Dynamic flight stability of hovering model insects: theory versus simulation using equations of motion coupled with Navier–Stokes equations

In the present paper, the longitudinal dynamic flight stability properties of two model insects are predicted by an approximate theory and computed by numerical sim- ulation. The theory is based on the averaged model （which assumes that the frequency of wingbeat is sufficiently higher than that of the body motion, so that the flapping wings＇ degrees of freedom relative to the body can be dropped and the wings can be replaced by wingbeat-cycle-average forces and moments）; the simulation solves the complete equations of motion coupled with the Navier-Stokes equations. Comparison between the theory and the simulation provides a test to the validity of the assumptions in the theory. One of the insects is a model dronefly which has relatively high wingbeat frequency （164 Hz） and the other is a model hawkmoth which has relatively low wingbeat frequency （26 Hz）. The results show that the averaged model is valid for the hawkmoth as well as for the dronefly. Since the wingbeat frequency of the hawkmoth is relatively low （the characteristic times of the natural modes of motion of the body divided by wingbeat period are relatively large） compared with many other insects, that the theory based on the averaged model is valid for the hawkmoth means that it could be valid for many insects.     

Wing/body kinematics measurement and force and moment analyses of the takeoff flight of fruitflies

In the paper, we present a detailed analysis of the takeoff mechanics of fruitflies which perform voluntary takeoff flights. Wing and body kinematics of the insects during takeoff were measured using high-speed video techniques.Based on the measured data, inertia force acting on the insect was computed and aerodynamic force and moment of the wings were calculated by the method of computational fluid dynamics. Subtracting the aerodynamic force and the weight from the inertia force gave the leg force. The following has been shown. In its voluntary takeoff, a fruitfly jumps during the first wingbeat and becomes airborne at the end of the first wingbeat. When it is in the air, the fly has a relatively large "initial" pitch-up rotational velocity(more than5 000°/s) resulting from the jumping, but in about 5 wingbeats, the pitch-up rotation is stopped and the fly goes into a quasi-hovering flight. The fly mainly uses the force of jumping legs to lift itself into the air(the force from the flapping wings during the jumping is only about 5%–10% of the leg force). The main role played by the flapping wings in the takeoff is to produce a pitch-down moment to nullify the large "initial" pitch-up rotational velocity(otherwise, the fly would have kept pitching-up and quickly fallen down).     

Lateral dynamic flight stability of hovering insects: theory vs. numerical simulation

In the present paper, the lateral dynamic flight stability properties of two hovering model insects are predicted by an approximate theory based on the averaged model, and computed by numerical simulation that solves the complete equations of motion coupled with the Navier-Stokes equations. Comparison between the theoretical and simulational results provides a test to the validity of the assumptions made in the theory. One of the insects is a model dronefly which has relatively high wingbeat frequency (164 Hz) and the other is a model hawkmoth which has relatively low wingbeat frequency (26 Hz). The following conclusion has been drawn. The theory based on the averaged model works well for the lateral motion of the dronefly. For the hawkmoth, relatively large quantitative differences exist between theory and simulation. This is because the lateral non-dimensional eigenvalues of the hawkmoth are not very small compared with the non-dimensional flapping frequency (the largest lateral non-dimensional eigenvalue is only about 10% smaller than the non-dimensional flapping frequency). Nevertheless, the theory can still correctly predict variational trends of the dynamic properties of the hawkmoth’s lateral motion.     

