## ABSTRACT

The dragonfly, *Anaxparthenope julius* (Brauer) was observed in free flight, and a theoretical analysis of flight performance at various speeds was carried out. The variation with time of forces and moments acting on wings and body in steady trimmed flight was calculated by the local circulation method. Measures of flight performance, such as top speed, cruising speed and maximum endurance speed, were estimated from a necessary power curve required in steady flight and from the estimated available power. The results show that without using any novel unsteady aerodynamic force generated by a separated flow over the wings, the dragonfly can make steady trimmed flight at various flight speeds, from hovering to top speed.

## INTRODUCTION

Application of the local circulation method (LCM) (Azuma *et al*. 1985) and a numerical computer simulation shows that a dragonfly, *Sympetrum frequens*, can make steady climbing flight without using an abnormally large lift coefficient (Norberg, 1975) or relying totally on unsteady aerodynamic forces (Savage *et al*. 1979). This result was obtained by considering the non-uniform and unsteady induced velocity distribution over a pair of stroke planes.

The induced velocity can be precisely calculated from the Biot–Savart law which is known in electromagnetic theory and is used to determine the induced velocity distribution generated by wake vortices (see any book of hydrodynamics such as Newman, 1977).

In the LCM, the timewise change of the induced velocity at any point in the stroke planes is given by multiplying the attenuation coefficient by the induced velocity generated by any preceding wing at the time that wing passed through that point. In Azuma *et al*. (1985), the effect of the trailing vortices on the derivation of the attenuation coefficient was introduced, but the effect of the shed vortices was neglected because the reduced frequency, which is a measure of the unsteadiness of the potential flow in the linear range of analysis and is given by k = *π*cf/ U = 1/AR*ψ*_{1} in hovering flight, is very small relative to the large-aspect-ratio wing of dragonflies (see the next section for all definitions).

Two novel mechanisms for lift generation in insects were described by Weis-Fogh (1973,1975). The first one, called ‘clap and fling’, utilized the separated flow around a pair of low-aspect-ratio wings (Lighthill, 1973, 1975; Maxworthy, 1979; Edward & Cheng, 1982). The second one, called ‘flip’, also utilized a pair of vortices, a bound vortex and a shed vortex of opposite sense, generated by a rapid pronation of the anterior portion of the respective large-aspect-ratio wings.

Savage *et al*. (1979) studied the role of the vortices generated by a twodimensional wing motion and unsteady effects on the lift generation and revealed that the lift is developed during a ‘pause’ in the downstroke preceding the supination and during the supination. Somps & Luttges (1985) made an experimental test to demonstrate the effect of unsteady separated flows and concluded that large lift forces are actually produced by unsteady flow–wing interaction. However, we do not agree with this interpretation because of the existence of physically unexplained phenomena in the experimental data (see Appendix A).

In this paper, we analyse the free flight of a dragonfly, *Anax parthenope julius*, from 16 mm ciné films taken in a wind tunnel, to calculate the flight performance at various flight speeds by applying the LCM in a form extended to introduce the unsteady aerodynamic effects in a conventional sense, and to show that flight can be performed without using the unsteady lift generated from the separated flow at least in steady trimmed flight.

## GEOMETRICAL CONFIGURATION

The general configuration of *Anax parthenope julius* is shown in Fig. 1 and geometrical characteristics of two dragonflies are given in Table 1. This dragonfly is considered to be one of the ‘high performance’ species: it has excellent manoeuvrability and is a most active predator. It has a large wing load and high beating frequency, enabling fast and skilful flight.

The beating can be represented both as the flapping (or heaving) motion of an elastic axis assumed to be a straight line roughly passing through a quarter chord of the aerofoil section at any spanwise station and as a feathering (or pitching motion about the elastic wing axis. These motions are performed actively through two (front and rear) joints at the respective wing root (von Lendenfeld, 1881). As shown in Fig. 1A,B, the flapping is assumed to be confined within a conical plane, the apex of which coincides with the front joint and the coning angle of which is defined by *β* although the actual flapping motion deviates slightly from the conical plane during the stroke (Fig. 2). The orbit of the three-quarter radius station of the elastic (or feathering) axis is called the ‘stroke plane’, which is considered to be normal to the cone axis.

