Many insects are well adapted to long-distance migration despite the larger energetic costs of flight for small body sizes. To optimize wing design for next-generation flying micro-robots, we analyse butterfly wing shapes and wing orientations at full scale using numerical simulations and in a low-speed wind tunnel at 2, 3.5 and 5 m s−1. The results indicate that wing orientations which maximize wing span lead to the highest glide performance, with lift to drag ratios up to 6.28, while spreading the fore-wings forward can increase the maximum lift produced and thus improve versatility. We discuss the implications for flying micro-robots and how the results assist in understanding the behaviour of the butterfly species tested.
Butterfly flight has long fascinated scientists and the public, but only in recent years have technological advances made it possible to at least approximate insect-sized flying robots [1–5]. The construction of such robots offers important challenges to materials scientists, engineers and physicists, but research is also driven by practical applications for search and rescue (e.g. in collapsed buildings or mines), or in other inhospitable environments such as those in extra-terrestrial atmospheres with low Reynolds numbers . Fortunately, the engineering challenges of micro-scale flight have already been solved by insects, so a first step in creating micro-scale flying robots is to examine the biological structures enabling flight in these animals. Efficient and sustained flight at the scale of butterflies poses unique problems. For example, the energetic cost of flight increases as the flying system decreases in size . Nevertheless, many insects and small birds successfully migrate over long distances despite their small size. Based on observations and mathematical modelling of their flight, it has been suggested that long-distance migration is made possible by adapting an alternating flapping–gliding flight mode which can minimize the energetic cost of locomotion [8,9].
While several authors describe the aerodynamic effects in air at Reynolds numbers between 105 and 106 [10–13], some of the conclusions of these studies might not be applicable to flight at Reynolds numbers below 104 due to the increased influence of viscous and boundary layer effects . The Reynolds number regime between 1000 and 10 000 is of particular interest because the smallest animals which perform intermittent gliding flight operate in this regime. Studies in biology have correlated wing morphology of flying insects and birds to flight performance on live animals [15–20] and using artificial models of butterflies and dragonflies [21–23]. A potential limitation of such studies using static models is that the wing shape is extracted from pictures without accounting for variability of wing orientation which the animal can do during flight. Of particular importance is the relative orientation of fore-wing and hind-wing which changes the overall planform of the wing significantly. Butterflies hold their wings in an orientation during gliding flight which may be very different from how they are pinned in a museum collection. In fact, museums display butterfly specimens with wings spread mainly for aesthetic reasons and to maximize the visibility of taxonomic characters rather than to mimic the aerodynamic functionality of the wing shape [24,25].
The effect of a change in fore-wing and hind-wing orientation on the overall wing shape is illustrated in figure 1 for a monarch butterfly. A very sensitive and controlled wind tunnel, which allows for low airspeeds of 2 m s−1 and a force resolution of 0.1 mN, is needed to perform force measurements on at-scale insect wings, consequently there are not many such measurements in the literature.
The flight speed of insects is one of the least known features of flight performance. Early measurements of butterfly speeds were made using a variety of methods and for many species, only single speed measurements have been taken [26–28]. Flight speeds for one of the best-known migrating butterflies, the monarch Danaus plexippus (Linnaeus) have been measured at 4.9–11.2 m s−1 according to the flight habit—gliding flight, cruising flight, speed flight, pre-nuptial flight or social flight ; but these were based on measurements made while driving a car alongside flying butterflies and cannot provide very exact airspeeds. Monarch flight speeds measured on a tethered flight mill varied from 0.39 to 2.36 m s−1 with an average of 1.08 m s−1 ; however, these speeds do not reflect flight speeds of butterflies in nature as the tether potentially interferes with free flight behaviour. A more accurate study by Dudley & Srygley  tracked 62 species of Neotropical butterflies flying across water by synchronizing a motorboat speed with the flying insects and measuring airspeed with a hand held anemometer. Although they did not record D. plexippus they did record D. eresimus Talbot and D. gilippus Bates, both of which are closely related to monarch butterflies and have similar wing shapes. They were recorded at mean airspeeds of 3.9 m s−1 and 3.6 m s−1, respectively. Importantly, flight speed can be influenced by a range of biological and physical factors including mass, size, age, gender, amount of feeding and water content, in addition to environmental factors such as temperature, humidity, solar radiation, wind speed and direction, and oxygen levels [26,27].
