Force production and flow structure of the leading edge vortex on flapping wings at high and low Reynolds numbers
,
Department of Integrative Biology, University of California, Berkeley, CA 94720, USA
Author for correspondence (e-mail:
flyman{at}caltech.edu)
Accepted 22 December 2003
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Summary |
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Key words: insect flight, Reynolds number, aerodynamics, flow visualization
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Introduction |
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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.
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Materials and methods |
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We calculated the translational force coefficients using the equations of
Ellington (1984) derived from a
blade element analysis:
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Flow visualization
We used digital particle image velocimetry (DPIV) to quantify the flow
structure around the wing while the wing translated at a 45° angle of
attack. When the flow around the wing reached a steady state, a commercial
software package controlling a dual Nd-YAG laser system (Insight v. 3.4; TSI
Inc., St Paul, MN, USA) pulsed two identically positioned light sheets,
approximately 2.5 mm thick, separated by 2 ms. Acamera positioned
perpendicular to these light sheets captured these two images. The fluid was
seeded with visualization particles prior to image capture by either forcing
air through a ceramic water filter stone or adding silver-coated glass beads
(mean diameter, 13 µm; Conduct-o-fil®; Potters Industries,
Inc., Valley Forge, PA, USA). When seeding with bubbles, we waited until
larger bubbles rose to the surface. The remaining bubbles, although slightly
positively buoyant, did not rise perceptively during capture of the paired
DPIV images. Forces measured with bubbles or beads in the tank were identical
to those measured in oil without additions, indicating that their introduction
did not alter the basic properties of the medium. The wing was centered within
two-dimensional DPIV images taken perpendicular to the long axis of the wing
(i.e. from the side) and images taken parallel to the long axis of the wing
(i.e. from the rear). The final data set consisted of 22 side views moving
from 0.24R to 1.08R (where R is the length of one
wing) in 1 cm increments. The laser and camera were then moved to capture 15
rear views starting 3 cm in front of the leading edge and continuing until 3
cm behind the trailing edge, also in 1 cm increments. We merged these two
views (side and rear) based on wing position within each slice creating a cube
containing ux, uy and
uz velocities. A second data set of rear views was
collected at high Re, with slices separated by only 0.5 cm to better
ascertain flow patterns.
For each image pair captured, a cross-correlation of pixel intensity peaks with 50% overlap of 64 pixelx64 pixel interrogation areas yielded a 30x30 array of vectors. Vector validation removed vectors greater than 3 standard deviations of the mean vector length in their respective images. Deleted values were filled by interpolation of a mean value from a 3x3 nearest neighbor matrix. Sub-pixel displacement accuracy was approximately 0.1 pixel, resulting in 2.5% uncertainty for mean pixel displacements of 4 pixels. A custom program written in MATLAB was used to calculate vorticity from the velocity fields that had been smoothed using a least-squares finite difference scheme. All force and flow descriptions were captured during a stroke that started from rest; there was no wake influence from prior strokes.
To provide a qualitative representation of the flow field, we built a wing
with a bubble rake consisting of a small plastic tube glued to the leading
edge. Along the basal two-fifths of the wing, the tubing was punctured with
small holes at approximately 1 cm intervals. This tube was attached to a pump
that created small bubble streams that allowed flow visualization as the wing
flapped. We used a Nikon D1X digital camera in continuous shooting mode 9
frames s1) to capture images throughout the translation
phase of flapping. We captured pictures of the wing at each Re when
its orientation was approximately parallel to the camera.
Force estimates
Two methodologies were employed to estimate the aerodynamic forces from the
velocity fields. The first and simplest method was based on the circulation
theorem and only yields an estimate of lift. In this method, the sectional
lift (L¢) at each spanwise position (z) was calculated
using:
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Results |
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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.
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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
(dz/dz) is equal and opposite to the rate
of change in chordwise circulation with respect to span
(d
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 (z) and chordwise (
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.
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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
KuttaZhukovski term in Noca,
1997
):
Aux
dA.
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 KuttaZhukovski term, can then be written
as:
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From this we can see that the sectional lift estimate based on the
circulation is actually contained within the KuttaZhukovski term of
Noca's formula. Thus, the contributions of the remaining terms to the
difference in lift, everything except
U0(z)
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 (uz). 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 ux 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 s1 at a spanwise position of 0.55R. This, value is significantly greater than the tip velocity of 0.31 m s1. 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).
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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.
5EH), 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.
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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. 6AC suggests a spiral flow. Sections through the forming tip vortex (Fig. 6CH) show a clear tip-to-base flow.
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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.
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Discussion |
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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.
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List of symbols |
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Acknowledgments |
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Footnotes |
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Present address: California Institute of Technology, Mail Code 138-78,
Pasadena, CA 91125, USA
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References |
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