The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

The enhanced aerodynamic performance of insects results from an interaction of three distinct yet interactive mechanisms: delayed stall, rotational circulation, and wake capture. Delayed stall functions during the translational portions of the stroke, when the wings sweep through the air with a large angle of attack. In contrast, rotational circulation and wake capture generate aerodynamic forces during stroke reversals, when the wings rapidly rotate and change direction. In addition to contributing to the lift required to keep an insect aloft, these two rotational mechanisms provide a potent means by which the animal can modulate the direction and magnitude of flight forces during steering maneuvers. A comprehensive theory incorporating both translational and rotational mechanisms may explain the diverse patterns of wing motion displayed by different species of insects.

https://science.sciencemag.org/content/sci/284/5422/1954.full.pdf

Wing rotation and the aerodynamic basis of insect flight.

Wing rotation and the aerodynamic basis of insect flight

Half of all insect species are dependent on living plant tissues, consuming about 10% of plant annual production in natural habitats and an even greater percentage in agricultural systems, despite sophisticated control measures. Plants are generally remarkably well-protected against insect attack, with the result that most insects are highly specialized feeders. The mechanisms underlying plant resistance to invading herbivores on the one side, and insect food specialization on the other, are the main subjects of this book. For insects these include food-plant selection and the complex sensory processes involved, with their implications for learning and nutritional physiology, as well as the endocrinological spects of life cycle synchronization with host plant phenology. In the case of plants exposed to insect herbivores, they include the activation of defence systems in order to minimize damage, as well as the emission of chemical signals that may attract natural enemies of the invading herbivores and maybe exploited by neighbouring plants that mount defences as well.

Insect-plant biology

INSECTS cannot fly, according to the conventional laws of aerodynamics: during flapping flight, their wings produce more lift than during steady motion at the same velocities and angles of attack1–5. Measured instantaneous lift forces also show qualitative and quantitative disagreement with the forces predicted by conventional aerodynamic theories6–9. The importance of high-life aerodynamic mechanisms is now widely recognized but, except for the specialized fling mechanism used by some insect species1,10–13, the source of extra lift remains unknown. We have now visualized the airflow around the wings of the hawkmoth Manduca sexta and a 'hovering' large mechanical model—the flapper. An intense leading-edge vortex was found on the down-stroke, of sufficient strength to explain the high-lift forces. The vortex is created by dynamic stall, and not by the rotational lift mechanisms that have been postulated for insect flight14–16. The vortex spirals out towards the wingtip with a spanwise velocity comparable to the flapping velocity. The three-dimensional flow is similar to the conical leading-edge vortex found on delta wings, with the spanwise flow stabilizing the vortex.

Leading-edge vortices in insect flight

The wing motion in free flight has been described for insects ranging from 1 to 100 mm in wingspan. To support the body weight, the wings typically produce 2–3 times more lift than can be accounted for by conventional aerodynamics. Some insects use the fling mechanism: the wings are clapped together and then flung open before the start of the downstroke, creating a lift-enhancing vortex around each wing. Most insects, however, rely on a leading-edge vortex (LEV) created by dynamic stall during flapping; a strong spanwise flow is also generated by the pressure gradients on the flapping wing, causing the LEV to spiral out to the wingtip. Technical applications of the fling are limited by the mechanical damage that accompanies repeated clapping of the wings, but the spiral LEV can be used to augment the lift production of propellers, rotors and micro-air vehicles (MAVs). Design characteristics of insect-based flying machines are presented, along with estimates of the mass supported, the mechanical power requirement and maximum flight speeds over a wide range of sizes and frequencies. To support a given mass, larger machines need less power, but smaller ones operating at higher frequencies will reach faster speeds.

The novel aerodynamics of insect flight: applications to micro-air vehicles.