In dragonflies, the cones related to the forewings are open towards the front whereas the cones related to the hindwings are open mainly towards the back. Therefore, the stroke planes of the forewings are located in front of the joints of the forewings and the stroke planes of the hindwings are located behind the joints of the hindwings. The tilt angles of the two stroke planes are defined by *γ*^{f} and *γ*^{h}.

## MODE OF WING BEAT

Free dragonfly flight was observed in a wind tunnel and filmed with a 16 mm high-speed ciné camera. By changing the wind speed in the tunnel the flight speed was altered. To get a clear image of the feathering motion and the twist distribution of the wings, three parallel stripes (1mm in width) were painted on each left wing at three spanwise stations (Fig. 1A). The width and thus the mass of the stripes were so small that the beating mode and frequency were substantially unchanged by them.

Observed modes of flight are given in Table 2. The orbits of the wing tips with respect to the body, which roughly show the stroke planes of the respective wings, and the movements of the wings (measured at the three-quarter span position) in inertial space, are shown in Fig. 2A–D for various flight speeds, V. The traces of the azimuth angles, *ψ*^{f} and *ψ*^{h}, which are defined by the flapping angles of the elastic axis projected to the stroke plane and measured from the horizontal line, are shown in Fig. 3A–D. They can be expressed by the first harmonic of a Fourier series:

However, as can be seen from Fig. 4A–D, the feathering angles at three span positions x = 0·25, 0·5 and 0·75, are expressed by the Fourier expansion series including higher harmonics as follows:

From these figures it can be seen that: (i) the beating frequency of the wings is almost unaltered (f = 29–32 Hz) with changes in flight speed; (ii) the tilt angle of the stroke planes with respect to the body axis gradually increases from about 40° to 70° as the flight speed increases; (iii) the phase difference between fore and hind pairs of wings is within 60°–90° and is not correlated with flight speed; (iv) the coning angle is about 8° in the forewing pair and about −2° in the hindwing pair and is almost constant (within 10% deviation) throughout the beating motion; (v) the flapping amplitude ranges from 25° to 40° and is not correlated with flight speed; (vi) the feathering amplitude ranges from 40° to 60° in the forewings and from 30° to 40° in the hindwings and is also not correlated with flight speed; (vii) the phase difference between the flapping and feathering motion is 90°, which is considered to be optimal for efficiency at low beating frequency (Azuma, 1981); and (viii) since the beating motion is performed through two joints at the respective wing roots, in the feathering motion the wing is twisted linearly for a given time – wash-out (twisted negatively towards the tip) in the downstroke and wash-in (twisted positively towards the tip) in the upstroke.

The wing beat modes were also observed in a smoke tunnel using a stroboscopic flash which made it possible to visualize the wake vortices of the beating wings and the modes of the wing motion. The wake vortices were visualized by the paraffin mist method (Watanabe *et al*. 1986) (Fig. 6A,B). A series of trailing and shed vortices generated, respectively, by the span and time (or azimuthal) change of bound vortices are clearly observed in wavy wake sheets of the respective wing pairs.

## AEROFOIL CHARACTERISTICS

To determine the aerodynamic characteristics of the dragonfly wing aerofoil, flight tests were conducted in a calm room using model gliders which had a main wing composed of a pair of dragonfly hindwings, as shown in Fig. 7A,B. It was necessary to introduce a small swept angle for the main wing and to install a tail wing for steady gliding flight with a positive angle of attack.

Fig. 8A–C shows plots of data obtained for the main wings of the model gliders in which the trimmed angle of attack was changed by shifting the centre of gravity with respect to the aerodynamic centre of the wing. The data for the gliders with a tail wing, after removal of the contribution of the tail wing, are represented by circles and the data for the gliders without a tail wing are represented by triangles. The data for which the angle of attack were measured are shown by solid symbols in Fig. 8A–C. The solid curves in Fig. 8 show typical characteristics of these threedimensional wings.

_{c}/

_{2}and a are the aspect ratio, the swept angle of the half chord line, and the lift slope of the three-dimensional wing, and

*δ*

_{1,}

*δ*

_{2},

*k*and Δ are parameters related to the planform of the wing and are presented by charts of DATCOM (Hoak & Ellison, 1968).