Effects of wing deformation on aerodynamic performance of a revolving insect wing

Flexible wings of insects and bio-inspired micro air vehicles generally deform remarkably during flapping flight owing to aerodynamic and inertial forces,which is of highly nonlinear fluid-structure interaction（FSI）problems.To elucidate the novel mechanisms associated with flexible wing aerodynamics in the low Reynolds number regime,we have built up a FSI model of a hawkmoth wing undergoing revolving and made an investigation on the effects of flexible wing deformation on aerodynamic performance of the revolving wing model.To take into account the characteristics of flapping wing kinematics we designed a kinematic model for the revolving wing in two-fold：acceleration and steady rotation,which are based on hovering wing kinematics of hawkmoth,Manduca sexta.Our results show that both aerodynamic and inertial forces demonstrate a pronounced increase during acceleration phase,which results in a significant wing deformation.While the aerodynamic force turns to reduce after the wing acceleration terminates due to the burst and detachment of leading-edge vortices（LEVs）,the dynamic wing deformation seem to delay the burst of LEVs and hence to augment the aerodynamic force during and even after the acceleration.During the phase of steady rotation,the flexible wing model generates more ver-tical force at higher angles of attack（40°–60°）but less horizontal force than those of a rigid wing model.This is because the wing twist in spanwise owing to aerodynamic forces results in a reduction in the effective angle of attack at wing tip,which leads to enhancing the aerodynamics performance by increasing the vertical force while reducing the horizontal force.Moreover,our results point out the importance of the fluid-structure interaction in evaluating flexible wing aerodynamics：the wing deformation does play a significant role in enhancing the aerodynamic performances but works differently during acceleration and steady rotation,which is mainly induced by inertial force in acceleration but by aerodynamic forces     

Effects of unsteady deformation of flapping wing on its aerodynamic forces

Effects of unsteady deformation of a flapping model insect wing on its aerodynamic force production are studied by solving the Navier-Stokes equations on a dynamically deforming grid.Aerodynamic forces on the flapping wing are not much affected by considerable twist,but affected by camber deformation.The effect of combined camber and twist deformation is similar to that of camber deformation.With a deformation of 6% camber and 20°twist(typical values observed for wings of many insects),lift is increased bv 10%～20%and lift-to-drag ratio by around 10%compared with the case of a rigid flat-plate wing.As a result.the deformation can increase the maximum lift coefficient of an insect.and reduce its power requirement for flight.For example,for a hovering bumblebee with dynamically deforming wings(6?mber and 20°twist),aerodynamic power required is reduced by about 16%compared with the case of rigid wings.     

A computational study of the wing–wing and wing–body interactions of a model insect

The aerodynamic interaction between the contralateral wings and between the body and wings of a model insect are studied, by using the method of numerically solving the Navier-Stokes equations over moving overset grids, under typical hovering and forward flight conditions. Both the interaction between the contralateral wings and the interaction between the body and wings are very weak, e.g. at hovering, changes in aerodynamic forces of a wing due to the present of the other wing are less than 3% and changes in aerodynamic forces of the wings due to presence of the body are less than 2%. The reason for this is as following. During each down- or up-stroke, a wing produces a vortex ring, which induces a relatively large jet-like flow inside the ring but very small flow outside the ring. The vortex rings of the left and right wings are on the two sides of the body. Thus one wing is outside vortex ring of the other wing and the body is outside the vortex rings of the left and right wings, resulting in the weak interactions.     

Wing kinematics measurement and aerodynamics of free-flight maneuvers in drone-flies

The time courses of wing and body kinematics of two free-flying drone-flies, as they performed saccades, were measured using 3D high-speed video, and the morpho- logical parameters of the wings and body of the insects were also measured. The measured wing kinematics was used in a Navier-Stokes solver to compute the aerodynamic forces and moments acting on the insects. The main results are as following. （1） The turn is mainly a 90° change of heading. It is made in about 10 wingbeats （about 55 ms）. It is of interest to note that the number of wingbeats taken to make the turn is approximately the same as and the turning time is only a little different from that of fruitflies measured recently by the same approach, even if the weight of the droneflies is more than 100 times larger than that of the fruitflies. The long axis of body is about 40° from the horizontal during the maneuver. （2） Although the body rotation is mainly about a vertical axis, a relatively large moment around the yaw axis （axis perpendicular to the long axis of body）, called as yaw moment, is mainly needed for the turn, because moment of inertial of the body about the yaw axis is much larger than that about the long axis. （3） The yaw moment is mainly pro- duced by changes in wing angles of attack： in a right turn, for example, the dronefly lets its right wing to have a rather large angle of attack in the downstroke （generally larger than 50°） and a small one in the upstroke to start the turn, and lets its left wing to do so to stop the turn, unlike the fruitflies who generate the yaw moment mainly by changes in the stroke plane and stroke amplitude.     

