

Fig. 8.

Calculated vertical (A) and horizontal (B) aerodynamic forces produced by the wings of a damselfly. Forces were calculated using the measured kinematic variables given in Table 2 and the local circulation method (see text for further details). Values are for damselfly C flying at 0.71 m s−1. The points are plotted relative to a wingbeat period T=29 ms. A cubic spline function was used for curve fitting. Mean on the right-hand side shows the time average of the aerodynamic forces produced by fore- and hindwings for a wingbeat period.

Fig. 8.

Calculated vertical (A) and horizontal (B) aerodynamic forces produced by the wings of a damselfly. Forces were calculated using the measured kinematic variables given in Table 2 and the local circulation method (see text for further details). Values are for damselfly C flying at 0.71 m s−1. The points are plotted relative to a wingbeat period T=29 ms. A cubic spline function was used for curve fitting. Mean on the right-hand side shows the time average of the aerodynamic forces produced by fore- and hindwings for a wingbeat period.

Comparison of Fig. 5 with Fig. 8 suggests that the forward (positive) thrust component is related to the generation of drag by the large feathering angle, rather like rowing a boat using a pair of oars. This is illustrated more clearly by Fig. 9. In the dragonfly (Fig. 9C,D; data from Azuma and Watanabe, 1988), the drag force never makes a contribution to the forward thrust as the drag component is always negative, whereas it reaches positive values during the second half of the upstroke in damselfly (Fig. 9A,B). This may be related to the fact that the stroke plane angle of the damselfly is much smaller than that of dragonfly (typical values for the dragonfly are 40–70 °; Azuma and Watanabe, 1988). We consider that the difference in the stroke plane angle between these two suborders is probably related to their different thoracic morphology (see Fig. 1).

Fig. 9.

Lift and drag components of the horizontal aerodynamic forces produced by a damselfly (A,B) or a dragonfly (C,D). Data for the dragonfly are from experiment 4 of Azuma and Watanabe (1988). The damselfly is the same as that shown in Fig. 8. The local circulation method was used for the calculation (see Azuma and Watanabe; 1988; Azuma, 1992, for further details). (A,C) Forewing; (B,D) hindwing. The data points are plotted relative to a wingbeat period T=29 ms for the damselfly and 36 ms for the dragonfly. A cubic spline function was used for curve fitting.

Fig. 9.

Lift and drag components of the horizontal aerodynamic forces produced by a damselfly (A,B) or a dragonfly (C,D). Data for the dragonfly are from experiment 4 of Azuma and Watanabe (1988). The damselfly is the same as that shown in Fig. 8. The local circulation method was used for the calculation (see Azuma and Watanabe; 1988; Azuma, 1992, for further details). (A,C) Forewing; (B,D) hindwing. The data points are plotted relative to a wingbeat period T=29 ms for the damselfly and 36 ms for the dragonfly. A cubic spline function was used for curve fitting.

Torque and power

The aerodynamic torque acting on the wings is shown in Fig. 10. Peaks in torque occur near the middle of the stroke, at which point the dynamic pressure is at its maximum value in both wings. Although the inertial component of torque could be quite large owing to the large moment of inertia of the wing itself and the added mass of the air, it is ignored here because (1) the observed motion of the wing tip, especially where the stroke reverses its direction, is not accurate enough to allow calculation of inertial torque, and (2) the inertial component will be offset by an elastic element presumably found in the driving system or in the muscles themselves (Azuma, 1992).

Fig. 10.

Aerodynamic torque calculated for damselfly C flying at a speed of 0.71 m s−1. The data points are plotted relative to a wingbeat period T=29 ms. A cubic spline function was used for curve fitting.

Fig. 10.

Aerodynamic torque calculated for damselfly C flying at a speed of 0.71 m s−1. The data points are plotted relative to a wingbeat period T=29 ms. A cubic spline function was used for curve fitting.

The power required for flight can be calculated using the LCM and the kinematic data given in Table 2. The calculated muscle-mass-specific power is shown in Fig. 11. The ratio of muscle mass to body mass was assumed to be 0.45 (May, 1991; Marden, 1987). Although these are steady trimmed flights, they cannot be assumed to be horizontal because the load factor, n=L/W (lift/weight), did not exactly equal 1.0, and the calculated power should therefore be corrected. The corrected values in Fig. 11 are obtained by subtracting the climbing power (=WV/sinΕ, the power required for climbing, where Ε is the flight path angle) from the calculated power before dividing by the muscle mass. The regression line passing through these points will give the power requirements for horizontal flight. The muscle-mass-specific power varied between 40 and 80 W kg−1, which is within the range obtained experimentally for Zygoptera during take-off (Ellington, 1991; Marden, 1987), but lower than a recent estimate of the maximum muscle-mass-specific power output of the dragonfly (100 W kg−1; May, 1991).

Fig. 11.

Power requirements for flight in the damselfly Ceriagrion melanurum. The power requirements were calculated from the measured kinematic variables given in Table 2 using the local circulation method (squares). Because the calculated lift did not equal the body weight for these flights, the calculated values were corrected by subtracting the climbing power (crosses). The regression line shown has the form Pn=−87.5V+111.

Fig. 11.