In this paper, we present a study of the flight performance of 42 wing shapes and the characterization of a low-speed wind tunnel which can measure complete aerodynamic polar curves of at-scale insect wings with online quality control to ensure precise and repeatable results. We then expand on the experimental results using numerical simulations to study the fluid dynamic effects that take place during flight and we interpret the results in the light of aerial robot development and behaviours of real butterflies. This paper builds on the initial wind tunnel results that were reported in a conference contribution . For each of the four butterfly species studied the fore-wing orientation, relative to the hind-wind, was varied at increments of 10° and the lift to drag ratios were measured at 42 angles of attack at 2, 3.5 and 5 m s−1.
In addition, we use computational fluid dynamics (CFD) simulations representative of these wind tunnel experiments, focusing on the geometries with fore-wing orientations which maximize wing chord and those which maximize span, to determine and visualize the flow parameters in order to gain a better understanding of the flow structures over these wings. Based on these experiments and simulations, we compare and discuss the gliding performance of the wing shapes from these four butterfly species. This study provides important experimental insight into the aerodynamic effects of different wing shapes. We also review the results from our previous work and interpret these data from a biological perspective on the lives of these four butterfly species. In particular, we correlate our findings to the wing orientation, and the flight speeds of in-flight measurements that have been reported in the biology literature. Furthermore, the gliding performance of these shapes can be used as a design benchmark for wing optimization in robotic flying insects.
2. Experimental methods
2.1. Wing shape selection
The selected butterflies include three species, which show a distinct flapping and gliding flight behaviour, and one species which is not known to use gliding as a major part of its flight repertoire but has interesting wing features which have been suggested to influence flight performance. We selected migrating monarchs, D. plexippus (Linnaeus) (Nymphalidae: Danainae: Danaini) which use a combination of flapping and gliding flight over long distances and have been documented to glide up to 80% of their flight time . In addition to the monarch, we evaluate the wing shapes of the orange plane, Pantoporia consimilis (Boisduval) (Nymphalidae: Limenitidinae: Neptini), which has a distinctive flight style of alternating flapping and gliding in short bursts ; the glasswing, Acraea andromacha (Fabricius) (Nymphalidae: Heliconiinae: Acraeini) that has a ‘lazy’ flight style, often gliding for extended periods on thermals along ridges and hilltops ; and for contrast the four-barred swordtail, Protographium leosthenes (Doubleday) (Papilionidae: Papilioninae: Leptocircini) which is a fast flier and rarely glides  (butterflies depicted in figure 2). The tested wing shapes are illustrated in figure 3. To isolate the effect of the wing shape from the influence of other parameters such as wing flexibility, camber, wing corrugation or surface topologies, we fabricate the wings out of rigid flat plates. The wing edge geometry can have a strong effect on the aerodynamic performance of a wing shape . We therefore choose to laser cut the wings from flat steel plates with a thickness of 150 µm (figure 4a) using a diode-pumped solid-state (DPSS) laser, which ensures a very clean and repeatable cut as illustrated in the scanning electron microscopy (SEM) image of the wing model edge (figure 4b). For normalization, we scale all wings to an area of 900 mm2.
