Introduction

One of the challenges in developing a comprehensive theory of biological flight is to understand how aerodynamic mechanisms change with body size. Recent work has focused on flows generated by flapping wings in a variety of different-sized animals, including fruit flies (Birch and Dickinson, 2001, 2003), hawkmoths (Ellington et al., 1996; VandenBerg and Ellington, 1997b; Willmott et al., 1997), butterflies (Brodsky, 1991; Ellington et al., 1996; VandenBerg and Ellington, 1997b; Willmott et al., 1997) and birds (Pennycuick et al., 2000; Spedding et al., 1984; Tucker, 1995). The body size of a flying animal determines, for the large part, the Reynolds number (Re) at which its wings operate. Thus, the diversity of insect sizes virtually guarantees that they experience Re regimes ranging from 10 to 10 000. In order to understand the aerodynamic constraints on insects of different size, it is important to determine how wing performance changes with Re.

Research has identified the phenomenon of dynamic stall as an essential aerodynamic mechanism responsible for the elevated performance of flapping wings. Because of its importance, the influence of body size on aerodynamic performance will be determined in large part by the effects of Re on the forces generated by dynamic stall. A prominent leading edge vortex (LEV), the hallmark of dynamic stall, has been observed on the leading edge of model Manduca wings at Re=5000 and model Drosophila wings at Re=150. In Drosophila, this enlarged area of vorticity is prominent at angles of attack above ∼12°, at which flow separates from the leading edge (Dickinson and Götz, 1993). The importance of the LEV was noted by Maxworthy in the context of Weis-Fogh's `clap-and-fling' mechanism (Maxworthy, 1979, 1981). The formation of an LEV was examined on both tethered and model dragonfly wings by Luttges and colleagues (Somps and Luttges, 1985; Saharon and Luttges, 1987; Reavis and Luttges, 1988). In a seminal study, Ellington and colleagues (Ellington et al., 1996) visualized an LEV on the wing of a live hawkmoth in tethered flight (Re∼4000). More recently, LEVs have been observed on butterfly wings in free flight (Srygley and Thomas, 2002). The topological structure of the LEV observed on butterfly wings during free flight differed somewhat from that observed on the robotic models. However, these experiments on real and model insects differed with respect to wing morphology, wing kinematics, Re and the presence or absence of a free stream flow. For these reasons, an explanation for the observed differences in flow structure remains obscure.

Two-dimensional studies, in which edge baffles inhibit spanwise flow, show that the growth of the LEV begins at the start of translation and continues until the vortex becomes unstable, detaches from the leading edge and is shed into the wake. Subsequently, a counter-rotating vortex forms at the trailing edge, which grows and sheds, followed by the build-up of another LEV. The process continues, leaving a wake of counter-rotating vortex pairs known as a von Kármán street (Schlichting, 1979). In three-dimensional flapping, however, the LEV is stable at both high (5000) and low (120) Re (Usherwood and Ellington, 2002b). Although viscous dissipation may play a role, this stability must arise in large part from the transport of vorticity into the wake. In model hawkmoth wings, axial flow through the vortex core forms a spiral vortex, which has been proposed as the mechanism of transport that drains energy from the LEV (VandenBerg and Ellington, 1997a,b; Willmott et al., 1997). However, experiments at Re=120 failed to find evidence for either strong axial flow within the LEV core or a spiral structure (Birch and Dickinson, 2001). In addition, attempts to limit flow with fences and edge baffles did not significantly alter flows, forces or shedding dynamics.

These differences suggest that the transport of vorticity that maintains prolonged attachment may take different forms at different Re. However, prior experiments at low and high Re were performed using different methods and different wing shapes. To measure the influence of Reynolds number on flow structure and force production more accurately, we performed two sets of experiments using a dynamically scaled robotic insect using identical kinematics and wing geometry. To change Re from 120 to 1400, we changed only the viscosity of the fluid in which the robot flapped. Our results indicate that the presence of axial flow in the vortex core on model Manduca wings and its absence on model Drosophila wings is an effect of Re and not an artefact of differences in experimental methodology or due to differences in wing morphology.