Fig. 9A shows the dragonfly aerofoil characteristics (or the aerodynamic characteristics of the two-dimensional wing, extrapolated from the above obtained data to hypothetical values in a large angle of attack range) as a function of angle of attack, in which the maximum lift coefficient is assumed to be C_{l-max} = 1·2. It is not clear why the minimum drag coefficient is slightly lower than those expected from the skin friction of a plate. Fig. 9B compares the data for polar curves for a locust wing, fruit-fly wings and a dragonfly wing obtained by Jensen (1956), Vogel (1967) and Newman *et al*. (1977), respectively. The two-dimensional pitching moment around the aerodynamic centre, which is assumed to be a line connected to the quarter-chord of the aerofoil along the span, is small and is therefore neglected. Although the present curve shown by a solid line gives a higher profile drag at large angles of attack than do the others, the above aerofoil data are utilized in the following calculations for both fore and hind pairs of wings.

## ANALYSIS BY MEANS OF THE LOCAL CIRCULATION METHOD

Let us consider first the flight performance of a dragonfly at various speeds. The spanwise variation of the airloading and the total aerodynamic forces and moments acting on the wings are calculated by the local circulation method (LCM). A detailed description of the method is given by Azuma *et al*. (1985), and only a brief explanation is given here.

A wing in beating motion is hypothesized to consist of a plurality of elliptical swings, and the respective elliptical wings are supposed to operate in the induced flow field generated by them. The timewise change of the induced velocity at any local (or spanwise and azimuthal) station is given by multiplying an attenuation coefficient, which is decided by the trailing and shed vortices left in a simple (not a free) wake model as a function of the spanwise distance, by the azimuth angle. Thus, the calculation is time-consuming and yet considered to be sufficiently accurate, without suffering from the computational divergence which sometimes appears in free-wake analysis.

From the actual wake vortices shown in Fig. 6A,B, a mathematical model of the wake sheets can be derived from the tracings of beating wings (Fig. 10). Then the attenuation coefficients are determined by the Bio–Savart law from the trailing vortices, which are assumed to be combined with the tip vortices of the respective wing tips, and from the shed vortices lying on the wake sheets.

Although the attenuation coefficient can be given as a function of span position and azimuth angle, the spanwise distance is fixed at the three-quarter position to simplify the calculations in the present analysis.

The method is based on the blade element analysis but is different from other previous studies as follows, (i) The aerodynamic coefficients are used in nonlinear forms as functions of angles of attack that include the stalled range, (ii) The induced velocity is not homogeneous on the stroke planes and is obtained by the LCM, which also includes the unsteady effect, (iii) The control inputs for free flight in computer simulation are the feathering angles along the wing span, the flapping angles, the tilt angles of the stroke planes, and the beating frequency for given flight speeds.

When the flight conditions and the wing motion are known, the airloading, and thus the total aerodynamic forces and moments as well as the inertial forces and moments, can be calculated by the LCM.

By referring to Fig. 11, the lift, drag and feathering moment acting on a wing element at spanwise station r with azimuth angle *ψ* are given as follows:

_{j,A}is the pitching moment acting on an apparent joint, which is a hypothetical joint at the root of the feathering (or elastic) axis, and is given by: Similarly, by assuming that every angular motion is small and neglecting coupling terms, the inertial forces and moments acting on a wing element can be given by: Where Then,. the total vertical force, horizontal force and pitching moment about the centre of gravity can be given by integrating the above elemental forces and moments along the wing span and by summing the individual components as follows: where superscripts f and h show the values for forewing and hindwing, respectively. Their mean values (or time averages) for one period of beating motion are given by: Neglecting the feathering component, the power required to beat the wing is given by: The vertical and horizontal forces are also normalized by the weight of the dragonfly and are expressed by the load factor, only the vertical component of which is given by

*n*in unit G, the ratio of the resulting acceleration and the acceleration due to gravity.

## PERFORMANCE

From the data given in Tables 2 and 3 for free flights, which were not exactly trimmed flight (load factor *n >* 1), the necessary (mechanical) power is obtained as shown by the circles in Fig. 12. By referring to these data and by selecting modified feathering angles as given in Table 4, the necessary power curve *versus* flight speed can be calculated (Fig. 12). The curve passes through lower values than those obtained from the free-flight tests, because the calculated power is based on almost completely trimmed flights *(n* ≈ 1).

