Insect flight

Leading edge vortex

Most insects use a method that creates a spiralling leading edge vortex. These flapping wings move through two basic half-strokes. The downstroke starts up and back and is plunged downward and forward. Then the wing is quickly flipped over (supination) so that the leading edge is pointed backward. The upstroke then pushes the wing upward and backward. Then the wing is flipped again (pronation) and another downstroke can occur. The frequency range in insects with synchronous flight muscles typically is 5 to 200 hertz (Hz). In those with asynchronous flight muscles, wing beat frequency may exceed 1000 Hz. When the insect is hovering, the two strokes take the same amount of time. A slower downstroke, however, provides thrust.^[8][9]

Identification of major forces is critical to understanding insect flight. The first attempts to understand flapping wings assumed a quasi-steady state. This means that the air flow over the wing at any given time was assumed to be the same as how the flow would be over a non-flapping, steady-state wing at the same angle of attack. By dividing the flapping wing into a large number of motionless positions and then analyzing each position, it would be possible to create a timeline of the instantaneous forces on the wing at every moment. The calculated lift was found to be too small by a factor of three, so researchers realized that there must be unsteady phenomena providing aerodynamic forces. There were several developing analytical models attempting to approximate flow close to a flapping wing. Some researchers predicted force peaks at supination. With a dynamically scaled model of a fruit fly, these predicted forces later were confirmed. Others argued that the force peaks during supination and pronation are caused by an unknown rotational effect that fundamentally is different from the translational phenomena. There is some disagreement with this argument. Through computational fluid dynamics, some researchers argue that there is no rotational effect. They claim that the high forces are caused by an interaction with the wake shed by the previous stroke.^[8][9]

Similar to the rotational effect mentioned above, the phenomena associated with flapping wings are not completely understood or agreed upon. Because every model is an approximation, different models leave out effects that are presumed to be negligible. For example, the Wagner effect, as proposed by Herbert A. Wagner in 1925,^[10] says that circulation rises slowly to its steady-state due to viscosity when an inclined wing is accelerated from rest. This phenomenon would explain a lift value that is less than what is predicted. Typically, the case has been to find sources for the added lift. It has been argued that this effect is negligible for flow with a Reynolds number that is typical of insect flight. The Reynolds number is a measure of turbulence; flow is laminar (smooth) when the Reynolds number is low, and turbulent when it is high.^[11] The Wagner effect was ignored, consciously, in at least one model.^[9] One of the most important phenomena that occurs during insect flight is leading edge suction. This force is significant to the calculation of efficiency. The concept of leading edge suction first was put forth by D. G. Ellis and J. L. Stollery in 1988 to describe vortex lift on sharp-edged delta wings.^[12] At high angles of attack, the flow separates over the leading edge, but reattaches before reaching the trailing edge. Within this bubble of separated flow is a vortex. Because the angle of attack is so high, a lot of momentum is transferred downward into the flow. These two features create a large amount of lift force as well as some additional drag. The important feature, however, is the lift. Because the flow has separated, yet it still provides large amounts of lift, this phenomenon is called stall delay, first noticed on aircraft propellers by H. Himmelskamp in 1945.^[13] This effect was observed in flapping insect flight and it was proven to be capable of providing enough lift to account for the deficiency in the quasi-steady-state models. This effect is used by canoeists in a sculling draw stroke.^[8][9]

All of the effects on a flapping wing may be reduced to three major sources of aerodynamic phenomena: the leading edge vortex, the steady-state aerodynamic forces on the wing, and the wing's contact with its wake from previous strokes. The size of flying insects ranges from about 20 micrograms to about 3 grams. As insect body mass increases, wing area increases and wing beat frequency decreases. For larger insects, the Reynolds number (Re) may be as high as 10000, where flow is starting to become turbulent. For smaller insects, it may be as low as 10. This means that viscous effects are much more important to the smaller insects.^[9][14][15]

downstroke

upstroke

Another interesting feature of insect flight is the body tilt. As flight speed increases, the insect body tends to tilt nose-down and become more horizontal. This reduces the frontal area and therefore, the body drag. Since drag also increases as forward velocity increases, the insect is making its flight more efficient as this efficiency becomes more necessary. Additionally, by changing the geometric angle of attack on the downstroke, the insect is able to keep its flight at an optimal efficiency through as many manoeuvres as possible. The development of general thrust is relatively small compared with lift forces. Lift forces may be more than three times the insect's weight, while thrust at even the highest speeds may be as low as 20% of the weight. This force is developed primarily through the less powerful upstroke of the flapping motion.^[9][16]

Clap and fling

The feathery wings of a thrips are unsuitable for leading edge vortex flight, but support clap and fling.