Wind induced deformation and vibration of a Platanus acerifolia leaf

Deformation and vibration of twig-connected single leaf in wind is investigated experimentally.Results showthat the Reynolds number based on wind speed and lengthof leaf blade is a key parameter to the aerodynamic problem.In case the front surface facing the wind and with an increase of Reynolds number,the leaf experiences static deformation,large amplitude and low frequency sway,reconfiguration to delta wing shape,flapping of tips,high frequencyvibration of whole leaf blade,recovery of delta wing shape,and twig-leaf coupling vibration.Abrupt changes from onestate to another occur at critical Reynolds numbers.In casethe back surface facing the wind,the large amplitude andlow frequency sway does not occur,the recovered delta wingshape is replaced by a conic shape,and the critical Reynoldsnumbers of vibrations are higher than the ones corresponding to the case with the front surface facing the wind.Apair of ram-horn vortex is observed behind the delta wingshaped leaf.A single vortex is found downstream of theconic shaped leaf.A lift is induced by the vortex,and thislift helps leaf to adjust position and posture,stabilize bladedistortion and reduce drag and vibration.     

NUMERICAL SIMULATION OF INSECT FLIGHT

In the non-inertial coordinates attached to the model wing, the two-dimensional unsteady flow field triggered by the motion of the model wing, similar to the flapping of the insect wings, was numerically simulated. One of the advantages of our method is that it has avoided the difficulty related to the moving-boundary problem. Another advantage is that the model has three degrees of freedom and can be used to simulate arbitrary motions of a two-dimensional wing in plane only if the motion is known. Such flexibility allows us to study how insects control their flying. Our results show that there are two parameters that are possibly utilized by insects to control their flight: the phase difference between the wing translation and rotation, and the lateral amplitude of flapping along the direction perpendicular to the average flapping plane.     

VISCOELASTIC CONSTITUTIVE MODEL RELATED TO DEFORMATION OF INSECT WING UNDER LOADING IN FLAPPING MOTION

Flexible insect wings deform passively under the periodic loading during napping flight. The wing flexibility is considered as one of the specific mechanisms on improving insect flight performance. The constitutive relation of the insect wing material plays a key role on the wing deformation, but has not been clearly understood yet. A viscoelastic constitutive relation model was established based on the stress relaxation experiment of a dragonfly wing (in vitro). This model was examined by the finite element analysis of the dynamic deformation response for a model insect wing under the action of the periodical inertial force in flapping. It is revealed that the viscoelastic constitutive relation is rational to characterize the biomaterial property of insect wings in contrast to the elastic one. The amplitude and form of the passive viscoelastic deformation of the wing is evidently dependent on the viscous parameters in the constitutive relation.     

Dynamic flight stability of hovering insects

The equations of motion of an insect with flapping wings are derived and then simplified to that of a flying body using the “rigid body” assumption. On the basis of the simplified equations of motion, the longitudinal dynamic flight stability of four insects (hoverfly, cranefly, dronefly and hawkmoth) in hovering flight is studied (the mass of the insects ranging from 11 to 1,648 mg and wingbeat frequency from 26 to 157 Hz). The method of computational fluid dynamics is used to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis are used to solve the equations of motion. The validity of the “rigid body” assumption is tested and how differences in size and wing kinematics influence the applicability of the “rigid body” assumption is investigated. The primary findings are: (1) For insects considered in the present study and those with relatively high wingbeat frequency (hoverfly, drone fly and bumblebee), the “rigid body” assumption is reasonable, and for those with relatively low wingbeat frequency (cranefly and howkmoth), the applicability of the “rigid body” assumption is questionable. (2) The same three natural modes of motion as those reported recently for a bumblebee are identified, i.e., one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. (3) Approximate analytical expressions of the eigenvalues, which give physical insight into the genesis of the natural modes of motion, are derived. The expressions identify the speed derivative M _u (pitching moment produced by unit horizontal speed) as the primary source of the unstable oscillatory mode and the stable fast subsidence mode and Z _w (vertical force produced by unit vertical speed) as the primary source of the stable slow subsidence mode. The project supported by the National Natural Science Foundation of China (10232010 and 10472008).     