By assuming an available power to musculature mass ratio of P_{a}/m_{m}^{=} 260W kg^{−1} (Weis-Fogh, 1975, 1977) and a musculature mass to total mass ratio of m_{m}/m = 0·25 (Greenwait, 1962), the available power can be estimated as P_{a} = 5·75×l0^{−2} W. Then the top speed (V_{max}) of this dragonfly is 1·2 ms^{−1} (Fig. 12). This value can be increased either by reducing the estimated drag area or by increasing the available power. The cruising flight speed (V_{cruise}, the maximum range speed), which is found at the point at which the necessary power curve is at a tangent to a line drawn through the origin or (dP/dV)_{min}, is 3·5 m s^{−1} (Fig. 12). The minimum power speed (V_{P min}, maximum endurance flight speed) is 1·7 m s^{−1}. The power required for hovering flight (P_{hov}) is estimated to be 3·6×10^{−2}W.

Fig. 13A–D shows the time variations of the vertical and horizontal components of the aerodynamic force, torque about the flapping axis and moment about the centre of gravity at V = 3·2 m s^{−1}. They were calculated by the LCM and by blade element theory based on the constant induced velocity distribution. For the calculation of the moment, the feathering moment around the elastic axis was assumed to be zero (C_{m} = 0) because of ambiguous aerodynamic characteristics due to complex aerofoil configuration. It can be seen that by assuming a constant induced velocity distribution some errors are introduced in these variations. However, as shown by the right-hand ordinate, their mean values are close to each other in the vertical component of the force. Although the forces are in a well-balanced condition, , and , the mean moment about the centre of gravity is not completely balanced but leaves a positive (pitch-up) value under the assumption of zero feathering moment, C_{m} = 0. However, the value in not too large and is within a range in which adjustment is possible by taking into account the feathering moment which is considered to be negative for a positive camber (upward convex) in the aerofoil.

Fig. 14 shows the inertial torque of the respective wings and the vertical component of the inertial force of the total wings in comparison with the aerodynamic component and the resultants. Although the respective inertial and aerodynamic components are of a comparable order of magnitude, the total torque and force are not large. The variation in the total vertical force is within OG and +3G in each beating cycle. As stated in Appendix A, it is important to recognize that the maximum value does not exceed *n =* 3G at the flight speed of 3·2 m s^{−1}.

Fig. 15 shows the spanwise load distribution of vertical and horizontal components of the total force acting on the fore-and hindwings.

Fig. 16 shows the timewise variation of angle of attack on the stroke planes of the wings in cruising flight. It is interesting that a region of large positive angle of attack is observed in the final stage of the downstroke near the wing tip of the forewing and a region of large negative angle of attack is observed in the early stage of the upstroke near the midspan of the forewing.

Fig. 17 shows the share of the lift and drag components in the mean vertical force, . At very low speed, including hovering, a contribution of the drag component on the vertical force cannot be neglected. This is not unconnected with the fact that the dragonfly flies with its stroke plane tilted from the horizontal even when hovering, although this increases the induced power required. This makes the transition from hovering flight to other flight modes easier.

Appendix A

Fig. 18 is compiled from data obtained from an experiment on the dragonfly *(Libellula luctuosa)* (Somps & Luttges, 1985). The following characteristics of wing motion and vertical forces (upward positive) can be obtained:

*Anax parthenope*, the vertical forces generated by the respective pairs of wings are represented by: and are shown in Fig. 19 for the aerodynamic and inertial components. Because of the above assumption, the result is only a rough estimation of the vertical force, but the aerodynamic component of the vertical force in the respective wings can be given over a wide range of the downstroke as described in the present paper.

However, in the experimental results obtained by Somps & Luttges (1985) (Fig. 18) there are two difficulties: the vertical force (over20G) and its mean value (5·3G) are both too large. These extremely large values are not realistic for the flight of living creatures, except when rapidly manoeuvring. The large values probably result from the fact that the inertial force related to the body mass was not completely removed from the measurement of the vertical force.

## Definitions

- AR
aspect ratio of wing, AR = b

^{2}/S - a
lift slope

- b
wing span (m)

- C
_{D}drag coefficient of three-dimensional wing

drag coefficient of body other than wings

induced drag coefficient

profile drag coefficient, .