2.2. Wind tunnel
The Harvard Microrobotics Lab wind tunnel is an open circuit Eiffel type wind tunnel (figure 5) oriented horizontally with a contraction ratio of 6.25 : 1 from Engineering Laboratory Design Inc. The square test section has a height and width of 30.5 cm and a length of 61 cm. The blockage caused by the butterfly wing models, defined as the ratio of the frontal area of the wing model to the cross-sectional area of the wind tunnel test section, is therefore a maximum of 0.9% when tested at 72° angle of attack. The test section has two test ports integrated in the side walls with a diameter of 100 mm each. The test port is outfitted with a rotation stage and a six-axis force/torque sensor which allows the angle of attack of the wing sample to change while measuring the lift and drag forces (figure 6). The six-axis force/torque sensor (Nano 17 from ATI, figure 6a) has a force measurement resolution of 3 mN and a torque resolution of 15 mN-mm. It is affixed to a Newport PR50CC mini DC rotary stage (figure 6b) which has an angular resolution of 0.01° for precise angular positioning of the wing samples. This stage is mounted on the the wind tunnel test section wall using a transparent poly(methyl methacrylate) interface port (figure 6c). The wing samples (figure 6d) were mounted to a 31.4 cm long carbon fibre lever arm (figure 6d) which is used to amplify the forces acting on the wing. The free end of this carbon tube is positioned on the force/torque sensor with the shape held in the centre of the test section (figure 6f). Real-time wind tunnel velocity verification was achieved using the TSI air velocity transducer (model 8455) mounted in the span-wise plane of the test section 45 cm downstream from the test section entrance (figure 5d). It has a resolution of 0.07% of the full-scale selected range (0–25 m s−1).
2.3. Wind tunnel characterization and quality control
The wind tunnel is controlled using a Dell Optiplex 980 mt host computer with an IBM Thinkcenter acting as the target machine using the Matlab Simulink XPC computational environment. The control schematic is shown in figure 5. The hardware control of the rotational stage, the wind tunnel fan speed and the TSI air velocity transducer are implemented using the Newport XPS-C2 2 axis controller.
Manufacturer testing found velocity uniformity to vary a maximum of ±0.98% from the mean free-stream velocity of 2 m s−1 over the area of the test section, excluding the boundary layer. To verify the precision of our wind tunnel and to characterize the boundary layer thickness, we performed span-wise velocity measurements at a target flow velocity of 0.4, 2.5 and 5 m s−1 (figure 7a). The results indicate that the flow field is uniform at a distance greater than 25 mm from the wall. To characterize the variation of the air speed, we performed a series of experiments at target velocities between 0.5 and 5 m s−1 over a time interval of 20 s at a sampling rate of 1000 Hz (figure 7b). It can be seen that the deviation from the target airspeed is less than 2% with an increased number of outliers at low flow velocities. Based on these two flow characterizations, we choose to perform our wing shape tests at a distance of 30 cm from the test section wall and at velocities of 2, 3.5 and 5 m s−1. Furthermore, we characterized the variation of the forces measured at each angle of attack (figure 7c). It can be seen that the force variation is uniform for the different angles of attack with a range of ±0.5%. During these measurements, the environment is controlled to minimize perturbations such as ground vibrations of people walking by or transient air movements. To ensure consistency and high quality of the force measurements even when running experiments for long periods, we implemented an online quality control scheme. The principle of this scheme is that at every angle of attack the airspeed is held constant and a series of force measurements are taken over a period of 20 s at a sampling rate of 1000 Hz. Based on these data, both the mean and the variation are recorded and evaluated. In the event that the variation of the data sequence is larger than a prefixed value, the measurement series is rejected and repeated until it meets the defined quality criteria. For the tests in this paper, we define the acceptable threshold such that 97% of the measurements have to be within a variation of only ±1% of the mean value.
3. Wind tunnel results
Each wing was evaluated at airspeeds of 2, 3.5 and 5 m s−1 which, also taking into consideration the variations in wing chord, corresponds to a Reynolds number range between 2597 and 12 632. The tested angles of attack range between 0° and 72° are evaluated during 20 s at each angle with a sampling rate of 1000 Hz.
