The mechanics of flight in the hawkmoth manduca sexta

The mechanics of flight in the hawkmoth manduca sexta

(Parte 1 de 8)

A detailed analysis of free flight in the hawkmoth Manduca sextaL. has revealed the kinematic changes as speed increases from hovering to fast forward flight (Willmott and Ellington, 1997). In this study, we investigate the aerodynamic significance of the observed kinematic variation, the power requirements for flight at different speeds and the nature of the constraints on maximum flight speed.

Insect flapping flight represents an unusual aerodynamic problem because of the inherent ‘unsteadiness’ and the low Reynolds number of the airflow. A large number of models for unsteady animal flight have been formulated, and these have been categorized and evaluated in recent reviews by Spedding (1992), Spedding and DeLaurier (1996) and Smith et al. (1996). The techniques range from those incorporating momentum and blade-element theories to those employing lifting-line or lifting-surface methods, but each requires nontrivial simplifying assumptions. Recent unsteady panel methods, for example, are more advanced than their predecessors in considering trailing wake vorticity and dynamic effects (Smith et al.1996), and interactions between wing deformation and the aerodynamic forces (Smith, 1996), but they still have important shortcomings. In particular, they do not yet incorporate the leading-edge vortices that have a profound influence on the flow around hawkmoth wings at all speeds (Willmott et al.1997) and which are likely to play a major role in insect flight aerodynamics (Ellington et al. 1996). At the low Reynolds numbers and high angles of incidence characteristic of insect flight, viscous effects in the near field are better addressed by Navier–Stokes solvers, which are now beginning to be applied to animal locomotion (e.g. Liu et al. 1996).

Advanced numerical techniques will, undoubtedly, become standard in animal flight investigations. Their introduction will, however, be delayed by the paucity of detailed and relevant empirical data for their validation and by the need to investigate fully the inherent errors in the new techniques and their applicability to other conditions (Visbal, 1986; Spedding, 1992). Accurate and reliable modelling of instantaneous forces

2723The Journal of Experimental Biology 200, 2723–2745 (1997) Printed in Great Britain ©The Company of Biologists Limited 1997

JEB0994

Mean lift coefficients have been calculated for hawkmoth flight at a range of speeds in order to investigate the aerodynamic significance of the kinematic variation which accompanies changes in forward velocity. The coefficients exceed the maximum steady-state value of 0.71 at all except the very fastest speeds, peaking at 2.0 or greater between 1 and 2ms-1. Unsteady high-lift mechanisms are therefore most important during hovering and slow forward flight. In combination with the wingtip paths relative to the surrounding air, the calculated mean lift coefficients illustrate how the relative contributions of the two halfstrokes to the force balance change with increasing forward speed. Angle of incidence data for fast forward flight suggest that the sense of the circulation is not reversed between the down- and upstrokes, indicating a flight mode qualitatively different from that proposed for lower-speed flight in the hawkmoth and other insects. The mid-downstroke angle of incidence is constant at 30–40° across the speed range. The relationship between power requirements and flight speed is explored; above 5ms-1, further increases in forward velocity are likely to be constrained by available mechanical power, although problems with thrust generation and flight stability may also be involved.

Hawkmoth wing and body morphology, and the differences between males and females, are evaluated in aerodynamic terms. Steady-state force measurements show that the hawkmoth body is amongst the most streamlined for any insect.

Key words: aerodynamics, hawkmoth, Manduca sexta, lift coefficient, power requirements, morphology.

Summary

Introduction

ALEXANDER P. WILLMOTT* ANDCHARLES P. ELLINGTON Department of Zoology, University of Cambridge, Downing Street, Cambridge CB2 3EJ, UK

Accepted 4 August 1997

*Present address: Kawachi Millibioflight Project, Japan Science and Technology Corporation (JST), Park Building 3F, 4-7-6 Komaba, Meguro-ku, Tokyo 153, Japan (e-mail: sandy@kawachi.jst.go.jp).

is not an option available at present. Instead, in this report, we concentrate in more general terms on the lift-generating requirements placed on the wings during flight at different speeds and, in particular, on the possible aerodynamic consequences of the kinematic trends described in Willmott and Ellington (1997).

The ‘mean coefficients’ method used here is a slightly modified momentum jet/blade-element approach of the type used by Dudley (1990), Dudley and Ellington (1990b), Cooper (1993) and Wakeling and Ellington (1997b). This technique has the advantage of being the most accessible of the current models – using data on the core kinematic parameters – and of being easily modified to incorporate new kinematics or wing morphology. Despite its simplifying assumptions, the mean coefficients model has been shown in a wide range of studies (e.g. Osborne, 1951; Dudley and Ellington, 1990b; Dudley, 1995) to be capable of producing meaningful estimates of aerodynamic forces and power requirements. It has also generated workable hypotheses linking flight kinematics and aerodynamics, thus meeting Spedding’s (1992) definition of a ‘useful’ aerodynamic model. Such models are necessarily refined as and when new techniques become available.