Materials and methods

The dynamically scaled robot used in this study has been described before (Dickinson et al., 1999; Sane and Dickinson, 2001) and its construction will be briefly summarized here. The robot consisted of six servo-motors and two coaxial arms immersed in a tank of mineral oil, although only one arm and wing were used in this study. The acrylic wing was cut in the planform of a Drosophila wing with a total length of 0.26 m when attached to the coaxial arm. At the base of the wing, a sensor measured parallel and perpendicular forces from which we calculated total force or separate lift and drag force components. Force data were collected at 100 Hz using a Measurement Computing PCI-DAS1000 Multifunction Analog Digital I/O board (Measurement Computing, Middleboro, MA, USA) and filtered off-line using a zero phase delay low-pass digital Butterworth filter with a cut-off frequency of 10 Hz, roughly 60 times the wing stroke frequency. The wing and arm apparatus were placed in a 1 m×1.5 m×3 m Plexiglas tank filled with 1.8 m³ of mineral oil (Chevron Superla^® white oil; Chevron Texaco Corp., San Ramon, CA, USA). We changed Re by working with oil of two different viscosities. For low Re, we used oil with a density of 0.88×10³ kg m^–3 and a kinematic viscosity of 120 centistokes (cSt). For high Re, we used oil with a density of 0.83×10³ kg m^–3 and a kinematic viscosity of 11 cSt.

We calculated the translational force coefficients using the equations of Ellington (1984) derived from a blade element analysis: (1) and (2) where C_L and C_D are the lift and drag coefficients, respectively, F_L and F_D are the measured lift and drag forces, respectively, ρ is density, is angular velocity, R is the length of one wing, S is wing surface area, and represents the nondimensional second moment of area, where r̂ is nondimensional wing length and ĉ is nondimensional chord length. Forces were averaged during the center third of a 240° stroke in which the wingtip moves approximately 10 mean cord lengths at constant velocity and angle of attack. To measure aerodynamic polars, we varied the angle of attack from –10° to 110° in 10° increments.

Results

Lift and drag coefficients during translation are consistent with prior measurements on three-dimensional wings (Dickinson et al., 1999). A wing starting from rest shows an initial force transient followed by constant force production (Fig. 1A). This stable force generation indicates prolonged attachment of the LEV for the full duration of the stroke. Mean coefficient values during one-third of translation (broken vertical lines in Fig. 1A) are plotted in the aerodynamic polars in Fig. 1B. Except at very low and very high angles of attack, the lift coefficients were higher at Re=1400. The drag coefficients at Re=1400 were less than the drag coefficients at Re=120 until an angle of attack of ∼30°, presumably due to the contribution of viscous skin friction at low angles of attack. At high angles of attack, however, coefficients at Re=1400 were greater as total drag becomes dominated by pressure. The net force coefficients were higher at Re=1400 than Re=120 for all angles of attack greater than 30° (Fig. 1C). In addition, the net force vector approaches the angle of 90° to the wing surface slightly faster at higher Re (Fig. 1D), indicating a higher relative contribution of viscous drag at Re=120.

In side view, during translation the flow structure around the wing shows two areas of opposite vorticity (Fig. 2A). Above the leading edge and spreading rearward over the upper side of the wing is a large area of clockwise (CW) vorticity indicative of the leading edge vortex (LEV). Along the undersurface of the wing there exists a region of vorticity of the opposite sense (CCW) that we call the under-wing vorticity layer. Qualitative inspection of the vorticity plots shows a region of comparatively greater vorticity near the core of the LEV at Re=1400, a result consistent with the force measurements.

Integrating vorticity values over the entire panel provides an approximation of the local circulation around the wing (Fig. 2B). Measured along the wing from the base to the tip, local circulation shows a general increase at both Re until approximately 0.6R, where it decreases due to separation of the LEV and formation of the tip vortex. Circulation is greater at Re=1400 over most of the wing.

To illustrate the relationship between the LEV and the tip vortex, a rectangular control volume of infinitesimal width enclosing a wing section is chosen such that the vorticity flux across the top, bottom and front surfaces is zero, leaving only flux across the sides and rear of the volume. This can be done by ensuring that the top, bottom and front faces are sufficiently far from the wing. The continuity law applied to vorticity requires that the sum of the fluxes across the sides and rear surfaces must be zero. Since the control volume has infinitesimal width, it can be shown that the rate of change in spanwise circulation with respect to span (dΓ_z/dz) is equal and opposite to the rate of change in chordwise circulation with respect to span (–dΓ_x/dz) along the back face of the control volume. Using the DPIV data, we can illustrate this relationship by comparing the spanwise circulation at each wing section to the chordwise circulation in the wake between the wing base at that section of the wing.

This comparison reveals a remarkable consistency between measures of spanwise (Γ_z) and chordwise (Γ_x) circulation (Fig. 3A). For each increase in spanwise circulation along the span of the wing, there is a corresponding decrease in the chordwise circulation within the wake, as required by continuity. Except for a constant offset, the similarity between spanwise circulation along the wing and chordwise circulation within the wake indicates that these two flows might be accurately represented by a continuous array of vortex filaments that bend back into the wake.