Insects in free flight were filmed at 5000 frames per second to determine the motion of their wings and bodies. General comments are offered on flight behaviour and manoeuvrability. Changes in the tilt of the stroke plane with respect to the horizontal provides kinematic control of manoeuvres, analogous to the type of control used for helicopters. A projection analysis technique is described that solves for the orientation of the animal with respect to a cam era-based coordinate system, giving full kinematic details for the longitudinal wing and body axes from single-view films. The technique can be applied to all types of flight where the wing motions are bilaterally symmetrical: forward, backward and hovering flight, as well as properly banked turns. An analysis of the errors of the technique is presented, and shows that the reconstructed angles for wing position should be accurate to within 1-2° in general. Although measurement of the angles of attack was not possible, visual estimations are given. Only 11 film sequences show flight velocities and accelerations that are small enough for the flight to be considered as ‘hovering’. Two sequences are presented for a hover-fly using an inclined stroke plane, and nine sequences of hovering with a horizontal stroke plane by another hover-fly, two crane-flies, a drone-fly, a ladybird beetle, a honey bee, and two bumble bees. In general, oscillations in the body position from its mean motion are within measurement error, about 1-2 % of the wing length. The amplitudes of oscillation for the body angle are only a few degrees, but the phase relation of this oscillation to the wingbeat cycle could be determined for a few sequences. The phase indicates that the pitching moments governing the oscillations result from the wing lift at the ends of the wingbeat, and not from the wing drag or inertial forces. The mean pitching moment of the wings, which determines the mean body angle, is controlled by shifting the centre of lift over the cycle by changing the mean positional angle of the flapping wings. Deviations of the wing tip path from the stroke plane are never large, and no consistent pattern could be found for the wing paths of different insects; indeed, variations in the path were even observed for individual insects. The wing motion is not greatly different from simple harmonic motion, but does show a general trend towards higher accelerations and decelerations at either end of the wingbeat, with constant velocities during the middle of half-strokes. Root mean square and cube root mean cube angular velocities are on average about 4 and 9% lower than simple harmonic motion. Angles of attack are nearly constant during the middle of half-strokes, typically 35° at a position 70 % along the wing length. The wing is twisted along its length, with angles of attack at the wing base some 10-20° greater than at the tip. The wings rotate through about 110° at either end of the wingbeat during 10-20 % of the cycle period. The mean velocity of the wing edges during rotation is similar to the mean flapping velocity of the wing tip and greater than the flapping velocity for more proximal wing regions, which indicates that vortex shedding during rotation is com parable with that during flapping. The wings tend to rotate as a flat plate during the first half of rotation, which ends just before, or at, the end of the half-stroke. The hover-fly using an inclined stroke plane provides a notable exception to this general pattern : pronation is delayed and overlaps the beginning of the downstroke. The wing profile flexes along a more or less localized longitudinal axis during the second half of rotation, generating the ‘flip’ profile postulated by Weis-Fogh for the hover-flies. This profile occurs to some extent for all of the insects, and is not exceptionally pronounced for the hover-fly. By the end of rotation the wings are nearly flat again, although a slight camber can sometimes be seen. Weis-Fogh showed that beneficial aerodynamic interference can result when the left and right wings come into contact during rotation at the end of the wingbeat. His ‘fling’ mechanism creates the circulation required for wing lift on the subsequent half-stroke, and can be seen on my films of the Large Cabbage White butterfly, a plum e moth, and the Mediterranean flour moth. However, their wings ‘peel’ apart like two pieces of paper being separated, rather than fling open rigidly about the trailing edges. A ‘partial fling’ was found for some insects, with the wings touching only along posterior wing areas. A ‘ near fling ’ with the wings separated by a fraction of the chord was also observed for m any insects. There is a continuous spectrum for the separation distance between the wings, in fact, and the separation can vary for a given insect during different manoeuvres. It is suggested that these variants on Weis-Fogh’s fling mechanism also generate circulation for wing lift, although less effectively than a complete fling, and that changes in the separation distance may provide a fine control over the amount of lift produced.