The model is used here to determine the mean lift coefficient at each of the flight speeds in the kinematic study and to estimate the associated aerodynamic and inertial power components. The coefficients are calculated under the assumption that they are constant throughout each halfstroke of the wingbeat. They thus represent the lowest possible lift coefficient required from the wings for a given set of wingbeat kinematics; if the coefficients are not constant (a more realistic scenario), then instantaneous values at certain points of the wingbeat cycle must exceed these mean values (Ellington, 1984a). A number of related questions are also addressed, such as the aerodynamic function of the upstroke and the aerodynamic and energetic significance of the asymmetric wingbeat. For comparative purposes, the lifting performance of hawkmoth wings and bodies under ‘steady’ flow conditions was measured. Finally, the morphology of adult Manduca sextawas investigated and the possible aerodynamic significance of the observed differences between the sexes, and between hawkmoths and other insects, is discussed.

Materials and methods

Measurement of steady-state wing and body forces

Steady-state lift and drag were measured using the optoelectronic force transducer described by Dudley and Ellington (1990b). A summary of its design and operation will be given here; details can be found in the earlier paper. The transducer measured the displacement in two orthogonal directions of a stainless-steel tube to which a test object was attached viaa mounting pin. The voltage output of the transducer had a linear relationship to the moment applied to the tube, and this was calibrated at the beginning and end of each set of measurements. A range of replaceable spring elements covered forces on the test object of up to 0.016N.

The transducer was positioned at the mouth of the open-jet wind tunnel used in the kinematic study (Willmott and Ellington, 1997).

Aerodynamic forces on the body

The wings were removed from freshly killed Manduca sexta whose body posture was similar to that seen in free flight. Owing to the problems of orienting the legs and antennae in a realistic position, and to the difficulty in determining such positions from the high-speed videos, these structures were also removed. The body was mounted at the top of a stainlesssteel entomological pin which passed through the left and right wingbases. The bodies were left to ‘set’ for 1 day before use in order to fix the abdomen in flight position.

Lift and drag measurements were made on six bodies for airspeeds from 1 to 5ms-1, and at the following angles of attack: -15°, 0°, 15°, 25°, 35°, 45°and 60°. The 0°angle was set with the longitudinal axis of the body parallel to the airflow; positive angles of attack refer to the body being pitched head-up with respect to the horizontal. In order to standardize the Reynolds numbers based on body length Reb for the different bodies, the slight differences in body length were corrected for by slight modifications in the airspeed. The Reynolds numbers used were 3150, 6040, 9150, 12270 and 15220, corresponding to speeds of approximately 1, 2, 3, 4 and 5ms - 1, respectively.

Aerodynamic forces on the wings

Before the wings were removed from the bodies of freshly killed moths, the left wing couple was moved into a realistic flight orientation. A small quantity of cyanoacrylate gel was applied between the dorsal surface of the hindwing and the ventral surface of the forewing, close to the wingbase, to hold the wings in the correct alignment. The wing couple was then detached carefully from the body by cutting through the wingbases as close as possible to the body. At this point, the anojugal flap folded under the hindwing. It was not possible to return this region to its correct position in fresh wings, and so the flap was removed by cutting along its fold line. The wing was mounted on a stainless-steel pin with the longitudinal axis of the wing aligned with the shaft of the pin: the head of the pin was removed and its final 5mm bent perpendicular to the main shaft to form a short hook which was then attached, using a small drop of beeswax, near to the base of the underside of the wings.

The Manduca sextawing is not a planar surface: a spanwise gradient of wing twist is inherent in its structure. The wing was mounted so that the chosen angle of attack corresponded to the inclination of the wing halfway between the wingbase and wingtip, close to the radius of the first moment of wing area which is where the steady-state aerodynamic force is assumed to act. The angle of attack in the proximal area of the wing was up to approximately 5°more positive than this value, whilst the angle in the distal region was up to 5°more negative. Lift and drag measurements were made at 10°angle of attack intervals from -50°to +70°, and at Reynolds numbers based on mean

2725The aerodynamics of hawkmoth flight wing chord Recof 1150, 30 and 5560, which corresponded to airspeeds of approximately 1, 3 and 5ms-1. The measurements at Rec=1150 were made on the day that the wings were removed. The remaining data were collected the following day.