Clap and fling is used in sea butterflies like Limacina helicina to "fly" through the water.

Clap and fling, or the Weis-Fogh mechanism, discovered by the Danish zoologist Torkel Weis-Fogh, is a lift generation method utilized during small insect flight.^[17] As insect sizes become less than 1 mm, viscous forces become dominant and the efficacy of lift generation from an airfoil decreases drastically. Starting from the clap position, the two wings fling apart and rotate about the trailing edge. The wings then separate and sweep horizontally until the end of the downstroke. Next, the wings pronate and utilize the leading edge during an upstroke rowing motion. As the clap motion begins, the leading edges meet and rotate together until the gap vanishes. Initially, it was thought that the wings were touching, but several incidents indicate a gap between the wings and suggest it provides an aerodynamic benefit.^[18][19]

Lift generation from the clap and fling mechanism occurs during several processes throughout the motion. First, the mechanism relies on a wing-wing interaction, as a single wing motion does not produce sufficient lift.^[20][21][22] As the wings rotate about the trailing edge in the flinging motion, air rushes into the created gap and generates a strong leading edge vortex, and a second one developing at the wingtips. A third, weaker, vortex develops on the trailing edge.^[19] The strength of the developing vortices relies, in-part, on the initial gap of the inter-wing separation at the start of the flinging motion. With a decreased gap inter-wing gap indicating a larger lift generation, at the cost of larger drag forces. The implementation of a heaving motion during fling,^[23] flexible wings,^[21] and a delayed stall mechanism were found to reinforce vortex stability and attachment.^[24] Finally, to compensate the overall lower lift production during low Reynolds number flight (with laminar flow), tiny insects often have a higher stroke frequency to generate wing-tip velocities that are comparable to larger insects.^[24]

The overall largest expected drag forces occur during the dorsal fling motion, as the wings need to separate and rotate.^[22] The attenuation of the large drag forces occurs through several mechanisms. Flexible wings were found to decrease the drag in flinging motion by up to 50% and further reduce the overall drag through the entire wing stroke when compared to rigid wings.^[21] Bristles on the wing edges, as seen in Encarsia formosa, cause a porosity in the flow which augments and reduces the drag forces, at the cost of lower lift generation.^[25] Further, the inter-wing separation before fling plays an important role in the overall effect of drag. As the distance increases between the wings, the overall drag decreases.^[18]

The clap and fling mechanism is also employed by the marine mollusc Limacina helicina, a sea butterfly.^[26][27] Some insects, such as the vegetable leaf miner Liriomyza sativae (a fly), exploit a partial clap and fling, using the mechanism only on the outer part of the wing to increase lift by some 7% when hovering.^[24]

• Clap and fling flight mechanism after Sane 2003
• 
  Clap 1: wings close over back
• 
  Clap 2: leading edges touch, wing rotates around leading edge, vortices form
• 
  Clap 3: trailing edges close, vortices shed, wings close giving thrust

• Black (curved) arrows: flow; Blue arrows: induced velocity; Orange arrows: net force on wing
• 
  Fling 1: wings rotate around trailing edge to fling apart
• 
  Fling 2: leading edge moves away, air rushes in, increasing lift
• 
  Fling 3: new vortex forms at leading edge, trailing edge vortices cancel each other, perhaps helping flow to grow faster (Weis-Fogh 1973)

Governing equations

A wing moving in fluids experiences a fluid force, which follows the conventions found in aerodynamics. The force component normal to the direction of the flow relative to the wing is called lift (L), and the force component in the opposite direction of the flow is drag (D). At the Reynolds numbers considered here, an appropriate force unit is 1/2(ρU²S), where ρ is the density of the fluid, S the wing area, and U the wing speed. The dimensionless forces are called lift (C_L) and drag (C_D) coefficients, that is:^[8]

${\displaystyle C_{\text{L}}(\alpha )={\frac {2L}{\rho U^{2}S}}\quad {\text{and}}\quad C_{\text{D}}(\alpha )={\frac {2D}{\rho U^{2}S}}.}$