Power requirements for flight in the damselfly Ceriagrion melanurum. The power requirements were calculated from the measured kinematic variables given in Table 2 using the local circulation method (squares). Because the calculated lift did not equal the body weight for these flights, the calculated values were corrected by subtracting the climbing power (crosses). The regression line shown has the form Pn=−87.5V+111.

Circulation and wake vortices

The circulation distribution Γ(η) along the wingspan can be calculated from the spanwise airloading λ(η) (lift per unit span, calculated using the LCM) and the velocity of the resultant flow (U) at that position. Since U is represented by flight speed V, the beating speed of the wing and the induced velocity v, Γ(η) is given as follows:
formula

where ρ is air density, ε is stroke plane angle and β is coning angle. An example of the calculated circulation distribution is shown in Fig. 12A for the forewing of damselfly C. During the downstroke, the circulation is positive at all spanwise positions and is maximal near the spanwise position η=0.8, whereas during the upstroke the circulation is mostly negative to generate positive airloading (consequently, positive vertical force, see Fig. 8A) in the inverse flow region where the flow is incident from the trailing edge.

Fig. 12.

The calculated circulation (Γ) distribution of a damselfly wing. (A) Effects of time and spanwise position on the circulation distribution around the right forewing of damselfly C. (B,C) Iso-circulation contours provided by Dr A. Tan. (B) Right forewing; (C) right hindwing. T, wingbeat period; t, time. Contour colour represents circulation strength.

Fig. 12.

The calculated circulation (Γ) distribution of a damselfly wing. (A) Effects of time and spanwise position on the circulation distribution around the right forewing of damselfly C. (B,C) Iso-circulation contours provided by Dr A. Tan. (B) Right forewing; (C) right hindwing. T, wingbeat period; t, time. Contour colour represents circulation strength.

Iso-circulation contours of the right fore- and hindwing are shown in Fig. 12B,C for frozen wakes in which the vortices are fixed in position as the wings move past. The difference between two adjacent contours, ΔΓ, is produced by the trailing vortex γx=∂Γ/∂y and the shed vortex γt=∂Γ/∂t=(∂Γ/∂x)U as:
formula
Therefore, the higher the contour density, the stronger the wake vortices. The colour of a contour represents the strength of the circulation. During the downstroke of the wing, the shed vortex changes from negative to positive values, whereas it changes from positive to negative during the upstroke. Strong or peak values of trailing vortices are observed at the wing tip and shed vortices are concentrated where the stroke changes direction.
     
  • a11, a12,

    coefficients used in the calculation of

  •  
  • a21, a22,

    feathering angle

  •  
  • aspect ratio (=b2/S)

  •  
  • b

    wing span

  •  
  • Cd

    drag coefficient of a two-dimensional wing or aerofoil

  •  
  • Cl

    lift coefficient of a two-dimensional wing or aerofoil

  •  
  • FH

    horizontal force, positive forward

  •  
  • FV

    vertical force, positive upward

  •  
  • f

    wingbeat frequency

  •  
  • g

    acceleration due to gravity

  •  
  • k kth

    harmonic component

  •  
  • L

    lift (mean vertical force)

  •  
  • l

    body length

  •  
  • m

    body mass

  •  
  • n

    load factor (=L/W)

  •  
  • Pn

    muscle-mass-specific power required for flight

  •  
  • Q

    torque about a joint

  •  
  • Re

    Reynolds number

  •  
  • S

    wing area

  •  
  • St

    total wing area

  •  
  • T

    wingbeat period

  •  
  • t

    time

  •  
  • U

    velocity of resultant flow (vector sum of flight speed, beating velocity and induced velocity)

  •  
  • V

    flight speed

  •  
  • W

    weight of body (=mg)

  •  
  • v

    induced velocity

  •  
  • x

    chordwise position

  •  
  • xj

    distance between the roots (joints) of the forewing and hindwing

  •  
  • y

    spanwise position along the feathering axis of the wing

  •  
  • α

    angle of attack of a wing section

  •  
  • β

    coning angle of the conical region in which the wing moves

  •  
  • χ

    angle between the painted stripe on the wing and the horizontal axis in an image

  •  
  • δh− δf

    phase shift of flapping motion between hind- and forewings

  •  
  • δ − ϕ

    phase shift between feathering motion and flapping motion

  •  
  • δk

    phase shift in kth harmonic component of azimuth angle

  •  
  • Ε

    flight path angle

  •  
  • ε

    stroke plane angle

  •  
  • ϕk

    phase shift in kth harmonic component of feathering angle

  •  
  • Γ

    circulation

  •  
  • γx

    trailing vortex

  •  
  • γt

    shed vortex

  •  
  • λ

    spanwise airloading

  •  
  • Θ

    body angle

  •  
  • θ

    feathering angle

  •  
  • θ0

    steady-state component of feathering angle

  •  
  • θ1

    first harmonic component of feathering angle

  •  
  • θt

    linear twist along non-dimensional spanwise position of wing

  •  
  • θ0.75

    wing twist at η=0.75

  •  
  • ρ

    air density

  •  
  • ω

    angular velocity of beating motion

  •  
  • ψ

    azimuth angle of flapping wing

  •  
  • ψ0

    steady-state component of azimuth angle

  •  
  • ψ1

    first harmonic component of azimuth angle

  •  
  • ψ

    time derivative of azimuth angle

  •  
  • η

    non-dimensional spanwise position, η=y/(b/2)

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