On the aerodynamic characteristics of hovering rigid and flexible hawkmoth-like wings

Insect wings are subjected to fluid, inertia and gravitational forces during flapping flight. Owing to their limited rigidity, they bent under the influence of these forces. Numerical study by Hamamoto et al. (Adv Robot 21(1–2):1–21, 2007) showed that a flexible wing is able to generate almost as much lift as a rigid wing during flapping. In this paper, we take a closer look at the relationship between wing flexibility (or stiffness) and aerodynamic force generation in flapping hovering flight. The experimental study was conducted in two stages. The first stage consisted of detailed force measurement and flow visualization of a rigid hawkmoth-like wing undergoing hovering hawkmoth flapping motion and simple harmonic flapping motion, with the aim of establishing a benchmark database for the second stage, which involved hawkmoth-like wing of different flexibility performing the same flapping motions. Hawkmoth motion was conducted at Re = 7,254 and reduced frequency of 0.26, while simple harmonic flapping motion at Re = 7,800 and 11,700, and reduced frequency of 0.25. Results show that aerodynamic force generation on the rigid wing is governed primarily by the combined effect of wing acceleration and leading edge vortex generated on the upper surface of the wing, while the remnants of the wake vortices generated from the previous stroke play only a minor role. Our results from the flexible wing study, while generally supportive of the finding by Hamamoto et al. (Adv Robot 21(1–2):1–21, 2007), also reveal the existence of a critical stiffness constant, below which lift coefficient deteriorates significantly. This finding suggests that although using flexible wing in micro air vehicle application may be beneficial in term of lightweight, too much flexibility can lead to deterioration in flapping performance in terms of aerodynamic force generation. The results further show that wings with stiffness constant above the critical value can deliver mean lift coefficient almost the same as a rigid wing when executing hawkmoth motion, but lower than the rigid wing when performing a simple harmonic motion. In all cases studied (7,800 ≤ Re ≤ 11,700), the Reynolds number does not alter the force generation significantly.     

The effects of corrugation and wing planform on the aerodynamic force production of sweeping model insect wings

The effects of corrugation and wing planform (shape and aspect ratio) on the aerodynamic force production of model insect wings in sweeping (rotating after an initial start) motion at Reynolds number 200 and 3500 at angle of attack 40° are investigated, using the method of computational fluid dynamics. A representative wing corrugation is considered. Wing-shape and aspect ratio (AR) of ten representative insect wings are considered; they are the wings of fruit fly, cranefly, dronefly, hoverfly, ladybird, bumblebee, honeybee, lacewing (forewing), hawkmoth and dragonfly (forewing), respectively (AR of these wings varies greatly, from 2.84 to 5.45). The following facts are shown. (1) The corrugated and flat-plate wings produce approximately the same aerodynamic forces. This is because for a sweeping wing at large angle of attack, the length scale of the corrugation is much smaller than the size of the separated flow region or the size of the leading edge vortex (LEV). (2) The variation in wing shape can have considerable effects on the aerodynamic force; but it has only minor effects on the force coefficients when the velocity at r ₂ (the radius of the second moment of wing area) is used as the reference velocity; i.e. the force coefficients are almost unaffected by the variation in wing shape. (3) The effects of AR are remarkably small: when AR increases from 2.8 to 5.5, the force coefficients vary only slightly; flowfield results show that when AR is relatively large, the part of the LEV on the outer part of the wings sheds during the sweeping motion. As AR is increased, on one hand, the force coefficients will be increased due to the reduction of 3-dimensional flow effects; on the other hand, they will be decreased due to the shedding of part of the LEV; these two effects approximately cancel each other, resulting in only minor change of the force coefficients. The project supported by the National Natural Science Foundation of China (10232010 and 10472008) and Ph. D. Student Foundation of Chinese Ministry of Education (20030006022) The English text was polished by Keren Wang.     