- C
_{L}lift coefficient of three-dimensional wing

- C
_{d}drag coefficient to two-dimensional wing,

- CG
centre of gravity

- C
_{l}lift coefficient of two-dimensional wing,

*C*_{l}*=*a*α*within unstalled region - C
_{l}._{max}maximum lift coefficient C

_{l}= C_{l max}beyond stall - C
_{m}moment coefficient of two-dimensional wing about the feathering or elastic axis

- c
chord length (m)

- D
drag (N)

- F
_{H}horizontal force, positive forward (N)

- F
_{v}vertical force, positive upward (N)

- f
beating frequency (Hz)

*f*drag area (m

^{2})- G
acceleration measured in units of gravity acceleration

*g*acceleration due to gravity,

*g =*9·80665 m s^{−2}- I
moment of inertia of a wing (kg·m

^{2}) k reduced frequency, k*= π*cf/U *k*aerodynamic parameter

- L
lift (N)

*L*body length

*l*spanwise airloading (N m

^{−1})- M
feathering moment, moment about feathering (elastic) axis (Nm)

- M
_{CG}pitching moment about the centre of gravity (positive for head-up) (Nm)

- M
_{j}pitching moment about an apparent joint on the root of the elastic axis of the wing (Nm)

- m
body mass (kg)

- m
_{m}musculature mass (kg)

*n*nth order of higher harmonics

*n*load factor

- P
necessary power (W)

- P
_{a}available power (W)

- P
_{i}induced power (W)

- P
_{o}profile power (W)

- P
_{p}parasite power (W)

- Q
torque about an apparent joint (Nm)

- R
distance from apparent joint to wing tip, R ≈ b/s (m)

*Re*Reynolds number

- r
spanwise station along the elastic axis (m)

- S
wing area (m

^{2}) - S
_{f}maximum sectional area of body (m

^{2}) - T
period of one beating cycle (s)

- t
time (s)

- U
- U
_{T}tangential component of velocity relative to a blade element (ms

^{−1}) - UP
perpendicular component of velocity relative to a blade element (ms

^{−1}) - V
flight speed (m s

^{−1}) - v
induced velocity (ms

^{−1}) *W*weight of body = mg (N)

- x
non-dimensional spanwise station, x = r/R

- x
_{CĠ}longitudinal distance of centre of gravity measured from front joint of forewing (m)

- x
_{j}distance between the first joints of forewing and hindwing

- V
_{CĠ}vertical distance of centre of gravity measured from hindwing angle of attack of wing section (degrees or radians)

*β*coning angle (degrees or radians)

*β**beating angle projected in horizontal plane (degrees)

- Γ
flight angle (degrees or radians)

*γ*tilt angle of stroke plane (degrees or radians)

*δ*_{1},*δ*_{2}, Δaerodynamic parameters related to the wing planform

*Δ*_{ψ}phase shift of flapping motion (degrees or radians)

*Δ*_{ϕ}phase shift of feathering motion (degrees or radians)

- Θ
body attitude (degrees)

*θ*feathering angle (degrees or radians)

*ρ*air density (kgm

^{−3})- Λ
_{C/2}swept angle to half-chord line (degrees or radians)

*ϕ*inflow angle (radians)

*θ*azimuth angle or flapping angle (degrees or radians)

*θ*_{1}amplitude of beating motion (degrees or radians)

*θ**beating angle projected in vertical plane (degrees)

*ω*angular velocity of beating motion,

*ω =*2*π*F (radians s^{−1})

## SUBSCRIPT OR SUPERSCRIPT

- o
oth harmonic component (or steady state) or two-dimensional values

- 1
the first harmonic component

- n
nth harmonic component

- 0·75R
three-quarter radius station

- A
aerodynamic component

- cruise
cruising flight

- f
forewing

- h
hindwing

- hov
hovering state

- I
inertial component

- max
maximum value

- min
minimum value

- P,min
power minimum

- root
wing root

- tip
wing tip

## ACKNOWLEDGEMENTS

The authors would like to thank Mr Masakatsu Takao, Mr Masayuki Kitamura, and other staff of the Japan Broadcasting Corporation (NHK) for stimulating us to try vortex-wake visualization by their scientific program on dragonflies. The authors are also indebted to Mr Shunji Oba, Iwata Agricultural High School, for catching dragonflies, and Professor Keiji Kawachi and Mr Isao Watanabe, The University of Tokyo, who assisted in the computer programming and the windtunnel test, respectively.

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