C_L and C_D are constants only if the flow is steady. A special class of objects such as airfoils may reach a steady state when it slices through the fluid at a small angle of attack. In this case, the inviscid flow around an airfoil can be approximated by a potential flow satisfying the no-penetration boundary condition. The Kutta-Joukowski theorem of a 2D airfoil further assumes that the flow leaves the sharp trailing edge smoothly, and this determines the total circulation around an airfoil. The corresponding lift is given by Bernoulli's principle (Blasius theorem):^[8]

${\displaystyle C_{\text{L}}=2\pi \sin \alpha \quad {\text{and}}\quad C_{\text{D}}=0.}$

The flows around birds and insects can be considered incompressible: The Mach number, or velocity relative to the speed of sound in air, is typically 1/300 and the wing frequency is about 10–103 Hz. Using the governing equation as the Navier-Stokes equation being subject to the no-slip boundary condition, the equation is:^[8]

${\displaystyle {\begin{aligned}{\frac {\partial \mathbf {u} }{\partial t}}+\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} &=-{\frac {\nabla p}{\rho }}+v\nabla ^{2}\mathbf {u} \\\nabla \cdot \mathbf {u} &=0\\\mathbf {u} _{\text{bd}}&=\mathbf {u} _{\text{s}}.\end{aligned}}}$

Where u(x, t) is the flow field, p the pressure, ρ the density of the fluid, ν the kinematic viscosity, u_bd the velocity at the boundary, and u_s the velocity of the solid. By choosing a length scale, L, and velocity scale, U, the equation can be expressed in nondimensional form containing the Reynolds number, R_e=uL/ν . There are two obvious differences between an insect wing and an airfoil: An insect wing is much smaller and it flaps. Using a dragonfly as an example, Its chord (c) is about 1 cm (0.39 in), its wing length (l) about 4 cm (1.6 in), and its wing frequency (f) about 40 Hz. The tip speed (u) is about 1 m/s (3.3 ft/s), and the corresponding Reynolds number about 103. At the smaller end, a typical chalcidoid wasp has a wing length of about 0.5–0.7 mm (0.020–0.028 in) and beats its wing at about 400 Hz. Its Reynolds number is about 25. The range of Reynolds number in insect flight is about 10 to 10⁴, which lies in between the two limits that are convenient for theories: inviscid steady flows around an airfoil and Stokes flow experienced by a swimming bacterium. For this reason, this intermediate range is not well understood. On the other hand, it is perhaps the most ubiquitous regime among the things we see. Falling leaves and seeds, fishes, and birds all encounter unsteady flows similar to that seen around an insect.^[8] The chordwise Reynolds number can be described by:

${\displaystyle Re={\frac {{\bar {c}}U}{v}}}$

${\displaystyle U=2\Theta fr_{g}}$ and ${\displaystyle r_{g}={\sqrt {{\frac {1}{s}}\int _{0}^{R}{r^{2}c(R)dr}}}}$

Where ${\displaystyle {\bar {c}}\ }$ is the average chord length, ${\displaystyle U}$ is the speed of the wing tip, ${\displaystyle \Theta }$ is the stroke amplitude, ${\displaystyle f}$ is the beat frequency, ${\displaystyle r_{g}}$ is the radius of gyration, ${\displaystyle s}$ is the wing area, and ${\displaystyle R}$ is the length of wing, including the wing tip.

In addition to the Reynolds number, there are at least two other relevant dimensionless parameters. A wing has three velocity scales: the flapping velocity with respect to the body (u), the forward velocity of the body (U₀), and the pitching velocity (Ωc). The ratios of them form two dimensionless variables, U₀/u and Ωc/u, the former is often referred to as the advance ratio, and it is also related to the reduced frequency, fc/U₀.^[8]

If an insect wing is rigid, for example, a Drosophila wing is approximately so, its motion relative to a fixed body can be described by three variables: the position of the tip in spherical coordinates, (Θ(t),Φ(t)), and the pitching angle ψ(t), about the axis connecting the root and the tip. To estimate the aerodynamic forces based on blade-element analysis, it is also necessary to determine the angle of attack (α). The typical angle of attack at 70% wingspan ranges from 25° to 45° in hovering insects (15° in hummingbirds). Despite the wealth of data available for many insects, relatively few experiments report the time variation of α during a stroke. Among these are wind tunnel experiments of a tethered locust and a tethered fly, and free hovering flight of a fruit fly.^[8]