The effects of corrugation and wing planform on the aerodynamic force production of sweeping model insect wings

The effects of corrugation and wing planform (shape and aspect ratio) on the aerodynamic force production of model insect wings in sweeping (rotating after an initial start) motion at Reynolds number 200 and 3500 at angle of attack 40° are investigated, using the method of computational fluid dynamics. A representative wing corrugation is considered. Wing-shape and aspect ratio (AR) of ten representative insect wings are considered; they are the wings of fruit fly, cranefly, dronefly, hoverfly, ladybird, bumblebee, honeybee, lacewing (forewing), hawkmoth and dragonfly (forewing), respectively (AR of these wings varies greatly, from 2.84 to 5.45). The following facts are shown. (1) The corrugated and flat-plate wings produce approximately the same aerodynamic forces. This is because for a sweeping wing at large angle of attack, the length scale of the corrugation is much smaller than the size of the separated flow region or the size of the leading edge vortex (LEV). (2) The variation in wing shape can have considerable effects on the aerodynamic force; but it has only minor effects on the force coefficients when the velocity at r ₂ (the radius of the second moment of wing area) is used as the reference velocity; i.e. the force coefficients are almost unaffected by the variation in wing shape. (3) The effects of AR are remarkably small: when AR increases from 2.8 to 5.5, the force coefficients vary only slightly; flowfield results show that when AR is relatively large, the part of the LEV on the outer part of the wings sheds during the sweeping motion. As AR is increased, on one hand, the force coefficients will be increased due to the reduction of 3-dimensional flow effects; on the other hand, they will be decreased due to the shedding of part of the LEV; these two effects approximately cancel each other, resulting in only minor change of the force coefficients.     

Aerodynamic and functional consequences of wing compliance

A growing body of evidence indicates that a majority of insects experience some degree of wing deformation during flight. With no musculature distal to the wing base, the instantaneous shape of an insect wing is dictated by the interaction of aerodynamic forces with the inertial and elastic forces that arise from periodic accelerations of the wing. Passive wing deformation is an unavoidable feature of flapping flight for many insects due to the inertial loads that accompany rapid stroke reversals—loads that well exceed the mean aerodynamic force. Although wing compliance has been implicated in a few lift-enhancing mechanisms (e.g., favorable camber), the direct aerodynamic consequences of wing deformation remain generally unresolved. In this paper, we present new experimental data on how wing compliance may affect the overall induced flow in the hawkmoth, Manduca sexta. Real moth wings were subjected to robotic actuation in their dominant plane of rotation at a natural wing beat frequency of 25 Hz. We used digital particle image velocimetry at exceptionally high temporal resolution (2,100 fps) to assess the influence of wing compliance on the mean advective flows, relying on a natural variation in wing stiffness to alter the amount of emergent deformation (freshly extracted wings are flexible and exhibit greater compliance than those that are desiccated). We find that flexible wings yield mean advective flows with substantially greater magnitudes and orientations more beneficial to lift than those of stiff wings. Our results confirm that wing compliance plays a critical role in the production of flight forces. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

High-speed stereo DPIV measurement of wakes of two bat species flying freely in a wind tunnel