The results of the measurements are represented in figure 8, showing a clear difference in performance of the various butterfly species as well as the variation of the fore-wing orientation. The results show some variation between measurements at adjacent angles of attack, even though averaging of different measurements for the same measurement point was used to reduce this. This is mainly due to the long rod used to amplify the forces measured also creating small vibrations and amplifying these. The maximum glide ratio, which is defined as the maximal value of the lift to drag polars, is achieved by the model A70, reaching a value of 6.26. It can be seen that there is a clear velocity dependence of the gliding ratio for all tested wings shapes.
Furthermore, it can be seen that there is an optimum as well for the fore-wing orientation angle. The results indicate that for all tests the fore-wing angle which maximizes the wing span offers the best gliding performance (D60, A70, L60, C60). The only exception from this tendency is the glasswing at 2 m s−1 although the difference in the gliding ratio of the model with maximal wing span is relatively small (6.7%).
Overall, the wing shapes tested perform well compared with the gliding ratios which have been reached by gliding micro-robots of similar size (3 in  and 5.6 in ) indicating that butterfly wing shapes can offer an increased flight performance which can be applied to novel flying robots. However, this is only the case for a wing orientation which maximizes the wing span and not for the wing shapes of butterflies typically displayed as museum specimens.
4. Numerical simulations
The numerical simulations of the flow over the butterfly wings enable the comparison of the aerodynamic trends observed during wind tunnel testing. In addition, they allow the visualization and determination of different flow parameters throughout the whole domain, which would be impossible to achieve without invasive techniques affecting the flow to be studied in the wind tunnel. These provide key information to deduce the reasons for the trends observed from the wind tunnel results. The simulation set-up, solution calculation and post processing was carried out with Workbench 17.0 (ANSYS, Inc.).
4.1. Numerical set-up
For this study, the butterfly wing shapes selected were those with the fore-wings fully forward (museum position) and those that maximize the wingspan, shown to provide the best glide ratio in the wind tunnel. These correspond to the following as named in figure 3: D0, D60, A0, A70, L0, L60, C0 and C60. All wing shapes were scaled to a surface area of 900 mm2 and extruded to a thickness of 150 µm, as was the case for the wind tunnel models. The wing geometry was rotated to be at an angle of attack of 10° when the flow in parallel to the domain in order to enable larger angles of attack while maintaining direct flow from the inlet. The three-dimensional computational unstructured grid was generated using ANSYS Meshing. The computational domain was set up with a symmetry plane at the wing root, to obtain a similar flow to that around a gliding butterfly, as shown in figure 9.
The computational meshes consist of between 1.6 and 1.8 million elements, and are significantly refined near the butterfly wing surface, as shown in figure 10. The refinement includes an inflation layer on the surface of the wing containing eight layers, with the first layer having a thickness of 35 µm, to achieve a y+ smaller than 1 over the entire wing. A mesh study was performed to check for grid independence for the results of lift and drag. Figure 11 shows the lift to drag results for meshes with different numbers of elements for L60 at 10° angle of attack. The different meshes were generated by first increasing the number of inflation layers and then refining the face meshes until the desired convergence was achieved. The difference in lift to drag, lift coefficient and drag coefficient between the most refined and second most refined mesh was computed for the L60 geometry at 10° and 20° angles of attack for the three different airspeeds. In all cases, these differences were less than 1.5%. The simulations for the rest of geometries were carried out with parameters similar to those for the finest mesh tested on L60.
To compute the numerical simulations, the solver used was ANSYS CFX, which solves the conservation of mass and momentum equations using the finite volume method, with a cell vertex formulation. The turbulence model used was Shear Stress Transport, which uses the k−ω model in the inner region of the boundary layer to predict separation while using the k−ε model in the free shear flow.
4.2. Numerical results
Each wing geometry was simulated at the same airspeeds as tested in the wind tunnel, of 2, 3.5 and 5 m s−1. For each speed, the angles of attack simulated were 0° to 20° in steps of 2° and up to 40° in steps of 5°.