Because they are relatively easy to measure, the wing-tip trajectories have been reported more frequently. For example, selecting only flight sequences that produced enough lift to support a weight, will show that the wing tip follows an elliptical shape. Noncrossing shapes were also reported for other insects. Regardless of their exact shapes, the plugging-down motion indicates that insects may use aerodynamic drag in addition to lift to support its weight.^[8]

Power input

One can now compute the power required to maintain hovering by, considering again an insect with mass m 0.1 g, average force, F_av, applied by the two wings during the downward stroke is two times the weight. Because the pressure applied by the wings is uniformly distributed over the total wing area, that means one can assume the force generated by each wing acts through a single point at the midsection of the wings. During the downward stroke, the center of the wings traverses a vertical distance d.^[14] The total work done by the insect during each downward stroke is the product of force and distance; that is,

${\displaystyle {\text{Work}}=F_{av}\times d={\text{2W}}_{\text{d}}\,\!}$

If the wings swing through the beat at an angle of 70°, then in the case presented for the insect with 1 cm long wings, d is 0.57 cm. Therefore, the work done during each stroke by the two wings is:^[14]

${\displaystyle {\text{Work}}=2\times 0.1\times 980\times 0.57=112{\text{erg}}\,\!}$

The energy is used to raise the insect against gravity. The energy E required to raise the mass of the insect 0.1 mm during each downstroke is:^[14]

${\displaystyle {\text{E}}={\text{mgh}}=0.1\times 980\times 10^{-2}=0.98{\text{erg}}\,\!}$

This is a negligible fraction of the total energy expended which clearly, most of the energy is expended in other processes. A more detailed analysis of the problem shows that the work done by the wings is converted primarily into kinetic energy of the air that is accelerated by the downward stroke of the wings. The power is the amount of work done in 1 s; in the insect used as an example, makes 110 downward strokes per second. Therefore, its power output P is, strokes per second, and that means its power output P is:^[14]

${\displaystyle {\text{P}}=112{\text{erg}}\times 110{\text{/s}}=1.23\times 10^{4}{\text{erg/s}}=1.23\times 10^{-3}{\text{W}}}$

Power output

In the calculation of the power used in hovering, the examples used neglected the kinetic energy of the moving wings. The wings of insects, light as they are, have a finite mass; therefore, as they move they possess kinetic energy. Because the wings are in rotary motion, the maximum kinetic energy during each wing stroke is:^[14]

${\displaystyle KE={\frac {1}{2}}I\omega _{\text{max}}^{2}}$

Here I is the moment of inertia of the wing and ω_max is the maximum angular velocity during the wing stroke. To obtain the moment of inertia for the wing, we will assume that the wing can be approximated by a thin rod pivoted at one end. The moment of inertia for the wing is then:^[14]

${\displaystyle I={\frac {m\ell ^{2}}{3}}}$

Where l is the length of the wing (1 cm) and m is the mass of two wings, which may be typically 10⁻³ g. The maximum angular velocity, ω_max, can be calculated from the maximum linear velocity, ν_max, at the center of the wing:^[14]

${\displaystyle \omega _{\text{max}}={\frac {v_{\text{max}}}{\ell /2}}}$

During each stroke the center of the wings moves with an average linear velocity ν_av given by the distance d traversed by the center of the wing divided by the duration Δt of the wing stroke. From our previous example, d = 0.57 cm and Δt = 4.5×10⁻³ s. Therefore:^[14]

${\displaystyle v_{av}={\frac {d}{\Delta t}}={\frac {0.57}{4.5\times 10^{-3}}}=127{\text{cm/s}}}$

The velocity of the wings is zero both at the beginning and at the end of the wing stroke, meaning the maximum linear velocity is higher than the average velocity. If we assume that the velocity oscillates (sinusoidally) along the wing path, the maximum velocity is twice as high as the average velocity. Therefore, the maximum angular velocity is:^[14]

${\displaystyle \omega _{\text{max}}={\frac {254}{\ell /2}}}$

And the kinetic energy therefore is:^[14]

${\displaystyle KE={\frac {1}{2}}I\omega _{max}^{2}=\left(10^{-3}{\frac {\ell ^{2}}{3}}\right)\left({\frac {254}{\ell /2}}\right)^{2}=43{\text{ erg}}}$

Since there are two wing strokes (the upstroke and downstroke) in each cycle of the wing movement, the kinetic energy is 2×43 = 86 erg. This is about as much energy as is consumed in hovering itself.^[14]