In order to establish the relationship between the structure of the flows and the aerodynamic performance of wings, two methodologies were employed to estimate the aerodynamic forces from the velocity fields. The first method was based on the circulation theorem and yielded only lift estimates. The second method, a two-dimensional steady version of a method developed by Noca et al. (1997), yielded both lift and drag estimates. For Re=120 and Re=1400, the estimates for lift from the circulation method were 0.3 N and 0.38 N, respectively. Comparing these estimates with the measured values of lift (0.44 N for Re=120 and 0.5 N for Re=1400) demonstrates that these simple estimates based on the circulation can account for approximately 70% of the lift produced by the wing. The lift estimated using the two-dimensional steady version of Noca's method was 0.37 N for Re=120 and 0.47 N for Re=1400, which is within 15% of the measured values. Whereas the lift estimates based on Noca's formula are closer to the measured values than those of the circulation method, both methods predict approximately the same difference in lift between the two Re. The reason for this can be found by considering the first term of equation 4, (referred to as the Kutta–Zhukovski term in Noca, 1997): ρ∫_Au×ωdA. Decomposing the velocity field into free stream [U₀=(U₀,0,0)^T] and perturbation (u¢) components so that u=U₀+u¢, the sectional lift component (y-component) of the Kutta–Zhukovski term, can then be written as: (5)

From this we can see that the sectional lift estimate based on the circulation is actually contained within the Kutta–Zhukovski term of Noca's formula. Thus, the contributions of the remaining terms to the difference in lift, everything except–ρ U₀(z)Γ_z(z), must be small. This suggests that a large percentage of the lift experienced by the wing ∼70%) at both Re (120 and 1400) is due to the spanwise circulation about the wing and that this difference in spanwise circulation at different Re can account for the differences in lift. The sectional lift predicted by the circulation method is shown in Fig. 3B.

The drag estimate from Noca's method yielded 0.18 N for Re=120, which was approximately 40% of the measured value, and 0.33 N for Re=1400, which was approximately 73% of the measured value. The reason for this inaccuracy in the drag estimates is unknown. The violation of two-dimensional flow, which we assumed in our calculation, is a likely candidate.

Flow visualizations display characteristic and consistent differences between Re. When fluid motion around the wing is viewed from the side, the LEV is evident at both Re=1400 and Re=120 (Fig. 4), although the flow pattern is more complicated at the higher Re. In Fig. 4, flow in three dimensions is shown by superimposing a pseudocolor plot to represent fluid velocity orthogonal to the field of the page (u_z). Next to each plot, we show the velocities in the x and z directions along a transect that passes through both the core of the LEV and the area of maximum axial flow. At Re=120, this tipward axial flow occurs over a broad region of the wing behind the LEV. There is no evidence of a peak in axial flow near the region of the vortex core, which is marked by the chordwise position at which u_x changes sign. By contrast, at Re=1400 an additional region of higher velocity axial flow within the core of the LEV is clearly visible, superimposed over a broad flow that is similar in structure to that present at Re=120. Furthermore, at Re=1400 the maximum axial flow within the core approaches velocities of 0.47 m s^–1 at a spanwise position of 0.55R. This, value is significantly greater than the tip velocity of 0.31 m s^–1. As the view moves more towards the tip, the LEV becomes less cohesive and at least two regions of intense axial flow are apparent over the upper surface of the wing (Fig. 4iii).

Fig. 5 shows four successive views from the rear, beginning slightly behind the leading edge, each moving rearward by 1 cm. Fluid motion at Re=120 (left column) appears smooth and similar to views previously published (Birch and Dickinson, 2001). After fluid moves up over the leading edge (Fig. 5A), the fluid becomes entrained in the clockwise-rotating tip vortex. This entrainment manifests itself as a more pronounced base-to-tip movement of fluid in slices closer to the trailing edge (Fig. 5C,D). In each slice, fluid movement appears smooth and cohesive, particularly the base-to-tip movement of fluid above the wing and the rotation of the incipient tip vortex. At Re=1400 (Fig. 5E–H), the flow structure seen in two-dimensional slices within the area of the LEV shows a pattern hinting at the structure of a spiral vortex. In Fig. 5F, the laser sheet intersects a region of high speed flow directed upward and distally. Moving rearward by 1 cm (Fig. 5G), the sheet continues to intersect a complicated base-to-tip movement but is now directed downward and distally. By Fig. 5H, the light sheet has moved behind the LEV and sections through an intense tip vortex.

In an attempt to more cleanly dissect the fluid structure within this region behind the leading edge, we performed another set of DPIV experiments where we positioned the camera to capture a closer view of the flow and separated slices by 0.5 cm (Fig. 6). The base-to-tip progression of a region of upward and distal directed flow in Fig. 6A–C suggests a spiral flow. Sections through the forming tip vortex (Fig. 6C–H) show a clear tip-to-base flow.