A comparison of the gliding efficiency of the four butterfly species, under the flow conditions stated above, is shown in figure 12. The results confirm that, as shown in the wind tunnel experiments, the glasswing achieves the highest lift to drag ratio of 6.28. The monarch is next at 6.17 followed by the orange plane and the four-barred swordtail with lift to drag ratios of 6.16 and 5.39, respectively. These compare with the wind tunnel results of 6.26 for the glasswing (0.32% lower than the CFD result), 6.17 for the orange plane (the same as for the CFD), 5.92 for the orange plane (4.05% lower) and 5.40 for the four-barred swordtail (0.19% higher). Figure 12 also shows that the highest lift to drag ratios are achieved at the highest airspeed. Most importantly, it can be observed that for angles of attack of 20° and below, the lift to drag ratio is improved for the geometries with the fore-wing extended to maximize the wingspan, with the largest difference around the optimum angle of attack, between 6° and 8°. These results are also in agreement with the experiments.
The variation of the coefficient of lift with angle of attack shown in figure 13 also presents some interesting trends. Most notably, it can be seen that for angles of attack below 20° the lift coefficient is significantly larger for geometries with the fore-wing extended. However, at higher angles of attack, the forward position of the fore-wing results in a higher lift, which in fact produces a significantly higher maximum lift coefficient than that for the fore-wing extended to maximum wingspan. Another interesting feature is the difference in stall behaviour. The extended fore-wing geometries gently stall after reaching a peak in lift coefficient, which is then slowly recovered. By contrast, those with forward fore-wings reach significantly higher angles of attack, of about 10° higher, before stalling and achieving a similar lift coefficient at 40°. These effects can be explained in a significant part by differences in the lift generation for the different fore-wing positions. While for the geometries with extended fore-wings most of the lift is generated in a conventional manner, with the faster air over the upper surface moving relatively parallel to the free-stream resulting in a suction force, much of that for the geometries with the fore-wings oriented forward is generated by a leading edge vortex (LEV), as shown in figures 14 and 15. These LEVs can be seen to generate a strong suction on the upper surface of the wing and energise the boundary layer over the wing preventing separation and delaying stall for forward oriented fore-wing geometry, as shown in figure 13. However, generating these vortices requires a large amount of energy, which results in a lower lift to drag ratio than conventional lift generation mechanisms.
A further notable feature to observe in figure 13 is that the four-barred swordtail has the highest lift coefficient despite having the lowest lift to drag ratio. A possible explanation could be that it produces several vortices due to its intricate shape, as shown in figure 14, as well as having a low aspect ratio in comparison to the other species. Finally, it can also be seen that higher lift coefficients are achieved at higher speeds for all geometries. However, this effect is significantly smaller than for the lift to drag ratio, especially at low angles of attack.
The variation of drag coefficient with angle of attack is displayed in figure 16. An interesting feature that can be seen is that, at angles of attack of less than 20°, the forward oriented fore-wing geometries experience lower drag, despite showing a significantly lower lift to drag ratio. This suggests that the higher lift to drag ratio at low angles of attack for the extended fore-wing geometries is driven by the lift coefficient. Nevertheless, the trend reverses at high angles of attack, when the strong LEV results in a high drag coefficient for the forward fore-wing geometries.
Figure 17 shows the relationship between the maximum lift to drag ratio and the aspect ratio for all the geometries tested. The figure suggests that the difference in aspect ratio between the wing geometries explains a significant part of the differences in lift to drag ratio between them although the differences in wing geometric features also have an influence on performance. For example, the four-barred swordtail at a fore-wing angle of 60° (aspect ratio of 3.47) has a lower maximum lift to drag ratio than the trend, while the orange plane and the monarch (aspect ratios of 4.07 and 4.47, respectively) have a slightly higher maximum lift to drag ratios than the trend. In general, higher airspeed shows higher maximum lift to drag ratio, as also shown in figure 12. A possible reason for this is that the higher Reynolds number at higher speed results in less separation and lower pressure drag, especially considering the low Reynolds numbers involved.