The Aerodynamics of Hovering Insect Flight. III. Kinematics

The conventional aerodynamic analysis of flapping animal flight invokes the `quasisteady assumption' to reduce a problem in dynamics to a succession of static conditions: it is assumed that the instantaneous forces on a flapping wing are equivalent to those for steady motion at the same instantaneous velocity and angle of attack. The validity of this assumption and the importance of unsteady aerodynamic effects have long been controversial topics. Weis-Fogh tested the assumption for hovering animal flight, where unsteady effects are most pronounced, and concluded that most insects indeed hover according to the principles of quasi-steady aerodynamics. The logical basis for his conclusion is reviewed in this paper, and it is shown that the available evidence remains ambiguous. The aerodynamics of hovering insect flight are re-examined in this series of six papers, and a conclusion opposite to Weis-Fogh's is tentatively reached. New morphological and kinematic data for a variety of insects are presented in papers II and III, respectively. Paper IV offers an aerodynamic interpretation of the wing kinematics and a discussion on the possible roles of different aerodynamic mechanisms. A generalized vortex theory of hovering flight is derived in paper V, and provides a method of estimating the mean lift, induced power and induced velocity for unsteady as well as quasi-steady flight mechanisms. The new data, aerodynamic mechanisms and vortex theory are all combined in paper VI for an analysis of the lift and power requirements and other mechanical aspects of hovering flight. A large number of symbols are needed for the morphological, kinematic and aerodynamic analyses. Most of them appear in more than one paper of the series, and so a single comprehensive table defining the major symbols from all of the papers is presented at the end of this paper.

The Aerodynamics of Hovering Insect Flight. I. The Quasi-Steady Analysis

Morphological parameters are presented for a variety of insects that have been filmed in free flight. The nature of the parameters is such that they can be divided into two distinct groups: gross parameters and shape parameters. The gross parameters provide a very crude, first-order description of the morphology of a flying animal: its mass, body length, wing length, wing area and wing mass. Another gross parameter of the wings is their virtual mass, or added mass, which is the mass of air accelerated and decelerated together with the wing at either end of the wingbeat. The wing motion during these accelerations is almost perpendicular to the wing surface, and the virtual mass is approximately given by the mass of air contained in an imaginary cylinder around the wing with the chord as its diameter. The virtual mass ranges from 0.3 to 1.3 times the actual wing mass, indicating that the total mass accelerated by the flight muscles can be more than twice the wing mass itself. Over the limited size range of insects in this study, the interspecific variation of non-dimensional forms of the gross parameters is much greater than any systematic allometric variation, and no interspecific correlations can be found. The new shape parameters provide quite a surprise, however: intraspecific coefficients of variation are very low, often only 1%, and interspecific allometric relations are extremely strong. Mechanical aspects of flight depend not only on the magnitude of gross morphological quantities, but also on their distributions. Non-dimensional radii are derived from the non-dimensional moments of the distributions; for example, the first radius of wing mass about the wing base gives the position of the centre of mass, and the second radius corresponds to the radius of gyration. The radii are called \`shape parameters' since they are functions only of the normalized shape of the distributions, and they provide a second-order description of the animal morphology. The various radii of wing area are strongly correlated, as are those of wing mass and of virtual mass: the higher radii for each quantity can all be expressed by allometric functions of the first radius. The overall shape of the distribution of a quantity can therefore be characterized by a single parameter, the position of the centroid of that quantity. The strong relations between the radii of wing area, mass and virtual mass hold for a diverse collection of insects, birds and bats. Thus flying animals adhere to \`laws of shape' regardless of biological differences. Aerodynamic and mechanical considerations are most likely to provide an understanding of these laws of shape, but an explanation has proved elusive so far. The detailed shape of a distribution can be reconstructed from the shape parameters by matching the moments of the observed distribution to those of a suitable analytical function. A Beta distribution is compared with the distribution of wing area, i.e. the shape of the wing, and a very good fit is found. With use of the laws of shape relating the higher radii to the first radius, the Beta distribution can be reduced to a function of only one parameter, thus providing a powerful tool for drawing a close approximation to the entire shape of a wing given only its centroid of area. Quite unexpectedly, the continuous spectrum of wing shapes can then be described in detail by a single parameter of shape.

Charles P. Ellington

Papers

Leading-edge vortices in insect flight

The novel aerodynamics of insect flight: applications to micro-air vehicles.

The Aerodynamics of Hovering Insect Flight. III. Kinematics

The Aerodynamics of Hovering Insect Flight. I. The Quasi-Steady Analysis

The Aerodynamics of Hovering Insect Flight. II. Morphological Parameters