Photographs of wings equipped with a bubble rake provide further evidence for the existence of a spiral vortex at Re=1400. Fig. 7 shows photographs at mid-downstroke for both Re=120 (A,B) and Re=1400 (C,D) after the wing tip has traveled approximately three chord lengths. Beneath the full-wing photographs are three close-ups of the LEV at approximately 0.3 s intervals, showing the development (or lack thereof) of the spiral flow. At Re=120, while the bubbles trace a straight flow within the LEV core, there exists a slight twist in the bubble lines, imperceptible to DPIV analysis. At Re=1400, this slight twist has developed into a distinct spiral.

Discussion

With the use of a dynamically scaled robot and DPIV, we have quantified both the forces and fluid motion around an insect wing flapping at Re=120 and Re=1400. Our results show that stable forces and flows develop in both cases. This stability reaches different equilibrium points, as measured by net force and circulation, and is generated by qualitatively different flows. At both Re=120 and Re=1400, we observed no evidence of von Kármán shedding, even though the amplitude of our strokes (270°) was beyond the morphological limit of any flapping animal ∼180°). This stability (Dickinson et al., 1999; Usherwood and Ellington, 2002a) requires that the generation of vorticity at the leading edge (forming the LEV) must be balanced by a transport of vorticity into the wake. Results from this experiment indicate that this equilibrium is maintained over a range of Re from 120 to 1400. Evidence for this equilibrium was presented in earlier experiments in which a wall baffle and wing fences were used to impede spanwise flow at the wing tip (Birch and Dickinson, 2001). None of these conditions initiated shedding. However, the presence of the wall increased the size of the LEV, and resultant forces were higher [cf. fig. 1b and fig. 2c in Birch and Dickinson, 2001; note that scales on the vorticity plots are mislabeled and should range from –28 s^–1 to 28 s^–1; rear view (fig. 1d) from– 20 s^–1 to 20 s^–1]. The role of induced flow (as mentioned in Birch and Dickinson, 2001) might be to decrease the rate at which vorticity is generated at the leading edge, but a transport mechanism is still required to maintain a stable equilibrium. Thus, neither inhibiting spanwise flow at Re=120 nor changing Re between 120 and 1400 affects the formation and stability of the LEV. It is not currently known whether inhibiting spanwise flow at Re=1400 would affect the formation and stability of the LEV.

While the LEV remains stable at both Re=120 and Re=1400, several conclusions can be drawn from the observed differences in flow structure. First, vortex transport via axial flow within the core of the LEV is not necessary for the stable attachment at Re=120. Forces and flows remain in equilibrium even when peak axial flow occurs behind the LEV, over the rear two-thirds of the wing (Fig. 4, first column), but not within the vortex core (Fig. 4, second column). A broad region of axial flow over the rear two-thirds of the wing is also present at Re=1400. However, in addition, we observed strong axial flow within the core of the LEV with velocities as high as 150% of wing tip speed. This secondary flow structure is clearly homologous to the spiral vortex identified by Ellington et al. (1996) and may contribute to the transport of vorticity into the wake. Second, flapping wings at Re=120 generate less circulation and lower forces, a result expected from the greater influence of viscosity. Furthermore, it is the decreased importance of viscosity that may account for the development of a spiral vortex at Re=1400. Whether this spiral flow is responsible for the elevation in circulation or whether both are independently related to the higher Re is not known.

How flow changes from the relatively simple pattern at Re=120 to spiral flow at Re=1400 is unclear. The emergence of the spiral vortex might be incremental or it might appear rapidly upon reaching some critical Re. Regardless of how it forms, such secondary flow structures are not unusual, especially when vortices transition to turbulent flow (see, for example, Berger, 1996; Leibovich, 1984). Even in experiments simulating two-dimensional conditions, vortices develop three-dimensional structures due to the asymmetries in the base vortex field or instabilities between consecutive vortices (Julien et al., 2003). Thus, it is not surprising that we observed the development of the spiral vortex at higher Re, considering the instabilities introduced via the velocity gradient and subsequent non-uniform pressure distribution along the wing, as well as the curved leading edge. Whether this structure is a precursor to turbulent breakdown of the LEV or an epiphenomenon of its generation unrelated to stability remains to be determined.

List of symbols

A

area enclosing the wing section

ĉ

nondimensional chord length

C_D

drag coefficient

C_L

lift coefficient

D¢(z)

sectional drag at each spanwise position

F_D

drag force

F_L

lift force

I

unit tensor

L¢(z)

sectional lift at each spanwise position

n

outward unit normal vector

R

length of one wing

r̂

nondimensional wing length

Re

Reynolds number

S

wing surface area

T

stress tensor

u

fluid velocity

U₀(z)

free stream velocity of the wing section

x

position vector

angular velocity

Γx

chordwise circulation within the wake

Γz

spanwise circulation of the wing section

ρ

density

ω

vorticity