5. Discussion and conclusion
This paper presents an aerodynamic study of gliding flight for a range of butterfly wing models of four species in different configurations for the purposes of gaining further insights into micro-robot wing design and improving the understanding of gliding butterfly behaviours. We focus on the aerodynamic effects of the wing shapes only, of different butterfly species as well as. The study uses results from a new low-speed wind tunnel and control architecture which allows for at-scale lift and drag force measurements of millimetre and centimetre size wings and CFD simulations to further understand the flow and help to understand the performance measured in the wind tunnel. Overall, the glasswing butterfly offers the best gliding performance with gliding ratios of up to 6.28, which is high compared with micro air vehicles of similar size. Not surprisingly, of the four species’ wing shapes tested, the glasswing butterfly most commonly uses a gliding flight mode. The results indicate that the wing configuration which maximizes the wing span is the most favourable configuration with regards to increasing the glide ratio. This is also the wing configuration adopted by gliding butterflies, although they may vary the angle of the wings above the horizontal plane depending on rate of descent. Perhaps unsurprisingly, the four-barred swordtail, which was the only species of those tested that is not regarded as a glider, presented the lowest maximum lift to drag ratio by a significant margin.
This study focuses on analysing the flow and comparing the performance of different wing shapes, which can be applied to a variety of flying micro-robots and can offer valuable insight into the behaviours of certain gliding butterflies. However, we acknowledge that other characteristics of insect wings such as camber , corrugations  or flexibility , among others, also significantly affect insect flight.
5.1. Fore-wing orientation
The optimal wing shape for migrating monarch butterflies, i.e. elongated fore-wings in a high aspect ratio configuration, has been discussed by Altizer & Davis . Although there are small differences in wing shape between migrating and non-migrating monarch populations, there is an overall similarity of wing shape among most of the Danaini, those in the genus Danaus in particular. Danaus species feed on Asclepiadaceae and are unpalatable to predators . As a result, Danaus butterflies are generally aposematic that is, they ‘advertise’ their unpalatability with conspicuous slow flight behaviour and bright markings. Indeed, the slow flight adaptations such as higher wing aspect ratio found in the Danaini may have preadapted D. plexippus for long distance migration, as well as other Danaini such as Tirumala, Parantica, Ideopsis and Euploea which are all known to migrate in the Old World .
Butterfly wings are decoupled as they do not possess a frenulum and retinaculum as in moths, and so they move their wings independently, especially when performing changes in direction. Butterfly hind-wings scoop air and provide extra force to quickly turn when chased . Butterflies can hover and rapidly change flight speed by startling shifts in wing-beat frequency, amplitude and stroke plane angle . Betts & Wootton  also noted that many butterflies appeared to unlink their wings during gliding flight and suggested that airflow between the fore-wings and hind-wings may improve the flow over the whole aerofoil and therefore delay stall at low speeds and/or high angles of attack. Decoupling increases wing area and lift potential, thus the angles at which butterflies unlink their fore and hind-wings may be a further consideration in the design of robot insect wings.
Our wind tunnel tests and numerical simulations also revealed that orientation of wings as shown in typical pinned museum specimens (forward orientation) gives lower overall performance but better performance at high angles of attack. Furthermore, butterflies can orient their fore-wings and hind-wings in an infinite number of three-dimensional positions either together or decoupled or even asymmetrically, so they may have more effective means of reducing stall. Two-dimensional robot wings do not have this capability, so although butterflies may not adopt a certain wing orientation, this does not mean that such an orientation should be excluded from consideration when designing fixed wing gliding robots. Forward wing orientation may be analogous to the forward winglets in fixed wing aircraft which improve stability and stall behaviour . Several butterfly species in the genus Charaxes (Nymphalidae: Charaxinae: Charaxini) with low aspect ratio wings are also good gliders over short distances. However, these species are stoutly built, powerful, fast flyers  and when they do glide it is usually at a speed much faster than commonly seen in aposematic ‘gliding’ species such as Danaus and Acraea. The aerodynamics that lead to this performance could be studied further.