Previous studies on wake flow visualization of live animals using DPIV have typically used low repetition rate lasers and 2D imaging. Repetition rates of around 10 Hz allow ~1 image per wingbeat in small birds and bats, and even fewer in insects. To accumulate data representing an entire wingbeat therefore requires the stitching-together of images captured from different wingbeats, and at different locations along the wing span for 3D-construction of wake topologies. A 200 Hz stereo DPIV system has recently been installed in the Lund University wind tunnel facility and the high-frame rate can be used to calculate all three velocity components in a cube, whose third dimension is constructed using the Taylor hypothesis. We studied two bat species differing in body size, Glossophaga soricina and Leptonycteris curasoa. Both species shed a tip vortex during the downstroke that was present well into the upstroke, and a vortex of opposite sign to the tip vortex was shed from the wing root. At the transition between upstroke/downstroke, a vortex loop was shed from each wing, inducing an upwash. Vorticity iso-surfaces confirmed the overall wake topology derived in a previous study. The measured dimensionless circulation, Γ/Uc, which is proportional to a wing section lift coefficient, suggests that unsteady phenomena play a role in the aerodynamics of both species.     

A simulation-based study on longitudinal gust response of flexible flapping wings

Winged animals such as insects are capable of flying and surviving in an unsteady and unpredictable aerial environment. They generate and control aerodynamic forces by flapping their flexible wings. While the dynamic shape changes of their flapping wings are known to enhance the efficiency of their flight, they can also affect the stability of a flapping wing flyer under unpredictable disturbances by responding to the sudden changes of aerodynamic forces on the wing. In order to test the hypothesis, the gust response of flexible flapping wings is investigated numerically with a specific focus on the passive maintenance of aerodynamic forces by the wing flexibility. The computational model is based on a dynamic flight simulator that can incorporate the realistic morphology, the kinematics, the structural dynamics, the aerodynamics and the fluid–structure interactions of a hovering hawkmoth. The longitudinal gusts are imposed against the tethered model of a hovering hawkmoth with flexible flapping wings. It is found that the aerodynamic forces on the flapping wings are affected by the gust, because of the increase or decrease in relative wingtip velocity or kinematic angle of attack. The passive shape change of flexible wings can, however, reduce the changes in the magnitude and direction of aerodynamic forces by the gusts from various directions, except for the downward gust. Such adaptive response of the flexible structure to stabilise the attitude can be classified into the mechanical feedback, which works passively with minimal delay, and is of great importance to the design of bio-inspired flapping wings for micro-air vehicles.     

Exploration of the rotational power consumption of a rigid flapping wing

The development of Micro Air Vehicles with flapping wings is inspired from the observation and study of natural flyers such as insects and birds. This article explores the rotational power consumption of a flapping wing using a mechanical flapper at Re ≃ 4,500. This mechanical flapper is simplified to a 2D translation and a rotation in a water tank. Moreover, the wing kinematics are reduced to a linear translation and a rotation for the purpose of our study. We introduce the notion of non-ideal flapper and associated non-ideal rotational power. Such non-ideal devices are defined as consuming power for adding and removing mechanical power to and from the flow, respectively. First, we use a traditional symmetrical wing kinematic which is a simplified kinematic inspired from natural flyers. The lift coefficient of this flapping is about C _L ≃ 1.5. This symmetrical wing kinematic is chosen as a reference. Further, wing kinematics with asymmetric rotations are then compared with this one. These new kinematics are built using a differential velocity defined according to the translational kinematics, a time lag and a distance, r _kp. The analogy of this distance is discussed as a key point to follow along the chord. First, the wing kinematics are varied keeping a similar shape for the profiles of the angular velocity. It is shown that when compared to the reference wing kinematic, a 10% reduction in the rotational power is obtained whilst the lift is reduced by 9%. Second, we release the limitation to a similar shape for the profiles of the angular velocity leading to a novel shape for the angular velocity profile named here as “double bump” profile. With these new wing kinematics, we show that a 60% reduction in the non-ideal rotational power can be achieved whilst the lift coefficient is only reduced by 1.7%. Such “double bump kinematics” could then be of interest to increase the endurance of Micro Air Vehicles.