Relative velocity and angles of incidence

The instantaneous relative velocity Urat any given spanwise location on the wing was determined from the resultant of the forward flight velocity V, the induced velocity w0, and the flapping velocity U at that point. The forward and flapping velocities were obtained from the free-flight kinematic data given in Willmott and Ellington (1997). The induced velocity was estimated using momentum jet theory under the assumptions that it was constant along the wingspan and over the course of the wingbeat, and that the actuator disc was approximately horizontal. The induced velocity under these conditions is vertical, and its magnitude w0at the actuator disc is given by the expression derived by Stepniewski and Keys

(1984) for helicopters in forward flight:

where w0,RFis the Rankine–Froude estimate for the hovering induced velocity:

where mis body mass, gis gravitational acceleration, ris the mass density of air, Fis stroke amplitude and Ris wing length. The factor kin equation 1 corrects for temporal and spatial variation in the wake; the value of 1.2 for ksuggested by Pennycuick (1975) for forward flight also corresponds well with estimates from vortex theory for hovering insect flight (Ellington, 1984d).

Fourier series approximations were fitted to the raw data for the positional angles. The functions for sweep angle fand elevation angle q(for definitions, see Willmott and Ellington, 1997) could be differentiated to obtain the components of the flapping velocity in the stroke plane and normal to the stroke plane, respectively. The series were truncated after the terms for the fourth harmonic because this resulted in a good, smoothed approximation to the raw data without introducing unrealistic high-order harmonics which led to excessive angular accelerations. The body was assumed to experience the same induced velocity as the wings. The relative velocity for the body Ur,b was calculated as the vector sum of the forward velocity Vand the induced velocity w0. The magnitude of the relative velocity was used to derive the Reynolds number for the body Reb:

components in three directions: tangential to the flapping velocity and parallel to the stroke plane, normal to this tangent and the longitudinal axis, and parallel to the longitudinal axis of the wing. The angle between the relative velocity and the stroke plane was determined from the first two components, and these angles were used to convert the geometric angles of rotation relative to the stroke plane aspfrom the kinematics study (Willmott and Ellington, 1997) into the more aerodynamically relevant angles of incidence ar. Positive values of this parameter, which is the angle between the wing chord and the relative velocity, indicate that the air was striking the ventral surface of the wing. These angles also represent the effective angles of incidence ar¢since the zero lift angle of the wings is approximately 0°(see below).

The angle between the relative velocity of the body and horizontal was subtracted from the body angle cto give the angle of incidence cr,bfor calculation of the aerodynamic forces on the body.

Aerodynamic forces and mean lift requirements

The most recent and detailed version of the mean coefficients method developed by Osborne (1951) is the model developed by C. P. Ellington and successfully applied to both horizontal and vertical flight in bumblebees (Cooper, 1993). An outline only of the approach will be presented here. The application of the model to hawkmoth flight differs from those in previous studies only in the rejection, based on the kinematic analysis (Willmott and Ellington, 1997), of the assumption that the wingbeat is sinusoidal and confined to the stroke plane. The asymmetry in the wingbeat was incorporated using the Fourier series described above. The inclusion of elevation angles necessitated changes to the expressions for the components, with respect to three mutually orthogonal directions in a gravitational coordinate system, of the relative velocity Urand of the normal to the relative velocity Un. These modifications are described in detail in Willmott (1995).

Reconstruction of wing shape

The wing was divided into 50 strips of equal spanwise width using a beta function to model the distribution of area in the wing planforms obtained in Willmott and Ellington (1997). Matching this function to the measured first and second moments of wing area results in a close approximation of Manduca sextawing shape (Ellington, 1984b) and gives an analytical function which is readily incorporated into the aerodynamic model. The root mean square difference between the real and predicted chord widths for the three moths was 3.3%.

Wing forces

At any given time in the wingbeat cycle, the instantaneous lift force Land drag force Dacting on an individual wing strip are given by:

where Urand c are the relative velocity and chord of the strip,

L c rU C=

2ρdrL,(4)

D c rU C=

2ρdrD,pro,(5)

Reb r,b=

Ulν

,(3)

w m

,(2)

w V kw V0

++0,RF, (1)

respectively, dris the spanwise width of the strip, CLis the lift coefficient, CD,prois the profile drag coefficient, and ris the density of air (1.19kgm-3at 23°C). The component of

Urparallel to the wing axis is ignored. CLand CD,proare unknown at this point, but are assumed to be constant throughout a halfstroke. The sign of CLduring the upstroke depends upon whether the oncoming air is striking the ventral surface (CLis positive) or the dorsal surface (CLis negative) of the wing.

(Parte 1 de 8)

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