Jantzen & Eisner  showed that fore-wings are most critical to butterfly flight. Butterflies with fore-wings removed were unable to fly; however, butterflies with hind-wings removed do not show any significant loss of flight functions other than a loss in both linear and turning acceleration, so that they are unable to execute rapid directional changes essential for predator avoidance. Thus, it is not surprising that modifications to fore-wing geometries had such a strong influence on glide performance.
5.2. Implications for flying micro-robots
The results we obtained showed that the gliding ratio is maximal at higher speeds and larger wing spans for the butterfly wing geometries at the Reynolds numbers tested. They also show the formation of interesting flow structures such as leading-edge votices similar to those in other studies . All these are important design considerations in the development of high performance flying micro-robots. More interesting is the ability of the butterflies to change the orientation of the fore-wing to achieve different benefits for different flight regimes. This could inspire micro-flying vehicles which have a similar wing configuration. These robots could benefit from increased aerodynamic efficiency by extending their fore-wings which would result in increased endurance range and maximum speed, and then have the ability to position their fore-wings forward to achieve increased lift at high angles of attack. This configuration would enable flying vehicles to glide at slower speeds and to perform higher-g manoeuvres.
In summary, it is clear that we can learn a lot about optimizing wing design for micro-robots by studying flying insects. In addition, we hope our findings may give valuable insight into the bio-mechanics of insect flight.
6. Future work
Future work could focus on a more detailed analysis of the aerodynamic effects on the wings using flow visualization techniques as well as a comparison of the butterfly wings to classical wing shapes currently used on flying robots and aeroplanes. Additional studies could focus on variations of the hind-wing orientations and interactions between the fore-wing and the hind-wing geometries, as well as on fabrication and testing of hierarchical wing surface structures to influence the boundary layer of the wing and increase its aerodynamic performance. Moreover, the findings of this work can give valuable insight into the bio-mechanics of the four butterfly species and could be compared to in flight measurements of live insects.
The flight performance of robotic flying insects may be improved by incorporating the variation in thickness of butterfly wing veins into the wing profile to increase aerodynamic coefficients particularly at slow speeds . Meng & Sun  studied the effects of corrugation on aerodynamic performance and conclude that corrugated wings can be approximated very well with flat plates and that the main benefit of corrugation is increased structural stability.
With respect to numerical simulations, it could be interesting to consider more detailed simulation set-ups such as direct numerical simulation or large eddy simulation to analyse the flow with more precision and to study unsteady effects during flapping flight or during other dynamic flight conditions. An example of this would be the study of aerodynamic three-dimensional unsteady effects by Lee et al. , which could be expanded with additional wing shapes of gliding butterflies at different fore-wing orientations.
R.E. contributed the biological knowledge and discussion related to butterflies; D.V., C.I. and M.S. prepared and performed the wind tunnel experiments; R.W. and M.K. advised and supervised the project and A.O.A. performed the numerical simulations and contributed to the discussion of results.
We have no competing interests.
This work is partially funded by the Wyss Institute for Biologically Inspired Engineering (ONR award no. N00014-10-1-0684), the EPSRC Centre for Doctoral Training in Fluid Dynamics across Scales (award no. EP/L016230/1) and the EPSRC Aquatic Micro Aerial Vehicles (AquaMAV): Bio-inspired air-water mobility for robotics (award no. EP/N009061/1).
We thank the President and Fellows of Harvard College for providing access to images from their butterfly database in the Museum of Comparative Zoology, which we used to extract the wing shapes. We also thank Dr James Weaver for taking the SEM images of the wing shape models and Raphael Zufferey for help with the figures and editing. Finally, we thank Dr Peter Vincent for advice on the CFD simulations.
One contribution of 19 to a theme issue ‘Coevolving advances in animal flight and aerial robotics’.
- © 2016 The Author(s)
Published by the Royal Society. All rights reserved.